AP Statistics Vocabulary

Absolute Value

The absolute value of a number is its distance from zero on the number line. For example, -7 is 7 units away from zero, so its absolute value would be 7. And 7 is also 7 units away from zero, so its absolute value would also be 7. Thus, the absolute value of a number refers to the magnitude of the number, without regard to its sign. The absolute value of -1 and 1 is 1, the absolute value of -2 and 2 is 2, the absolute value of -3 and 3 is 3, and so on.

Example: "To compute the distance between two points on a number line, the programmer writes int distance = Math.abs(pointA - pointB);, which ensures the result is always non-negative regardless of the order of subtraction."

Accuracy

Accuracy refers to how close a sample statistic is to a population parameter . Thus, if you know that a sample mean is 99 and the true population mean is 100, you can make a statement about the sample accuracy. For example, you might say the sample mean is accurate to within 1 unit.

Example: "A student measures the acceleration due to gravity as 9.78 m/s², which is very close to the accepted value of 9.80 m/s², demonstrating high          even if the measurements were not perfectly precise."

Addition Rule

The rule of addition applies to the following situation. We have two events from the same sample space, and we want to know the probability that either event occurs. Rule of Addition. If events A and B come from the same sample space, the probability that event A and/or event B occur is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur. P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Note: Invoking the fact that P( A ∩ B ) = P( A )P( B | A ), the Addition Rule can also be expressed as: P(A ∪ B) = P(A) + P(B) - P(A)P( B | A ).

Example: "Since events A (rolling a 2) and B (rolling an even number) are not mutually exclusive, we apply the         : P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 1/6 + 3/6 − 1/6 = 3/6 = 0.5."

Alpha

With respect to estimation problems , alpha refers to the likelihood that the true population parameter lies outside the confidence interval . Alpha is usually expressed as a proportion. Thus, if the confidence level is 95%, then alpha would equal 1 - 0.95 or 0.05. With respect to hypothesis tests , alpha refers to significance level , the probability of making a Type I error .

Example: "The researcher set          at 0.05, meaning she was willing to accept a 5% chance of committing a Type I error by rejecting a true null hypothesis."

Alternative Hypothesis

There are two types of statistical hypotheses. Null hypothesis . The null hypothesis, denoted by H 0 , is usually the hypothesis that sample observations result purely from chance. Alternative hypothesis . The alternative hypothesis, denoted by H 1 or H a , is the hypothesis that sample observations are influenced by some non-random cause. For example, suppose we wanted to determine whether a coin was fair and balanced. A null hypothesis might be that half the flips would result in Heads and half, in Tails. The alternative hypothesis might be that the number of Heads and Tails would be very different. Symbolically, these hypotheses would be expressed as H 0 : p = 0.5 H a : p ≠ 0.5 Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this result, we would be inclined to reject the null hypothesis. That is, we would conclude that the coin was probably not fair and balanced.

Example: "In a study testing whether a new drug lowers blood pressure, the          H_a: μ < 120 mmHg indicates that the researcher believes the drug produces a reduction in mean blood pressure."

Analysis of Variance

Also referred to as ANOVA, is a method of testing whether or not the means of three or more populations are equal. The method is applicable if: all populations of interest are normally distributed. the populations have equal standard deviations. samples (not necessarily of the same size) are randomly and independently selected from each population. The test statistic for analysis of variance is the F-ratio.

Example: "To compare the mean exam scores of students taught by three different methods, the researcher used         , checking first that the populations were approximately normal and had roughly equal standard deviations."

ANOVA

Analysis of variance (ANOVA) refers to a set of techniques for comparing sample statistics (means, proportions, etc.) among two or more groups. If the comparison reveals a statistically significant difference, the researcher concludes that the corresponding population parameter in one or more groups are different. Researchers use analysis of variance to test causal relationships in controlled experiments. In a controlled experiment, an experimenter manipulates an independent variable (a potential cause) and measures the effect on a dependent variable . The goal of the experiment is to determine whether the independent variable has a causal effect on the dependent variable. Analysis of variance provides objective decision rules for determining whether observed differences between groups are attributable to random chance or to the independent variable(s) manipulated by the experimenter.

Example: "The one-way          yielded an F-statistic of 4.87 with a p-value of 0.012, leading the researcher to reject the null hypothesis that all four group means are equal."

Average

A number that describes the central tendency of the data; there are a number of specialized averages, including the arithmetic mean, weighted mean, median, mode, and geometric mean.

Example: "The          SAT math score for the sampled students was 530, but because the distribution was right-skewed due to a few extremely high scores, the median of 510 was a more appropriate measure of center."

Back-to-Back Stemplots

Back-to-back stemplots are a graphic option for comparing data from two populations. The center of a back-to-back stemplot consists of a column of stems, with a vertical line on each side. Leaves representing one dataset extend from the right, and leaves representing the other dataset extend from the left. Boys   Girls 7 1 1 4 6 4 5 8 1 2 2 2 8 9 3 4 7 9 2 5 8 1 3 0 1 2 3 4 5 6 7 1 2 6 8 3 4 4 6 6 8 9 4 3 6 4 The back-to-back stemplot above compactly describes the amount of cash (in dollars) carried by a random sample of teenage boys and girls. The boys carried more cash than the girls - a median of $42 for the boys versus $36 for the girls. Both distributions were roughly bell-shaped, although there was more variation among the boys. And finally, there were neither gaps nor outliers in either group.

Example: "The teacher created          to compare the quiz scores of the morning and afternoon classes, placing the shared stems in the center with morning class leaves extending left and afternoon class leaves extending right."

Balanced Design

In experimental design , a balanced design has an equal number of observations in all treatment groups. In contrast, an unbalanced design has an unequal number of observations in some treatment groups. Balance is not required for analysis of variance with a single factor (i.e., one-way analysis of variance). But balance can be helpful for analysis of variance with two or more factors because: Balanced designs are less vulnerable to violations of the equal variance assumption. Unbalanced designs can produce correlated factors, making it hard to interpret results clearly.

Example: "Because the researcher assigned exactly 25 subjects to each of the four treatment groups, the experiment had a         , which simplified the subsequent ANOVA calculations."

Bar Chart

A bar chart is made up of columns or rows plotted on a graph. Here is how to read a bar chart made up of columns. The columns are positioned over a label that represents a categorical variable . The height of the column indicates the size of the group defined by the column label. And here is how to read bar chart made up of rows. The rows extend from labels that represent a categorical variable. The length of the row indicates the size of the group defined by the row label. The bar chart below is in "column" format. It shows average per capita income for the four "New" states - New Jersey, New York, New Hampshire, and New Mexico. Per Capita Income   $36,000 $24,000 $12,000           New Jersey New Hampshire New York New Mexico.

Example: "The          displaying the distribution of preferred study methods showed that 45% of students preferred flashcards, making it the tallest bar and the modal category."

Bayes Rule

Bayes' rule (also known as Bayes' theorem) is a useful tool for calculating conditional probability . Bayes rule can be expressed as: Bayes' rule. Let A 1 , A 2 , ... , A n be a set of mutually exclusive events that together form the sample space S. Let B be any event from the same sample space, such that P(B) > 0. Then, P(A|B) =   P(A) P(B|A) P(B) where P(A) is the probability of Event A. P(B) is the probability of Event B. P(A|B) is the conditional probability of Event A, given Event B. P(B|A) is the conditional probability of Event B, given Event A.

Example: "Using         , the doctor calculated the probability that a patient who tested positive actually had the disease: P(Disease | Positive) = [P(Positive | Disease) × P(Disease)] / P(Positive)."

Bayes Theorem

Bayes' theorem (also known as Bayes' rule) is a useful tool for calculating conditional probability . Bayes theorem can be expressed as: Bayes' theorem. Let A 1 , A 2 , ... , A n be a set of mutually exclusive events that together form the sample space S. Let B be any event from the same sample space, such that P(B) > 0. Then, P(A|B) =   P(A) P(B|A) P(B) where P(A) is the probability of Event A. P(B) is the probability of Event B. P(A|B) is the conditional probability of Event A, given Event B. P(B|A) is the conditional probability of Event B, given Event A.

Example: "         allowed the quality control engineer to update the probability that a defective item came from Machine A, given that the prior probability of Machine A producing a defect was 0.03 and the overall defect rate was 0.05."

Bernoulli Trials

An experiment with the following characteristics: There are only two possible outcomes called “success” and “failure” for each trial. The probability p of a success is the same for any trial (so the probability q = 1 − p of a failure is the same for any trial).

Example: "Each free throw attempt by a basketball player can be modeled as a Bernoulli trial, since there are only two outcomes (make or miss) and the probability of success remains constant at 0.78 across all attempts."

Bias

Bias refers to the tendency of a measurement process to over- or under-estimate the value of a population parameter . In survey sampling, for example, bias would be the tendency of a sample statistic to systematically over- or under-estimate a population parameter.

Example: "Because the survey was conducted only at a gym, the sample systematically overrepresented physically active individuals, introducing          into the estimate of the population's average weekly exercise time."

Bias Due to Measurement Error

A poor measurement process can also lead to bias. In survey research, the measurement process includes the environment in which the survey is conducted, the way that questions are asked, and the state of the survey respondent.

Example: "Respondents who underreported their daily caloric intake on a self-report survey introduced         , causing the sample mean to consistently underestimate the true population mean."

Bias Due to Unrepresentative Samples

A good sample is representative. This means that each sample point represents the attributes of a known number of population elements.

Example: "Sampling only from urban zip codes when studying nationwide income levels introduced         , since rural households with systematically lower incomes were excluded."

Bias in Survey Sampling

In survey sampling, bias refers to the tendency of a sample statistic to systematically over- or under-estimate a population parameter.

Example: "Voluntary response bias emerged in the online poll because only strongly opinionated individuals chose to participate, causing the sample statistic to systematically misrepresent the broader population's views."

Biased Estimate

When the expected value of the sampling distribution of a statistic is not equal to a population parameter , that statistic is said to be a biased estimate of the parameter.

Example: "The sample range is a          of the population range because its expected value consistently underestimates the true range, unlike the sample mean, which is an unbiased estimate of the population mean."

Bimodal Distribution

Distributions of data can have few or many peaks. Distributions with one clear peak are called unimodal , and distributions with two clear peaks are called bimodal , as illustrated in the figures below. Unimodal distribution Bimodal distribution.

Example: "The histogram of test scores displayed a         , with peaks near 60 and 90, suggesting the class consisted of two distinct subgroups with very different levels of preparation."

Binomial Distribution

A discrete random variable (RV) which arises from Bernoulli trials; there are a fixed number, n, of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X is defined as the number of successes in n trials. The notation is: X~B(n,p). The mean is μ = np and the standard deviation is σ = npq npq . The probability of exactly x successes in n trials is P ( X = x ) = n x p x q n − x P(X=x)= n x p x q n − x .

Example: "If a student randomly guesses on each of 20 multiple-choice questions with four options, the number of correct answers follows a          with n = 20 and p = 0.25."

Binomial Experiment

A binomial experiment is a statistical experiment that has the following properties: The experiment consists of n repeated trials. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. The probability of success, denoted by p , is the same on every trial. The trials are independent ; that is, the outcome on on trial does not affect the outcome on other trials. Here is an example of a binomial experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: The experiment consists of repeated trials. We flip a coin 2 times. Each trial can result in just two possible outcomes - heads or tails. The probability of success is constant - 0.5 on every trial. The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.

Example: "Randomly selecting 15 voters and recording whether each supports a particular candidate qualifies as a         , provided the population is large enough that the trials are approximately independent."

Binomial Formula

Binomial probability refers to the probability that a binomial experiment results in exactly x successes. Given x (the number of successes in a binomial experiment), n (the number of trials in a binomial experiment), and P (the probability of success on any individual trial), we can compute the binomial probability based on the binomial formula: Binomial Formula. Suppose a binomial experiment consists of n trials and the probability of success on an individual trial is p . Then the binomial probability that the experiment results in x successes is: P(X = x) = n C x * p x * (1 - p) n - x or P(X = x) = { n! / [ x! (n - x)! ] } * p x * (1 - p) n - x where n C x is the number of combinations of n things taken x at a time.

Example: "Using the          P(X = 3) = C(10, 3)(0.4)^3(0.6)^7, the student calculated the probability of getting exactly 3 heads in 10 flips of a biased coin with P(heads) = 0.4."

Binomial Probability

Binomial probability refers to the probability that a binomial experiment results in exactly x successes. The following notation is helpful when we talk about binomial probability. x : The number of successes that result from the binomial experiment. n : The number of trials in the binomial experiment. p : The probability of success on an individual trial. q : The probability of failure on an individual trial. (This is equal to 1 - p .) b( x ; n, p ): Binomial probability - the probability that an n -trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is p . n C r : The number of combinations of n things, taken r at a time. Given x , n , and p , we can compute the binomial probability based on the following formula: Binomial Formula. Suppose a binomial experiment consists of n trials and the probability of success on an individual trial is p . Then the binomial probability that the experiment results in x successes is: P(X = x) = n C x * p x * (1 - p) n - x or P(X = x) = { n! / [ x! (n - x)! ] } * p x * (1 - p) n - x where n C x is the number of combinations of n things taken x at a time.

Example: "The          of obtaining exactly 7 successes in 12 trials with p = 0.5 is calculated as C(12,7)(0.5)^7(0.5)^5 ≈ 0.193."

Binomial Random Variable

A binomial random variable refers to the number of successes in a binomial experiment .

Example: "Let X be the number of defective items in a random sample of 30 products from a production line where 8% of all items are defective; X is a          with n = 30 and p = 0.08."

Bivariate Data

Statistical data are often classified according to the number of variables being studied. Univariate data . When we conduct a study that looks at only one variable, we say that we are working with univariate data. Suppose, for example, that we conducted a survey to estimate the average weight of high school students. Since we are only working with one variable (weight), we would be working with univariate data. Bivariate data . When we conduct a study that examines the relationship between two variables, we are working with bivariate data. Suppose we conducted a study to see if there were a relationship between the height and weight of high school students. Since we are working with two variables (height and weight), we would be working with bivariate data.

Example: "The researcher collected          by recording both the number of hours each student studied and their corresponding exam score, then used a scatterplot and least-squares regression to explore the relationship."

Blinding

In an experiment , if subjects in the control group know that they are receiving a placebo , the placebo effect will be reduced or eliminated; and the placebo will not serve its intended control purpose. Blinding is the practice of not telling subjects whether they are receiving a placebo. In this way, subjects in the control and treatment groups experience the placebo effect equally. Often, knowledge of which groups receive placebos is also kept from analysts who evaluate the experiment. This practice is called double blinding. It prevents the analysts from "spilling the beans" to subjects through subtle cues; and it assures that their evaluation is not tainted by awareness of actual treatment conditions.

Example: "         was essential in the clinical trial so that patients receiving the placebo would not alter their reported symptoms based on the knowledge that they were not receiving the active treatment."

Blocking

Blocking is the technique used in a randomized block experiment to sort experimental units into homogeneous groups, called blocks . The goal of blocking is to create blocks such that dependent variable scores are more similar within blocks than across blocks. For example, consider an experiment designed to test the effect of different teaching methods on academic performance. In this experiment, IQ is a potential nuisance variable. That is, even though the experimenter is primarily interested in the effect of teaching methods, academic performance will also be affected by student IQ. To control for the unwanted effects of IQ, we might include IQ as a blocking variable in a randomized block experiment. We would assign students to blocks, such that students within the same block have the same (or similar) IQ's. Then, we would randomly assign students within blocks to different teaching methods. By holding IQ constant within blocks, we can attribute within-block differences in academic performance to differences in teaching methods, rather than to differences in IQ.

Example: "The researcher used          by gender before randomly assigning participants to treatments, ensuring that variability due to gender differences would not obscure the effect of the new tutoring program."

Blocking Variable

In a randomized block experiment , a good blocking variable has four distinguishing characteristics: It is included as a factor in the experiment. It is not of primary interest to the experimenter. It affects the dependent variable. It is unrelated to independent variables in the experiment. A blocking variable is a potential nuisance variable - a source of undesired variation in the dependent variable. By explicitly including a blocking variable in an experiment, the experimenter can tease out nuisance effects and more clearly test treatment effects of interest. Warning: If a blocking variable does not affect the dependent variable or if it is strongly related to an independent variable, a randomized block design may not be the best choice. Other designs may be more efficient.

Example: "In a study on fertilizer effectiveness, the type of soil served as a          because it affected crop yield but was not the primary factor of interest, so the researcher grouped plots by soil type before randomizing treatments."

Boxplot

A boxplot , sometimes called a box and whisker plot , is a type of graph used to display patterns of quantitative data. A boxplot splits the dataset into quartiles . The body of the boxplot consists of a "box" (hence, the name), which goes from the first quartile (Q1) to the third quartile (Q3). Within the box, a vertical line is drawn at the Q2, the median of the dataset. Two horizontal lines, called whiskers , extend from the front and back of the box. The front whisker goes from Q1 to the smallest non-outlier in the dataset, and the back whisker goes from Q3 to the largest non-outlier. Smallest non-outlier Q1 Q2 Q3   Largest non-outlier             . .                                         -600 -400 -200 0 200 400 600 800 1000 1200 1400 1600 If the dataset includes one or more outliers , they are plotted separately as points on the chart. In the boxplot above, two outliers precede the first whisker (on the left side of the plot).

Example: "The side-by-side          revealed that the interquartile range for Group B was twice as large as for Group A, indicating much greater variability in Group B's response times even though both groups had similar medians."

Categorical Variable

Variables can be classified as categorical (aka, qualitative) or quantitative (aka, numerical). Categorical. Categorical variables take on values that are names or labels. The color of a ball (e.g., red, green, blue) or the breed of a dog (e.g., collie, shepherd, terrier) would be examples of categorical variables. Quantitative. Quantitative variables are numerical. They represent a measurable quantity. For example, when we speak of the population of a city, we are talking about the number of people in the city - a measurable attribute of the city. Therefore, population would be a quantitative variable.

Example: "Political party affiliation (Democrat, Republican, Independent) is a         , so the researcher summarized it with a frequency table and bar chart rather than computing a mean."

Census

A census is a study that obtains data from every member of a population . In most studies, a census is not practical, because of the cost and/or time required.

Example: "Rather than relying on a sample, the school district conducted a          by surveying every enrolled student, eliminating sampling error but requiring significantly more time and resources."

Center

Graphically, the center of a distribution is located at the median of the distribution. This is the point in a graphic display where about half of the observations are on either side. In the chart below, the height of each column indicates the frequency of observations. Here, the observations are centered over 4.

Example: "The          of the distribution of sample means was located at approximately 72, which matched the known population mean, confirming that the sample mean is an unbiased estimator of the population mean."

Central Limit Theorem

The central limit theorem states that the sampling distribution of a statistic (sample mean, sample proportion, etc.) will be normal or nearly normal, if the sample size is large enough. How large is "large enough"? The answer depends on three factors. Requirements for accuracy. The more closely the sampling distribution needs to resemble a normal distribution, the more sample points will be required. The shape of the underlying population. The more closely the original population resembles a normal distribution, the fewer sample points will be required. The nature of the statistic. The guidelines for a sample mean are different from the guidelines for a sample proportion. For a sample mean, some statisticians say that a sample size of 30 is large enough when the population distribution is roughly bell-shaped. Others recommend a sample size of at least 40. But if the original population is distinctly not normal (e.g., is badly skewed, has multiple peaks, and/or has outliers), researchers like the sample size to be even larger. For a sample proportion, it is safe to assume that the shape of the sampling distribution will be approximately normal when the following conditions are true: Population size (N) is at least 10 times sample size (n). The sampling method is simple random sampling . n * p ≥ 10, where p is the sample proportion. n * (1 - p) ≥ 10. Note: When the sample proportion p equals 0.5, the last two conditions require that at least 20 observations be sampled from a population for the sampling distribution to be approximatley normal. When the sample proportion p is more extreme than 0.5, more observations are required.

Example: "Even though individual household incomes are strongly right-skewed, the          assures us that the sampling distribution of the sample mean from samples of size n = 100 will be approximately normal."

Chi-Square Distribution

Suppose we conduct the following statistical experiment . We select a random sample of size n from a normal population, having a standard deviation equal to σ. We find that the standard deviation in our sample is equal to s . Given these data, we can compute a statistic, called chi-square , using the following equation: Χ 2 = [ ( n - 1 ) * s 2 ] / σ 2 The distribution of the chi-square statistic is called the chi-square distribution. The chi-square distribution is defined by the following probability density function : Y = Y 0 * ( Χ 2 ) ( v/2 - 1 ) * e - Χ 2 / 2 where Y 0 is a constant that depends on the number of degrees of freedom, Χ 2 is the chi-square statistic, v = n - 1 is the number of degrees of freedom , and e is a constant equal to the base of the natural logarithm system (approximately 2.71828). Y 0 is defined, so that the area under the chi-square curve is equal to one. In the figure above, the red curve shows the distribution of chi-square values computed from all possible samples of size 3, where degrees of freedom is n - 1 = 3 - 1 = 2. Similarly, the green curve shows the distribution for samples of size 5 (degrees of freedom equal to 4); and the blue curve, for samples of size 11 (degrees of freedom equal to 10).

Example: "When testing whether a die is fair, the test statistic follows a          with 5 degrees of freedom (one less than the number of outcomes), and we reject the null hypothesis if χ² exceeds the critical value."

Chi-Square Goodness of Fit Test

A chi-square goodness of fit test attempts to answer the following question: Are sample data consistent with a hypothesized distribution? The test is appropriate when the following conditions are met: The sampling method is simple random sampling . The population is at least 10 times as large as the sample. The variable under study is categorical . The expected value for each level of the variable is at least 5. Here is how to conduct the test. Define hypotheses. For a chi-square goodness of fit test, the hypotheses take the following form. H 0 : The data are consistent with a specified distribution. H a : The data are not consistent with a specified distribution. Typically, the null hypothesis specifies the proportion of observations at each level of the categorical variable. The alternative hypothesis is that at least one of the specified proportions is not true. Specify significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used. Find degrees of freedom. The degrees of freedom (DF) is equal to the number of levels (k) of the categorical variable minus one: DF = k - 1 . Compute expected frequency counts. The expected frequency counts at each level of the categorical variable are equal to the sample size times the hypothesized proportion from the null hypothesis E i = np i where E i is the expected frequency count for the i th level of the categorical variable, n is the total sample size, and p i is the hypothesized proportion of observations in level i . Find test statistic. The test statistic is a chi-square random variable (Χ 2 ) defined by the following equation. Χ 2 = Σ [ (O i - E i ) 2 / E i ] where O i is the observed frequency count for the i th level of the categorical variable, and E i is the expected frequency count for the i th level of the categorical variable. Find P-value . The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. This involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Example: "A          was used to determine whether the observed distribution of M&M colors in a bag matched the manufacturer's claimed proportions, yielding a p-value of 0.23 and failing to reject the null hypothesis."

Chi-Square Statistic

Suppose we select a random sample from a normal population. The chi-square statistic can be computed using the following equation: Χ 2 = [ ( n - 1 ) * s 2 ] / σ 2 where n is the sample size, σ is the population standard deviation, s is the sample standard deviation equals, and Χ 2 is the chi-square statistic.

Example: "The          χ² = Σ[(Observed − Expected)² / Expected] was computed for each cell of the contingency table; a large value indicates a substantial discrepancy from what is expected under the null hypothesis."

Chi-Square Test for Homogeneity

The chi-square test of homogeneity is applied to a single categorical variable . It is used to compare the distribution of frequency counts across different populations. It answers the following question: Are frequency counts distributed identically across different populations? The test procedure is appropriate when the following conditions are met: For each population, the sampling method is simple random sampling . The population is at least 10 times as large as the sample. The variable under study is categorical . If sample data are displayed in a contingency table (Populations x Category levels), the expected frequency count for each cell of the table is at least 5. This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. State the hypothesis. Every hypothesis test requires a null hypothesis and an alternative hypothesis . Suppose that data were sampled from r populations, and assume that the categorical variable had c levels. At any specified level of the categorical variable, the null hypothesis states that each population has the same proportion of observations. Thus, H 0 : P level 1 of population 1 = P level 1 of population 2 = . . . = P level 1 of population r H 0 : P level 2 of population 1 = P level 2 of population 2 = . . . = P level 2 of population r . . . H 0 : P level c of population 1 = P level c of population 2 = . . . = P level c of population r The alternative hypothesis (H a ) is that at least one of the null hypothesis statements is false. Formulate an analysis plan. The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the significance level and the test method (i.e., the chi-square test of homogeneity). Analyze sample data. Using sample data from the contingency tables, find the degrees of freedom, expected frequency counts, test statistic, and the P-value associated with the test statistic. The analysis described in this section is illustrated in the sample problem at the end of this lesson. Degrees of freedom. The degrees of freedom (DF) is equal to: DF = (r - 1) * (c - 1) where r is the number of populations, and c is the number of levels for the categorical variable. Expected frequency counts. The expected frequency counts are computed separately for each population at each level of the categorical variable, according to the following formula. E r,c = (n r * n c ) / n where E r,c is the expected frequency count for population r at level c of the categorical variable, n r is the total number of observations from population r, n c is the total number of observations at treatment level c , and n is the total sample size. Test statistic. The test statistic is a chi-square random variable (Χ 2 ) defined by the following equation. Χ 2 = Σ [ (O r,c - E r,c ) 2 / E r,c ] where O r,c is the observed frequency count in population r for level c of the categorical variable, and E r,c is the expected frequency count in population r for level c of the categorical variable. P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above. Interpret results. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. This involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Example: "To determine whether the distribution of preferred music genres was the same across three age groups, the researcher conducted a          on the contingency table of counts."

Chi-Square Test for Independence

A chi-square test for independence is applied when you have two categorical variables from a single population. It is used to determine whether there is a significant association between the two variables. The test consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. State the hypotheses. A chi-square test for independence is conducted on two categorical variables. Suppose that Variable A has r levels, and Variable B has c levels. The null hypothesis states that knowing the level of Variable A does not help you predict the level of Variable B. That is, the variables are independent. The alternative hypothesis states that the variables are not independent. Formulate analysis plan. The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify a significance level and should identify the chi-square test for independence as the test method. Analyze sample data. Using sample data, find the degrees of freedom, expected frequencies, test statistic, and the P-value associated with the test statistic. Degrees of freedom. The degrees of freedom (DF) is equal to: DF = (r - 1) * (c - 1) where r is the number of levels for one catagorical variable, and c is the number of levels for the other categorical variable. Expected frequencies. The expected frequency counts are computed separately for each level of one categorical variable at each level of the other categorical variable. Compute r*c expected frequencies, according to the following formula. E r,c = (n r * n c ) / n where E r,c is the expected frequency count for level r of Variable A and level c of Variable B, n r is the total number of sample observations at level r of Variable A, n c is the total number of sample observations at level c of Variable B, and n is the total sample size. Test statistic. The test statistic is a chi-square random variable (Χ 2 ) defined by the following equation. Χ 2 = Σ [ (O r,c - E r,c ) 2 / E r,c ] where O r,c is the observed frequency count at level r of Variable A and level c of Variable B, and E r,c is the expected frequency count at level r of Variable A and level c of Variable B. P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above. Interpret results. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. This involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Example: "A          was applied to a sample of 500 adults to assess whether there is a significant association between smoking status and education level, two categorical variables collected from the same population."

Cluster

Cluster sampling refers to a sampling method that has the following properties. The population is divided into N groups, called clusters . The researcher randomly selects n clusters to include in the sample. The number of observations within each cluster M i is known. The total number of observations M = M 1 + M 2 + M 3 + ... + M N-1 + M N . Each element of the population can be assigned to one, and only one, cluster.

Example: "In the national reading survey, schools served as         ; the researchers randomly selected 50 schools and then assessed all students within each chosen school rather than sampling students individually."

Cluster Sampling

Cluster sampling refers to a type of sampling method . With cluster sampling, the researcher divides the population into separate groups, called clusters. Then, a simple random sample of clusters is selected from the population. The researcher conducts his analysis on data from the sampled clusters. Compared to simple random sampling and stratified sampling , cluster sampling has advantages and disadvantages. For example, given equal sample sizes, cluster sampling usually provides less precision than either simple random sampling or stratified sampling. On the other hand, if travel costs between clusters are high, cluster sampling may be more cost-effective than the other methods.

Example: "Because it was impractical to list every household in the city, the researchers used          by randomly selecting 30 city blocks and surveying all households on the selected blocks."

Coefficient of Determination

The coefficient of determination (denoted by R 2 ) is a key output of regression analysis. It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable. The coefficient of determination is the square of the correlation (r) between predicted y scores and actual y scores; thus, it ranges from 0 to 1. With linear regression, the coefficient of determination is also equal to the square of the correlation between x and y scores. An R 2 of 0 means that the dependent variable cannot be predicted from the independent variable. An R 2 of 1 means the dependent variable can be predicted without error from the independent variable. An R 2 between 0 and 1 indicates the extent to which the dependent variable is predictable. An R 2 of 0.10 means that 10 percent of the variance in Y is predictable from X ; an R 2 of 0.20 means that 20 percent is predictable; and so on. The formula for computing the coefficient of determination for a linear regression model with one independent variable is given below. Coefficient of determination. The coefficient of determination (R 2 ) for a linear regression model with one independent variable is: R 2 = Σ (ŷ i - ȳ) 2 / Σ (y i - ȳ) 2 where y i is the value of the dependent variable for observation i, ŷ i is the predicted value of the dependent variable for observation i, and ȳ is the mean of observed values of the dependent variable. If you know the linear correlation (r) between the independent variable and the dependent variable, then the coefficient of determination (R 2 ) is easily computed using the following formula: R 2 = r 2 .

Example: "The          R² = 0.81 indicates that 81% of the variability in students' final exam scores can be explained by the linear relationship with the number of hours they studied."

Combination

A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC. Computing the number of combinations. The number of Combinations of n objects taken r at a time is: n C r = n(n - 1)(n - 2) ... (n - r + 1)/r! = n! / r!(n - r)! = n P r / r! where n P r is the number of Permutations of n objects taken r at a time. Note that AB and BA are considered to be one combination, because the order in which objects are selected does not matter. This is the key distinction between a combination and a permutation . A combination focuses on the selection of objects without regard to the order in which they are selected. A permutation, in contrast, focuses on the arrangement of objects with regard to the order in which they are arranged.

Example: "When choosing 3 students from a class of 25 to serve on a committee where order does not matter, the number of possible selections is C(25, 3) = 25! / (3! × 22!) = 2,300         ."

Complement

The complement of an event is the event not occurring. Thus, the complement of Event A is Event A not occurring. The probability that Event A will not occur is denoted by P(A').

Example: "If the probability of rain tomorrow is P(Rain) = 0.35, then the probability of its         , no rain, is P(Rain') = 1 − 0.35 = 0.65."

Completely Randomized Design

A completely randomized design is probably the simplest experimental design , in terms of data analysis and convenience. With this design, subjects are randomly assigned to treatments. Treatment Placebo Vaccine 500 500 A completely randomized design layout for a hypothetical medical experiment is shown in the table to the right. In this design, the experimenter randomly assigned subjects to one of two treatment conditions. They received a placebo or they received a cold vaccine. The same number of subjects (500) are assigned to each treatment condition (although this is not required). The dependent variable is the number of colds reported in each treatment condition. If the vaccine is effective, subjects in the "vaccine" condition should report significantly fewer colds than subjects in the "placebo" condition. A completely randomized design relies on randomization to control for the effects of extraneous variables. The experimenter assumes that, on averge, extraneous factors will affect treatment conditions equally; so any significant differences between conditions can fairly be attributed to the independent variable.

Example: "In a         , 60 subjects were randomly assigned to one of three diet plans with 20 subjects each, and weight loss after 8 weeks was compared across groups using a one-way ANOVA."

Conditional Distribution

A two-way table (also called a contingency table) is used to examine relationships between categorical variables. The entries in the cells of a two-way table can be frequency counts or relative frequencies (just like a one-way table ). Dance Sports TV Total Men 0.04 0.20 0.16 0.40 Women 0.32 0.12 0.16 0.60 Total 0.36 0.32 0.32 1.00 Relative Frequency of Table The table above shows preferences for leisure activities in the form of relative frequencies. The relative frequencies in the body of the table are called conditional frequencies or the conditional distribution . The table above shows relative frequencies for the whole table. Two-way tables can also show relative frequencies separately for the for rows and for columns. The table below shows relative frequencies for rows. Dance Sports TV Total Men 0.10 0.50 0.40 1.00 Women 0.53 0.20 0.27 1.00 Total 0.36 0.32 0.32 1.00 Relative Frequency of Row And the next table shows relative frequencies for columns. Dance Sports TV Total Men 0.11 0.62 0.50 0.40 Women 0.89 0.38 0.50 0.60 Total 1.00 1.00 1.00 1.00 Relative Frequency of Column.

Example: "The          of preferred activity given that the respondent is female showed that 80% of women preferred dancing, compared to only 10% in the overall marginal distribution."

Conditional Frequency

A two-way table (also called a contingency table) is used to examine relationships between categorical variables. The entries in the cells of a two-way table can be frequency counts or relative frequencies (just like a one-way table ). Dance Sports TV Total Men 0.04 0.20 0.16 0.40 Women 0.32 0.12 0.16 0.60 Total 0.36 0.32 0.32 1.00 Relative Frequency of Table The table above shows preferences for leisure activities in the form of relative frequencies. The relative frequencies in the body of the table are called conditional frequencies or the conditional distribution . The table above shows relative frequencies for the whole table. Two-way tables can also show relative frequencies separately for the for rows and for columns. The table below shows relative frequencies for rows. Dance Sports TV Total Men 0.10 0.50 0.40 1.00 Women 0.53 0.20 0.27 1.00 Total 0.36 0.32 0.32 1.00 Relative Frequency of Row And the next table shows relative frequencies for columns. Dance Sports TV Total Men 0.11 0.62 0.50 0.40 Women 0.89 0.38 0.50 0.60 Total 1.00 1.00 1.00 1.00 Relative Frequency of Column.

Example: "The          of selecting a student who prefers science given that the student is in 11th grade was 0.42, calculated by dividing the joint frequency by the marginal frequency for 11th grade."

Conditional Probability

The probability that event A occurs, given that event B has occurred, is called a conditional probability. The conditional probability of A, given B, is denoted by the symbol P(A|B).

Example: "Given that a randomly selected card from a standard deck is red, the          that it is a heart is P(Heart | Red) = 13/26 = 0.5, since exactly half of all red cards are hearts."

Confidence Interval

Statisticians use a confidence interval to express the degree of uncertainty associated with a sample statistic . A confidence interval is an interval estimate combined with a probability statement. For example, suppose a statistician conducted a survey and computed an interval estimate, based on survey data. The statistician might use a confidence level to describe uncertainty associated with the interval estimate. He/she might describe the interval estimate as a "95% confidence interval". This means that if we used the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter to fall within the interval estimates 95% of the time. Confidence intervals are preferred to point estimates , because only confidence intervals indicate (a) the precision of the estimate and (b) the uncertainty of the estimate.

Example: "Based on a random sample of 150 voters, the statistician computed a 95%          for the true proportion who support the ballot measure as (0.48, 0.62)."

Confidence Level

In survey sampling, different samples can be randomly selected from the same population; and each sample can often produce a different confidence interval . Some confidence intervals include the true population parameter; others do not. A confidence level refers to the percentage of all possible confidence intervals that can be expected to include the true population parameter. For example, suppose all possible samples were selected from the same population using the same sampling plan , and a confidence interval were computed for each sample. A 95% confidence level implies that 95% of the confidence intervals would include the true population parameter. Often, researchers choose 90%, 95%, or 99% confidence levels; but any percentage can be used.

Example: "A 95%          means that if the same sampling procedure were repeated many times, approximately 95% of the resulting confidence intervals would contain the true population parameter."

Confounding

Confounding occurs when the experimental controls do not allow the experimenter to reasonably eliminate plausible alternative explanations for an observed relationship between independent and dependent variables. Consider this example. A drug manufacturer tests a new cold medicine with 200 volunteer subjects - 100 men and 100 women. The men receive the drug, and the women do not. At the end of the test period, the men report fewer colds. This experiment implements no controls at all! As a result, many variables are confounded, and it is impossible to say whether the drug was effective. For example, gender is confounded with drug use. Perhaps, men are less vulnerable to the particular cold virus circulating during the experiment, and the new medicine had no effect at all. Or perhaps the men experienced a placebo effect . This experiment could be strengthened with a few controls. Women and men could be randomly assigned to treatments. One treatment could receive a placebo, with blinding . Then, if the treatment group (i.e., the group getting the medicine) had sufficiently fewer colds than the control group, it would be reasonable to conclude that the medicine was effective in preventing colds.

Example: "In the observational study linking coffee consumption to heart disease, physical activity level was a          variable because it was associated with both coffee drinking habits and cardiovascular outcomes."

Contingency Table

A two-way table (also called a contingency table ) is a useful tool for examining relationships between categorical variables. The entries in the cells of a two-way table can be frequency counts or relative frequencies (just like a one-way table ). Dance Sports TV Total Men 2 10 8 20 Women 16 6 8 30 Total 18 16 16 50 The two-way table above shows the favorite leisure activities for 50 adults - 20 men and 30 women. Because entries in the table are frequency counts, the table is a frequency table .

Example: "The          organizing survey responses by both gender and preferred social media platform allowed the researcher to compute expected counts and perform a chi-square test for independence."

Continuous Probability Distribution

If a random variable is a continuous variable , its probability distribution is called a continuous probability distribution . A continuous probability distribution differs from a discrete probability distribution in several ways. The probability that a continuous random variable will assume a particular value is zero. As a result, a continuous probability distribution cannot be expressed in tabular form. Instead, an equation or formula is used to describe a continuous probability distribution. The equation used to describe a continuous probability distribution is called a probability density function (pdf). All probability density functions satisfy the following conditions: The random variable Y is a function of X; that is, y = f(x). The value of y is greater than or equal to zero for all values of x. The total area under the curve of the function is equal to one. The charts below show two continuous probability distributions. The first chart shows a probability density function described by the equation y = 1 over the range of 0 to 1 and y = 0 elsewhere. y = 1 The next chart shows a probability density function described by the equation y = 1 - 0.5x over the range of 0 to 2 and y = 0 elsewhere. The area under the curve is equal to 1 for both charts. y = 1 - 0.5x The probability that a continuous random variable falls in the interval between a and b is equal to the area under the pdf curve between a and b . For example, in the first chart above, the shaded area shows the probability that the random variable X will fall between 0.6 and 1.0. That probability is 0.40. And in the second chart, the shaded area shows the probability of falling between 1.0 and 2.0. That probability is 0.25.

Example: "The heights of adult women follow a         , and probabilities are found by computing the area under the normal curve between specified values rather than summing discrete point probabilities."

Continuous Random Variable

A random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV.

Example: "The exact time it takes a student to complete a standardized test is a         , since it can take any value within the allowed interval and is not restricted to whole numbers."

Continuous Variable

If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable ; otherwise, it is called a discrete variable . Some examples will clarify the difference between discrete and continuous variables. Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.

Example: "The volume of water in a rain gauge is a          because it can take any non-negative value within its range, unlike the number of rainy days in a month, which is discrete."

Control Group

In an experiment , a control group is a baseline group that receives no treatment or a neutral treatment. To assess treatment effects, the experimenter compares results in the treatment group to results in the control group.

Example: "In the drug trial, the          received a placebo so that any improvement observed in the treatment group could be attributed to the drug rather than the passage of time or the placebo effect."

Convenience Sample

A convenience sample is one of the main types of non-probability sampling methods . A convenience sample is made up of people who are easy to reach. Consider the following example. A pollster interviews shoppers at a local mall. If the mall was chosen because it was a convenient site from which to solicit survey participants and/or because it was close to the pollster's home or business, this would be a convenience sample.

Example: "The researcher obtained a          by surveying students in the school cafeteria during lunch, which was quick and easy but likely introduced bias by underrepresenting students who eat off campus."

Correlation

Correlation coefficients measure the strength of association between two variables. The most common correlation coefficient, called the Pearson product-moment correlation coefficient , measures the strength of the linear association between variables. The sign and the absolute value of a Pearson correlation coefficient describe the direction and the magnitude of the relationship between two variables. The value of a correlation coefficient ranges between -1 and 1. The greater the absolute value of a correlation coefficient, the stronger the linear relationship. The strongest linear relationship is indicated by a correlation coefficient of -1 or 1. The weakest linear relationship is indicated by a correlation coefficient equal to 0. A positive correlation means that if one variable gets bigger, the other variable tends to get bigger. A negative correlation means that if one variable gets bigger, the other variable tends to get smaller. Keep in mind that the Pearson correlation coefficient only measures linear relationships. Therefore, a correlation of 0 does not mean zero relationship between two variables; rather, it means zero linear relationship. (It is possible for two variables to have zero linear relationship and a strong curvilinear relationship at the same time.) A formula for computing a Pearson correlation coefficient is given below. Correlation coefficient. The correlation r between two variables is: r = Σ (xy) / sqrt [ ( Σ x 2 ) * ( Σ y 2 ) ] where Σ is the summation symbol, x = x i - x , x i is the x value for observation i, x is the mean x value, y = y i - y , y i is the y value for observation i, and y is the mean y value. Fortunately, you will rarely have to compute a correlation coefficient by hand. Many software packages (e.g., Excel) and most graphing calculators have a correlation function that will do the job for you.

Example: "The strong positive          (r = 0.91) between hours of sunlight and daily ice cream sales does not imply causation; outdoor temperature is a lurking variable that likely drives both."

Correlation Coefficient

Correlation coefficients measure the strength of association between two variables. The most common correlation coefficient, called the Pearson product-moment correlation coefficient, measures the strength of the linear association between variables measured on an interval or ratio scale.

Example: "The Pearson          of r = −0.78 indicates a strong negative linear association between the number of absences and final course grade, meaning students who miss more class tend to earn lower grades."

Covariance

Covariance is a measure of the extent to which corresponding elements from two sets of ordered data move in the same direction. We use the following formula to compute covariance. Cov( X , Y ) = Σ ( X i - X ) ( Y i - Y ) / N = Σ x i y i / N where N is the number of scores in each set of data X is the mean of the N scores in the first dataset X i is the i the raw score in the first set of scores x i is the i th deviation score in the first set of scores Y is the mean of the N scores in the second dataset Y i is the i the raw score in the second set of scores y i is the i th deviation score in the second set of scores Cov( X , Y ) is the covariance of corresponding scores in the two sets of data.

Example: "A positive          between study time and test scores indicates that the two variables tend to move in the same direction, but unlike the correlation coefficient,          is not standardized and depends on units of measurement."

Critical Value

The critical value is a factor used to compute the margin of error, as shown in the equations below. Margin of error = Critical value x Standard deviation of the statistic Margin of error = Critical value x Standard error of the statistic When the sampling distribution of the statistic is normal or nearly normal, the critical value can be expressed as a t score or as a z-score . To find the critical value, follow these steps. Compute alpha (α): α = 1 - (confidence level / 100) Find the critical probability (p*): p* = 1 - α/2 To express the critical value as a z-score, find the z-score having a cumulative probability equal to the critical probability (p*). To express the critical value as a t statistic, follow these steps. Find the degrees of freedom (df). Often, df is equal to the sample size minus one. The critical t statistic (t*) is the t statistic having degrees of freedom equal to df and a cumulative probability equal to the critical probability (p*). Should you express the critical value as a t statistic or as a z-score? There are several ways to answer this question. As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the t statistic is preferred. Nevertheless, many introductory texts and the Advanced Placement Statistics Exam use the z-score exclusively. On this website, we provide sample problems that illustrate both approaches. You can use the Normal Distribution Calculator to find the critical z-score, and the t Distribution Calculator to find the critical t statistic. You can also use a graphing calculator or standard statistical tables (found in the appendix of most introductory statistics texts).

Example: "For a two-tailed z-test at the α = 0.05 significance level, the          are z* = ±1.96; if the computed test statistic falls outside this range, the researcher rejects the null hypothesis."

Cumulative Frequency

In a dataset, the cumulative frequency for a value x is the total number of scores that are less than or equal to x .

Example: "In the frequency table for test scores, the          for a score of 80 was 72, meaning 72 out of 90 students scored at or below 80 points."

Cumulative Frequency Plot

A cumulative frequency plot is a way to display cumulative information graphically. It shows the number, percentage, or proportion of observations in a dataset that are less than or equal to particular values. In a dataset, the cumulative frequency for a value x is the total number of scores that are less than or equal to x . The charts below illustrate the difference between frequency and cumulative frequency. Both charts show scores for a test administered to 300 students. In the first chart, column height shows frequency - the number of students in each test score grouping. For example, about 30 students received a test score between 51 and 60.   Frequency   100 80 60 40 20               41-50 51-60 61-70 71-80 81-90 91-100 In the next chart, column height shows cumulative frequency - the number of students up to and including each test score. This is a cumulative frequency chart. It shows that 30 students received a test score of at least 50; 60 students received a score of at least 60; 120 students received a score of at least 70; and so on.   Cumulative frequency   300 240 180 120 60               50 60 70 80 90 100.

Example: "The          (ogive) showed that approximately 70% of runners finished the 5K race in 30 minutes or less, with the curve rising most steeply between the 25- and 35-minute marks."

Cumulative Probability

A cumulative probability refers to the probability that the value of a random variable falls within a specified range. Frequently, cumulative probabilities refer to the probability that a random variable is less than or equal to a specified value. Consider a coin flip experiment. If we flip a coin two times, we might ask: What is the probability that the coin flips would result in one or fewer heads? The answer would be a cumulative probability. It would be the probability that the coin flip results in zero heads plus the probability that the coin flip results in one head. Thus, the cumulative probability would equal: P(X 1) = P(X = 0) + P(X = 1) = 0.25 + 0.50 = 0.75 The table below shows both the probabilities and the cumulative probabilities associated with this experiment. Number of heads Probability Cumulative Probability 0 0.25 0.25 1 0.50 0.75 2 0.25 1.00.

Example: "Using the standard normal table, the          P(Z ≤ 1.65) = 0.9505 tells us that about 95% of values in a standard normal distribution fall at or below z = 1.65."

Cumulative Relative Frequency

The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.

Example: "The          of 0.84 at a salary of $70,000 indicates that 84% of sampled employees earn $70,000 or less, which is useful for estimating percentile ranks."

Data

A set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage).

Example: "The researcher collected both qualitative          (students' favorite subjects) and quantitative continuous          (their GPA) to explore whether academic interest is associated with academic performance."

Decision Rule

A decision rule is a procedure that a researcher uses to decide whether to accept or reject the null hypothesis . For example, a researcher might hypothesize that a population mean is equal to 10. He/she might collect a random sample of observations to test this hypothesis. The decision rule might be to accept the hypothesis if the sample mean were close to 10 (say, between 9 and 11), and to reject the hypothesis if the sample mean were not close to 10 (say, less than 9 or greater than 11).

Example: "The          stated: reject H₀ if the p-value is less than 0.05; since the computed p-value was 0.031, the researcher rejected the null hypothesis and concluded the treatment had a statistically significant effect."

Degrees of Freedom

The term degrees of freedom (df) refers to the number of independent sample points used to compute a statistic minus the number of parameters estimated from the sample points. For example, consider the sample estimate of the population variance ( s 2 ):   n Σ i=1 ( X i - X ) 2 s 2 = ( n - 1 ) where X i is the score for observation i in the sample, X is the sample estimate of the population mean, and n is the number of observations in the sample. The formula is based on n independent sample points and one estimated population parameter ( X ). Therefore, the number of degrees of freedom is n minus one. For this example, df = n - 1.

Example: "When estimating the population variance using s² = Σ(xᵢ − x̄)² / (n − 1), the          equal n − 1 because one parameter, the sample mean, was estimated from the data before computing deviations."

Dependent Events

If two events are NOT independent, then we say that they are dependent.

Example: "Drawing two cards from a deck without replacement produces          because the probability of the second card being an ace changes depending on whether the first card drawn was an ace."

Dependent Variable

In an experimental design, the independent variable is the variable that is manipulated by the experimenter to determine its relationship to an observed phenomenon, called the dependent variable. More generally, the independent variable is the “cause,” while dependent variable is the “effect” of the independent variable. The dependent variable is also called the response variable.

Example: "In the experiment examining the effect of sleep deprivation on reaction time, reaction time was the          because it was the outcome measured in response to the manipulated amounts of sleep."

Deviation Score

A deviation score is the difference between a raw score and the mean. d i = x i - x where d i is the deviation score for the i th observation in a set of observations x i is the raw score for the i th observation in a set of observations x is the mean of all the observations in a set of observations.

Example: "A student who scored 88 on an exam with a class mean of 75 has a          of dᵢ = 88 − 75 = +13, indicating her score was 13 points above the mean."

Discrete Probability Distribution

If a random variable is a discrete variable , its probability distribution is called a discrete probability distribution . An example will make this clear. Suppose you flip a coin two times. This simple statistical experiment can have four possible outcomes: HH, HT, TH, and TT. Now, let the random variable X represent the number of Heads that result from this experiment. The random variable X can only take on the values 0, 1, or 2, so it is a discrete random variable. The probability distribution for this statistical experiment appears below. Number of heads Probability 0 0.25 1 0.50 2 0.25 Because the random variable is discrete, the above table is an example of a discrete probability distribution.

Example: "The          for the number of heads in three fair coin flips assigns probabilities P(X=0)=0.125, P(X=1)=0.375, P(X=2)=0.375, and P(X=3)=0.125, which sum to 1."

Discrete Variable

If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable ; otherwise, it is called a discrete variable . Some examples will clarify the difference between discrete and continuous variables. Suppose the fire department mandates that all fire fighters must weigh between 150 and 250 pounds. The weight of a fire fighter would be an example of a continuous variable; since a fire fighter's weight could take on any value between 150 and 250 pounds. Suppose we flip a coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable.

Example: "The number of customers entering a store each hour is a          because it can only take non-negative integer values and cannot equal, for example, 17.5."

Disjoint

Two events are disjoint if they cannot occur simultaneously.

Example: "Rolling a 3 and rolling a 5 on a single die roll are          events because both cannot occur simultaneously, so P(3 or 5) = P(3) + P(5) = 1/6 + 1/6 = 1/3 by the simplified addition rule."

Dotplot

A dotplot is a type of graphic display used to compare frequency counts within categories or groups. As you might guess, a dotplot is made up of dots plotted on a graph. Here is how to interpret a dotplot. Each dot can represent a single observation from a set of data, or a specified number of observations from a set of data. The dots are stacked in a column over a category, so that the height of the column represents the relative or absolute frequency of observations in the category. Here is an example to show what a dotplot looks like and how to interpret it. Suppose 30 first graders are asked to pick their favorite color. Their choices can be summarized in a dotplot, as shown below. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Red Orange Yellow Green Blue Indigo Violet Each dot represents one student, and the number of dots in a column represents the number of first graders who selected the color associated with that column. For example, Red was the most popular color (selected by 9 students), followed by Blue (selected by 7 students). Selected by only 1 student, Indigo was the least popular color.

Example: "The          of daily high temperatures for January showed most values clustered between 30°F and 45°F, with two isolated dots at 68°F indicating unusually warm days that may warrant further investigation as outliers."

Double Bar Chart

A double bar chart is similar to a regular bar chart , except that it provides two pieces of information for each category rather than just one. Often, the charts are color-coded with a different colored bar representing each piece of information. Above, a double bar chart shows customer satisfaction ratings for different cars, broken out by gender. The blue rows represent males; the red rows, females. Both groups prefer the Japanese cars to the American cars, with Honda receiving the highest ratings and Ford receiving the lowest ratings. Moreover, both genders agree on the rank order in which the cars are rated. As a group, the men seem to be tougher raters; they gave lower ratings to each car than the women gave.

Example: "The          compared the pass rates of male and female students across four subject exams, making it easy to see that the gender gap was largest in mathematics and smallest in English."

Double Blinding

In an experiment , if subjects in the control group know that they are receiving a placebo , the placebo effect will be reduced or eliminated; and the placebo will not serve its intended control purpose. Blinding is the practice of not telling subjects whether they are receiving a placebo. In this way, subjects in the control and treatment groups experience the placebo effect equally. Often, knowledge of which groups receive placebos is also kept from analysts who evaluate the experiment. This practice is called double blinding . It prevents the analysts from "spilling the beans" to subjects through subtle cues; and it assures that their evaluation is not tainted by awareness of actual treatment conditions.

Example: "The clinical trial used          so that neither the patients nor the physicians administering treatment knew who received the active drug versus the placebo, preventing unconscious bias from influencing the measured outcomes."

Effect Size

In a hypothesis test , the power of the test refers to the probability that the test will reject the null hypothesis when the null hypothesis is false. To compute the power of the test, one offers an alternative guess about the "true" value of the population parameter, assuming that the null hypothesis is false. That true value is called the critical value. The effect size is the difference between the critical value and the value specified in the null hypothesis. For example, suppose the null hypothesis states that a population mean is equal to 100. A researcher might ask: What is the probability of rejecting the null hypothesis if the true population mean is equal to 90? In this example, the critical value would be 90. The effect size would be 90 - 100, which equals -10.

Example: "The          of Cohen's d = 0.8 indicated a large practical difference between the two group means, suggesting the treatment effect was meaningful and would be detectable even with a moderately sized sample."

Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule that describes how data is distributed in a normal distribution . It states that: About 68% of the data falls within 1 standard deviation (σ) of the mean (μ). About 95% of the data falls within 2 standard deviations (2σ) of the mean (μ). About 99.7% of the data falls within 3 standard deviations (3σ) of the mean (μ). The empirical rule provides a quick way to estimate the spread of data around the mean. It is widely used in statistics to understand the variability of data and to identify outliers.

Example: "Because adult male heights are approximately normally distributed with μ = 70 inches and σ = 3 inches, the          tells us that about 95% of men have heights between 64 and 76 inches, within two standard deviations of the mean."

Equally Likely

Each outcome of an experiment has the same probability.

Example: "When rolling a fair six-sided die, all six outcomes are          with probability 1/6 each, so the probability of rolling an odd number is simply 3 × (1/6) = 0.5."

Estimation

In statistics, estimation refers to the process by which one makes inferences about a population, based on information obtained from a sample. Often, we use sample statistics (e.g., mean, proportion) to estimate population parameters (e.g., mean, proportion).

Example: "Using         , a student might approximate the kinetic energy of a moving car (mass ~1000 kg, speed ~30 m/s) as roughly 450,000 J without needing precise measurements."

Estimator

In statistics, the terms estimate and estimator have distinct meanings: Estimator. This is a rule or formula used to estimate a population parameter , based on sample data. Estimate. This is the actual numerical value obtained when the estimator is applied to a specific sample . It is a realized value of the estimator based on observed data. For example,the formula x̄ = Σ (x i )/n is an estimator for a population mean (μ). The sample mean (x̄ ) = 5 is an estimate of a population mean (μ).

Example: "The sample mean x̄ is a commonly used          of the population mean μ because it is unbiased, meaning E(x̄) = μ, and its standard error decreases as sample size increases."

Event

A subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a sample space and is usually denoted by S. An event is an arbitrary subset in S. It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital letters such as A, B, C, and so on.

Example: "When rolling two dice, the          'sum equals 7' is a subset of the sample space containing all 36 equally likely outcomes and includes exactly six pairs: {(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}."

Expected Value

The mean of the discrete random variable X is also called the expected value of X. Notationally, the expected value of X is denoted by E(X). Use the following formula to compute the mean of a discrete random variable. E(X) = Σ [ x i * P(x i ) ] where x i is the value of the random variable for outcome i, and P(x i ) is the probability that the random variable will be equal to outcome i.

Example: "If a game pays $10 with probability 0.2 and $0 with probability 0.8, the          E(X) = 10(0.2) + 0(0.8) = $2, meaning a player earns an average of $2 per game over many repetitions."

Experiment

An experiment is a controlled study in which the researcher attempts to understand cause-and-effect relationships. The study is "controlled" in the sense that the researcher controls (1) how subjects are assigned to groups and (2) which treatments each group receives. In the analysis phase, the researcher compares group scores on some dependent variable . Based on the analysis, the researcher draws a conclusion about whether the treatment ( independent variable ) had a causal effect on the dependent variable.

Example: "In a well-designed         , students were randomly assigned to either a computer-based or traditional instruction group, and post-test scores were compared to assess whether instructional method caused a difference in learning."

Experimental Design

The term experimental design refers to a plan for assigning subjects to treatment conditions. A good experimental design serves three purposes. Causation . It allows the experimenter to make causal inferences about the relationship between independent variables and a dependent variable . Control . It allows the experimenter to rule out alternative explanations due to the confounding effects of extraneous variables (i.e., variables other than the independent variables). Variability . It reduces variability within treatment conditions, which makes it easier to detect differences in treatment outcomes.

Example: "The          specified that 90 subjects would be randomly divided into three equal groups each receiving a different drug dose, with the change in blood pressure as the response variable."

Experimental Unit

Any individual or object to be measured.

Example: "In a study testing three different fertilizers on wheat crops, each individual plot of land was an         , and the grain yield per plot was the measured response variable."

Explanatory Variable

The independent variable in an experiment; the value controlled by researchers.

Example: "In the regression analysis predicting fuel efficiency from vehicle weight, vehicle weight was the          whose values were used to explain variability in miles per gallon."

Exponential Distribution

A continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital, notation: X ~ Exp(m). The mean is μ = 1 m 1 m and the standard deviation is σ = 1 m σ = --> 1 m . The probability density function is f(x) = me–mx, x ≥ 0 and the cumulative distribution function is P(X ≤ x) = 1 – e–mx.

Example: "The time between successive arrivals at an emergency room can be modeled using an         , where if the average arrival rate is λ = 4 per hour, the mean time between arrivals is 1/λ = 15 minutes."

Extraneous Variable

Every experiment exists to examine a cause-and-effect relationship. With respect to the relationship under investigation, an experimental design needs to account for three types of variables: Dependent variable. The dependent variable is the outcome being measured, the effect in a cause-and-effect relationship. Independent variables. An independent variable is a variable that is explicitly included in an experiment, because the experimenter believes it is a potential cause in a cause-and-effect relationship. Extraneous variables. An extraneous variable is any other variable that could affect the dependent variable, but is not explicitly included in the experiment.

Example: "Room temperature was an          in the memory experiment because it could affect participants' concentration and test performance, even though it was not the independent variable being manipulated."

Extrapolation

Extrapolation is the process of using a regression equation to predict values of the response variable for values of the explanatory variable that are outside the range of the data used to fit the model. Extrapolation can lead to unreliable predictions.

Example: "Using the regression equation ŷ = 12 + 3x to predict the salary of someone with 40 years of experience, when the data only covered 0–20 years, is an example of          that may produce unreliable predictions."

F Distribution

The distribution of all possible values of the f statistic is called an F distribution . The curve of the F distribution depends on the degrees of freedom , v 1 and v 2 . When describing an F distribution, the number of degrees of freedom v 1 is always stated first. Thus, f (5, 9) would refer to an F distribution with v 1 = 5 and v 2 = 9 degrees of freedom; whereas f (9, 5) would refer to an F distribution with v 1 = 9 and v 2 = 5 degrees of freedom. Note that the curve represented by f (5, 9) would differ from the curve represented by f (9, 5). The F distribution has the following properties: The mean of the distribution is equal to v 1 / ( v 2 - 2 ). The variance is equal to [ v 2 2 * ( v 1 + 2 ) ] / [ v 1 * ( v 2 - 2 ) * ( v 2 - 4 ) ].

Example: "In a one-way ANOVA comparing five group means, the test statistic follows an          with 4 numerator degrees of freedom and 45 denominator degrees of freedom, written F(4, 45)."

F Statistic

The f statistic , also known as an f value , is a random variable that has an F distribution . Here are the steps required to compute an f statistic : Select a random sample of size n 1 from a normal population, having a standard deviation equal to σ 1 . Select an independent random sample of size n 2 from a normal population, having a standard deviation equal to σ 2 . The f statistic is the ratio of s 1 2 /σ 1 2 and s 2 2 /σ 2 2 . The following is an equation commonly used to compute an f statistic: f = [ s 1 2 /σ 1 2 ] / [ s 2 2 /σ 2 2 ] where σ 1 is the standard deviation of population 1, s 1 is the standard deviation of the sample drawn from population 1, σ 2 is the standard deviation of population 2, and s 1 is the standard deviation of the sample drawn from population 2.

Example: "The         , computed as the ratio of the between-group mean square to the within-group mean square, was F = 6.34, which exceeded the critical value of 3.24 at α = 0.05, leading to rejection of the null hypothesis."

Factor

In an experiment, a factor (also called an independent variable) is an explanatory variable manipulated by the experimenter. Each factor has two or more levels (i.e., different values of the factor). Combinations of factor levels are called treatments . The table below shows independent variables, factors, levels, and treatments for a hypothetical experiment. Vitamin C 0 mg 250 mg 500 mg Vitamin E 0 mg Treatment 1 Treatment 2 Treatment 3 400 mg Treatment 4 Treatment 5 Treatment 6 /* Small screen font size for table */ @media (max-width: 500px) { table.fontsize th, table.fontsize td {font-size:8pt;} } In this hypothetical experiment, the researcher is studying the possible effects of Vitamin C and Vitamin E on health. There are two factors - dosage of Vitamin C and dosage of Vitamin E. The Vitamin C factor has three levels - 0 mg per day, 250 mg per day, and 500 mg per day. The Vitamin E factor has 2 levels - 0 mg per day and 400 mg per day. The experiment has six treatments. Treatment 1 is 0 mg of E and 0 mg of C, Treatment 2 is 0 mg of E and 250 mg of C, and so on.

Example: "In a two-         experiment studying plant growth, fertilizer type and watering frequency were both         , each with two levels, creating four distinct treatment combinations randomly assigned to plots."

Factorial

In general, n objects can be arranged in n ( n - 1)( n - 2) ... (3)(2)(1) ways. This product is represented by the symbol n !, which is called n factorial . By convention, 0! = 1. Thus, 0! = 1; 2! = (2)(1) = 2; 3! = (3)(2)(1) = 6; 4! = (4)(3)(2)(1) = 24; 5! = (5)(4)(3)(2)(1) = 120; and so on.

Example: "To determine the number of ways to arrange 5 different books on a shelf, we compute 5! = 5 × 4 × 3 × 2 × 1 = 120, since each position can be filled by any remaining unchosen book."

Factorial Experiment

A factorial experiment allows researchers to study the joint effect of two or more factors on a dependent variable . With a full factorial design, the experiment includes a treatment group for every combination of factor levels. Therefore, the number of treatment groups is the product of factor levels. For example, consider the full factorial design shown below:   A 1 A 2 B 1 B 2 B 3 B 1 B 2 B 3 C 1 Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 C 2 Group 7 Group 8 Group 9 Group 10 Group 11 Group 12 C 3 Group 13 Group 14 Group 15 Group 16 Group 17 Group 18 C 4 Group 19 Group 20 Group 21 Group 22 Group 23 Group 24 Factor A has two levels, factor B has three levels, and factor C has four levels. Therefore, the full factorial design has 2 x 3 x 4 = 24 treatment groups. Note: Unlike full factorial experiments, fractional factorial experiments include only a subset of possible treatment groups.

Example: "In a          with two factors—teaching method (lecture vs. discussion) and class size (small vs. large)—all four combinations were tested, enabling the researcher to detect any interaction effects between the factors."

First Quartile

The value that is the median of the of the lower half of the ordered data set.

Example: "In an ordered dataset of exam scores, a          of Q1 = 62 means that 25% of students scored at or below 62 points, and this value forms the left edge of the box in a boxplot."

Five-Number Summary

A five-number summary consists of the minimum value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum value of a dataset. It is commonly used to describe the distribution of a dataset and construct boxplots.

Example: "The          of commute times—{8, 22, 35, 47, 94} minutes—revealed a large gap between Q3 and the maximum, indicating a right-skewed distribution with at least one extreme high outlier."

Frequency

The number of times a value of the data occurs.

Example: "The          of students scoring in the 80–89 range was 18, meaning 18 out of 60 students achieved scores in that interval, making it the most common score range in the class."

Frequency Count

A frequency count is a measure of the number of times that an event occurs.

Example: "The          of 14 heads in 30 coin flips was used to compute a relative frequency of 14/30 ≈ 0.467, which was then compared to the theoretical probability of 0.5 in a chi-square goodness of fit test."