Unit 1: Exploring One-Variable Data

Statistics

The science of learning from data while being honest about variability.

Variability

The natural tendency of data values to differ from one another; a core idea that must be acknowledged when analyzing data.

Descriptive statistics

Methods for organizing, displaying, and summarizing data (center, spread, shape, position measures).

Inferential statistics

Using sample data to draw conclusions about a larger population (developed more in later units).

Individual

The “who” the data describe (a person, school, game, day, etc.).

Variable

A characteristic measured on each individual (e.g., height, major, commute time).

Data

The recorded values of a variable for the individuals in a study.

One-variable data

Data in which each individual contributes one measurement (or one category label) for a single variable.

Categorical (qualitative) variable

A variable whose values are category names or group labels (e.g., blood type, brand of phone).

Quantitative variable

A variable whose values are numerical measurements or counts for which arithmetic makes sense (e.g., age, commute time).

Numeric-looking label

A number used only as an identifier/category (e.g., ZIP code, jersey number), so it is categorical, not quantitative.

Discrete quantitative variable

A quantitative variable that takes a finite or countable set of values, often with gaps (e.g., number of AP classes).

Continuous quantitative variable

A quantitative variable that can take infinitely many values with no gaps (e.g., height, weight).

Distribution

What values a variable takes and how often it takes them (for quantitative data, includes shape, center, spread, and unusual features).

Frequency

The count of individuals in a category (categorical) or in a bin/interval (quantitative).

Relative frequency

The proportion of individuals in a category/bin; computed as count divided by total n.

Frequency table

A table listing each category (or value/bin) and its count.

Relative frequency table

A table listing each category (or value/bin) and its proportion (or percent) of the total.

Bar chart

A display for categorical data with one bar per category showing frequency or relative frequency; bars are separated.

Histogram

A display for quantitative data that groups values into bins (intervals) and uses touching bars to show frequencies/relative frequencies.

Bin

An interval of numerical values used to group quantitative data in a histogram.

Dotplot

A graph placing one dot per data value on a number line (stacking repeats); best for small-to-moderate data sets.

Stemplot (stem-and-leaf plot)

A display that splits each number into a stem (leading digits) and leaf (final digit) to show shape while preserving exact values.

Stemplot key

A note explaining how to read stems and leaves (e.g., “2|7 means 27”) to avoid ambiguity.

Time plot (time series plot)

A graph of a quantitative variable over time with time on the horizontal axis; used when order over time is meaningful.

Cumulative relative frequency plot (ogive)

A graph showing how relative frequency accumulates from smaller to larger values; useful for reading medians and quartiles (0.50, 0.25, 0.75).

SOCS

Checklist for describing quantitative distributions: Shape, Outliers/unusual features, Center, Spread (in context).

Unimodal

A distribution with one clear peak.

Bimodal

A distribution with two clear peaks, often suggesting two subgroups mixed together.

Symmetric distribution

A distribution whose left and right sides are roughly mirror images.

Skewed right

A distribution with a long tail to the right (toward higher values); mean is typically greater than the median.

Skewed left

A distribution with a long tail to the left (toward lower values); mean is typically less than the median.

Outlier

An unusually large or small value compared to the overall pattern; often flagged using the 1.5·IQR rule (not automatic proof of error).

Quartiles (Q1, Q2, Q3)

Values that split ordered data into quarters: Q1 is the 25th percentile, Q2 is the median (50th), Q3 is the 75th percentile.

Interquartile range (IQR)

A resistant measure of spread for the middle 50% of data: IQR = Q3 − Q1.

1.5·IQR rule

Outlier rule using fences: values below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR) are flagged as outliers.

Five-number summary

Minimum, Q1, median, Q3, maximum; the basis for constructing a boxplot.

Boxplot (modified boxplot)

Graph based on the five-number summary: box from Q1 to Q3, line at median, whiskers to most extreme non-outliers, and points for outliers.

Population

The entire group of individuals of interest in a study.

Sample

A subset of the population from which data are actually collected.

Parameter

A numerical summary describing a population (e.g., population mean μ, population standard deviation σ).

Statistic

A numerical summary computed from a sample (e.g., sample mean x̄, sample standard deviation s).

Mean

The arithmetic average; the “balance point” of a distribution and is sensitive to outliers.

Median

The middle value (or average of the two middle values) in ordered data; resistant to outliers.

Standard deviation

A measure of variability: a typical distance of values from the mean (s for sample, σ for population); sensitive to outliers because deviations are squared.

Shifting (add/subtract a constant)

Transforming data by y = x + c; adds c to measures of center (mean/median) but leaves measures of spread (SD/IQR) unchanged.

Rescaling (multiply by a constant)

Transforming data by y = ax; multiplies measures of center by a and measures of spread by |a| (a negative a reflects the distribution).

z-score

A standardized value giving how many standard deviations x is from the mean: z = (x − μ)/σ (or z = (x − x̄)/s for sample summaries).

Normal distribution N(μ,σ)

A bell-shaped, symmetric density model determined by mean μ (center) and standard deviation σ (spread); mean equals median at the center.

Empirical Rule (68–95–99.7 rule)

For a Normal distribution: about 68% of observations fall within 1σ of μ, about 95% within 2σ, and about 99.7% within 3σ.