SAT Math — Problem-Solving & Data Analysis: Learn-It-From-Scratch Notes

Ratio

A comparison of two quantities by division; the order matters, and ratios can compare part to part or part to whole.

Rate

A ratio that compares quantities with different units, often using the word per, such as miles per hour or dollars per pound.

Unit rate

A rate with 1 in the denominator, used to compare options easily, such as cost per 1 ounce.

Proportional relationship

A relationship in which one quantity is always a constant multiple of the other, written as y = kx; its graph is a straight line through the origin.

Constant of proportionality

The constant k in the equation y = kx; for nonzero x, it equals y/x.

Dimensional analysis

A unit-conversion method that multiplies by conversion factors written as fractions equal to 1 so unwanted units cancel.

Percent

A fraction out of 100; for example, 35 percent means 35/100 or 0.35.

Percentage point change

The arithmetic difference between two percentages, such as a change from 40 percent to 55 percent being 15 percentage points.

Mean

The arithmetic average of a data set, found by dividing the sum of the values by the number of values.

Median

The middle value in an ordered data set, or the average of the two middle values if there is an even number of data points.

Interquartile range (IQR)

A measure of spread equal to Q3 minus Q1; it describes the spread of the middle 50 percent of the data.

Standard deviation

A measure of the typical distance of data values from the mean; larger standard deviation means greater spread.

Box plot

A graph that displays the five-number summary: minimum, Q1, median, Q3, and maximum.

Scatterplot

A graph of paired (x, y) data used to examine direction, form, strength, and outliers in a relationship.

Slope

In a linear model y = mx + b, the amount y changes for each 1-unit increase in x.

Residual

The difference between the actual y-value and the predicted y-value from a model, computed as actual minus predicted.

Extrapolation

Using a model to make a prediction outside the range of observed data; it is usually less reliable than predicting within the data range.

Conditional probability

The probability that event A occurs given that event B has occurred, written as P(A | B).

Independent events

Events for which knowing one happened does not change the probability of the other; equivalently, P(A | B) = P(A).

Margin of error

The amount added to and subtracted from a sample statistic to create a plausible interval for a population parameter.

Random sampling

Selecting a sample so members of the population have a chance to be chosen, which helps reduce bias and improve representativeness.

Random assignment

Assigning participants to treatment groups by chance, which helps balance confounding variables and supports causal conclusions.

Confounding variable

A variable related to both the explanatory variable and the response variable that can distort an apparent relationship.

Observational study

A study that records data without imposing treatments; it can show association but generally cannot prove causation.

Randomized controlled experiment

An experiment with random assignment and usually a control group; it is a strong design for making causal claims.