The least-squares regression line is the line that fits a scatter plot best by making the sum of the squared residuals as small as possible. Its equation is ŷ = a + bx, where b is the slope and a is the y-intercept. You use it to predict a value of y from a value of x.
The correlation coefficient, r, measures the direction and strength of a linear relationship between two quantitative variables. It is always between −1 and 1. A positive r means y tends to rise as x rises, and a negative r means y tends to fall. The closer r is to 1 or −1, the closer the points are to a line.
Comparing distributions means describing how two or more groups differ in shape, outliers, center and spread, using comparison words and the context. Graph the groups on the same scale, then write sentences like “Route 1’s median time is 2 minutes longer than Route 2’s, but its IQR is less than half as large.”
Transforming data means changing every value in a data set the same way, such as adding 5 points to every score or converting inches to centimeters. Adding or subtracting a constant shifts the center but leaves the spread alone. Multiplying or dividing by a constant changes both the center and the spread by that factor. The shape stays the same.
A percentile tells what percent of the data are at or below a value: the 75th percentile is greater than or equal to 75% of the data. A z-score tells how many standard deviations a value is from the mean: z = (x − mean) ÷ standard deviation. Both describe where one value stands in a distribution.
The 1.5 × IQR rule is the standard way to find outliers. Find the interquartile range, IQR = Q3 − Q1, and multiply it by 1.5. Any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is an outlier. These two cutoffs are called the lower and upper fences.
Standard deviation measures how spread out data are around the mean. To find it, subtract the mean from each value, square those deviations, add them, divide by n − 1 (for a sample), and take the square root. A small standard deviation means the values sit close to the mean; a large one means they are spread out.
Describing a distribution means telling what the graph of a quantitative variable shows: its shape, any outliers, its center and its spread, written in the context of the data. A short way to remember this is SOCS. For example: “Commute times are skewed right, with a median of 27.5 minutes and a range of 21 minutes.”
A stem-and-leaf plot (or stemplot) shows quantitative data by splitting each value into a stem and a leaf. The stem is every digit except the last, and the leaf is the last digit. Leaves are written in order beside their stem, so the plot shows the shape of the data and keeps every value.
Bar graphs and pie charts are the two main graphs for categorical data. A bar graph uses separated bars whose heights show the count or percent in each category. A pie chart splits a circle into slices, one per category, sized by its share of the whole. Use a pie chart only when the parts make up one whole.