Summary statistics: center, position and variability

What the exam asks

In Section I (Multiple Choice), expect questions asking you to locate the median or quartiles from a histogram or cumulative table, compare standard deviations of two graphs by eye, or calculate a new mean and standard deviation after a unit conversion (such as converting Celsius to Fahrenheit). In Section II (Free Response), for example in Question 2 (Practices 3 and 4), you may be asked to calculate summary statistics, show the structure of your work, and interpret the standard deviation or interquartile range in context with proper units.

Core ideas

Measures of center

• Sample mean ($\bar{x}$): The arithmetic average of the $n$ observations. This formula is on the AP formula sheet: $$\bar{x} = \frac{1}{n}\sum x_i$$
• Median: The middle value when the data are ordered from smallest to largest. It splits the data so that about 50% of values fall at or below it. If $n$ is odd, the median is the single middle value at position $\frac{n+1}{2}$. If $n$ is even, it is the average of the two middle values.

Measures of position

• Minimum and maximum: The smallest and largest observed values in the data set.
• Percentiles: The $p$th percentile is a value with about $p\%$ of the observations at or below it.
• First quartile ($Q_1$) and third quartile ($Q_3$): is the median of the lower half of the ordered data (the 25th percentile), and is the median of the upper half (the 75th percentile). When is odd, exclude the overall median before splitting the data into halves. Together, and form the boundaries of the of the data.