AP Statistics · Topic 1.7 · Unit 1

AP Stats 1.7: Summary Statistics

By Jude Wallis · Published July 28, 2026

Center is the mean or median; position is quartiles and percentiles; spread is the range, IQR, or standard deviation. The median and IQR resist outliers, while the mean, range, and standard deviation do not. Sample standard deviation divides by n minus 1.

AP Statistics: Unit 1 (topics 1.7). Topic 1.7 (Summary Statistics for One Quantitative Variable) sits in Unit 1 of the redesigned AP Statistics course (effective Fall 2026, first exam May 2027). Unit 1 is the heaviest weighted unit at 20-30% of the multiple-choice section.

Measures of center and position

Topic 1.7 is where you calculate the numbers that describe a quantitative distribution. The two common measures of center are the mean and the median. The sample mean is the sum of the values divided by how many there are, written $\bar{x} = \frac{\sum x_i}{n}$ . The median is the middle value once the data are ordered: the single middle value when $n$ is odd, or the average of the two middle values when $n$ is even.

Position matters too. The minimum and maximum are the smallest and largest values. The first quartile $Q_1$ is the median of the lower half of the ordered data, and the third quartile $Q_3$ is the median of the upper half; on this site the middle value is excluded from both halves, the TI-84 convention. About 25% of values fall at or below $Q_1$ and about 75% at or below $Q_3$ , so more generally the $p$ th percentile is the value with $p\%$ of the data at or below it.

Measures of variability and outliers

Three measures describe spread. The range is $\text{max} - \text{min}$ . The interquartile range is $\text{IQR} = Q_3 - Q_1$ , the width of the middle 50%. The standard deviation is a typical distance of the values from their mean; the sample standard deviation is

s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

and its square $s^2$ is the sample variance. Dividing by $n-1$ , not $n$ , is what the AP formula sheet uses for a sample.

Two outlier rules appear in the CED. A value is an outlier if it is more than $1.5 \times \text{IQR}$ above $Q_3$ or below $Q_1$ , or, by the second rule, more than 2 standard deviations above or below the mean. Work them in the 1.5 IQR rule and standard deviation by hand.

Resistance and units

Some statistics shrug off outliers and some do not. The median and IQR are resistant (robust), so outliers barely move them. The mean, range, and standard deviation are nonresistant, so a single extreme value can pull them a long way. That is why you report the median and IQR for skewed data and the mean and standard deviation for roughly symmetric data; see mean vs median.

Changing the units of measurement changes the values of the statistics, so a mean in inches becomes a different number in centimeters. Summary statistics also let you compare center, variability, shape, and outliers across two or more independent samples, and they can justify a claim about the variable in context.

Full summary statistics with an outlier check

Seven daily high temperatures, in degrees Fahrenheit, are 60, 62, 65, 68, 70, 72, 91. Find the mean, median, quartiles, IQR, and range, then check for outliers with the 1.5 IQR rule.

The data are already ordered and $n = 7$ . Sum them: $60 + 62 + 65 + 68 + 70 + 72 + 91 = 488$ .
Mean: $\bar{x} = 488 / 7 = 69.71$ (to two decimals).
Median: the middle (4th) value is 68.
Lower half (values below the median, middle value excluded): 60, 62, 65. Its median is $Q_1 = 62$ .
Upper half: 70, 72, 91. Its median is $Q_3 = 72$ .
IQR: $Q_3 - Q_1 = 72 - 62 = 10$ . Range: $91 - 60 = 31$ .
Fences: $1.5 \times \text{IQR} = 1.5 \times 10 = 15$ . Upper fence $= 72 + 15 = 87$ ; lower fence $= 62 - 15 = 47$ .
Compare: $91 > 87$ , so 91 is an outlier. No value is below 47.

Mean $\approx 69.71^{\circ}$ F, median $68^{\circ}$ F, $Q_1 = 62$ , $Q_3 = 72$ , IQR $= 10$ , range $= 31$ . The value 91 lies above the upper fence of 87, so it is an outlier by the 1.5 IQR rule. The mean sitting above the median is consistent with the right skew that outlier creates.

Frequently asked questions

Why does the sample standard deviation divide by n minus 1?

Dividing by $n-1$ instead of $n$ corrects for using the sample mean in place of the unknown population mean, giving an unbiased estimate of the variance. The AP formula sheet uses $n-1$ for a sample.

Which outlier rule should I use, 1.5 IQR or 2 standard deviations?

Both are in the CED. The 1.5 IQR rule pairs with the five-number summary and boxplots; the 2 standard deviations rule pairs with the mean and standard deviation. Use whichever the question's summaries point to.