Descriptive Statistics Cheat Sheet

A printable reference covering mean, median, mode, range, variance, standard deviation, IQR, five-number summaries, box plots, and z-scores for grades 9-12.

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Descriptive statistics gives students tools to organize, summarize, and describe data clearly. This cheat sheet covers measures of center, measures of spread, five-number summaries, box plots, and z-scores. Students need these ideas to compare data sets, recognize unusual values, and communicate patterns without listing every data point. These skills are used throughout statistics, science labs, business, sports analysis, and everyday data interpretation. The main measures of center are the mean, median, and mode, while common measures of spread include range, interquartile range, variance, and standard deviation. The mean is found with $\bar{x} = \frac{\sum x_i}{n}$ , and the sample standard deviation is $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ . A five-number summary uses the minimum, $Q_1$ , median, $Q_3$ , and maximum to build a box plot. A z-score, $z = \frac{x - \bar{x}}{s}$ , tells how many standard deviations a value is from the mean.

Key Facts

The mean of a data set is $\bar{x} = \frac{\sum x_i}{n}$ , where $\sum x_i$ is the sum of all data values and $n$ is the number of values.
The median is the middle value when data are ordered, or the average of the two middle values when $n$ is even.
The range measures total spread and is calculated by $\text{range} = \text{maximum} - \text{minimum}$ .
The interquartile range measures the spread of the middle half of the data and is $\text{IQR} = Q_3 - Q_1$ .
The sample variance is $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$ , and the sample standard deviation is $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ .
The population variance is $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$ , and the population standard deviation is $\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$ .
A five-number summary is $\text{min}, Q_1, \text{median}, Q_3, \text{max}$ , and it is used to draw a box plot.
A z-score is $z = \frac{x - \bar{x}}{s}$ for a sample, and it shows whether a value is above or below the mean and by how many standard deviations.

Vocabulary

Mean: The mean is the arithmetic average found by adding all data values and dividing by the number of values.
Median: The median is the middle value of an ordered data set, or the average of the two middle values if there is an even number of values.
Mode: The mode is the data value or values that occur most often in a data set.
Interquartile Range: The interquartile range is the difference between the third quartile and first quartile, written as $\text{IQR} = Q_3 - Q_1$ .
Standard Deviation: Standard deviation measures the typical distance of data values from the mean.
Z-Score: A z-score tells how many standard deviations a data value is above or below the mean.

Common Mistakes to Avoid

Forgetting to order the data before finding the median or quartiles is wrong because position-based statistics require the values to be arranged from least to greatest.
Dividing by $n$ instead of $n - 1$ for sample variance is wrong because $s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$ corrects for using a sample mean.
Confusing range with interquartile range is wrong because range uses the maximum and minimum, while $\text{IQR} = Q_3 - Q_1$ uses only the middle half of the data.
Treating every data set as if the mean is the best center is wrong because outliers and skewed data can pull the mean away from a typical value.
Interpreting a negative z-score as an error is wrong because $z < 0$ simply means the value is below the mean.

Practice Questions

1 Find the mean, median, mode, and range for the data set $6, 8, 8, 10, 13, 15$ .
2 For the data set $3, 5, 7, 9, 12, 14, 18, 20$ , find $Q_1$ , the median, $Q_3$ , and $\text{IQR}$ .
3 A test score is $x = 88$ , the class mean is $\bar{x} = 76$ , and the standard deviation is $s = 6$ . Find the z-score using $z = \frac{x - \bar{x}}{s}$ .
4 A data set is strongly right-skewed because of a few very large values. Explain whether the mean or median better represents the center and why.