Quartiles, Percentiles & IQR Cheat Sheet

A printable reference covering quartiles, percentiles, five-number summaries, interquartile range, and outlier fences for grades 7-12.

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Quartiles, percentiles, and the interquartile range help students describe where data values fall within a distribution. This cheat sheet gives quick rules for ordering data, finding position values, and summarizing spread. Students need these tools to compare data sets, read box plots, and identify unusual values. The focus is on clear formulas and consistent steps that work for both small lists and larger data sets. The most important ideas are the median, quartiles, percentiles, and IQR. Quartiles split ordered data into four parts, while percentiles describe the percent of data at or below a value. The interquartile range is $IQR = Q_3 - Q_1$ , which measures the spread of the middle $50\%$ of the data. Outlier fences use $Q_1 - 1.5(IQR)$ and $Q_3 + 1.5(IQR)$ to flag values that are unusually low or high.

Key Facts

Always order the data from least to greatest before finding quartiles, percentiles, or the interquartile range.
The median, or $Q_2$ , is the middle value of an ordered data set, and for an even number of values it is the average of the two middle values.
The first quartile $Q_1$ is the median of the lower half of the data, and the third quartile $Q_3$ is the median of the upper half of the data.
The interquartile range is $IQR = Q_3 - Q_1$ , and it measures the spread of the middle $50\%$ of the data.
A percentile tells the percent of data values that are less than or equal to a given value.
One common percentile position formula is $L = \frac{p}{100}(n+1)$ , where $p$ is the percentile and $n$ is the number of data values.
The lower outlier fence is $Q_1 - 1.5(IQR)$ , and the upper outlier fence is $Q_3 + 1.5(IQR)$ .
The five-number summary is the minimum, $Q_1$ , median, $Q_3$ , and maximum.

Vocabulary

Quartile: A quartile is a value that divides an ordered data set into four parts with about $25\%$ of the data in each part.
Percentile: A percentile is a location measure that tells what percent of data values are less than or equal to a given value.
Interquartile Range: The interquartile range is the difference between the third quartile and first quartile, given by $IQR = Q_3 - Q_1$ .
Median: The median is the middle value of an ordered data set, also called $Q_2$ .
Five-Number Summary: A five-number summary lists the minimum, $Q_1$ , median, $Q_3$ , and maximum of a data set.
Outlier Fence: An outlier fence is a boundary found with $Q_1 - 1.5(IQR)$ or $Q_3 + 1.5(IQR)$ to help identify unusual values.

Common Mistakes to Avoid

Forgetting to order the data first is wrong because quartiles and percentiles depend on position in a sorted list.
Including the median in both halves when the method says to exclude it can change $Q_1$ and $Q_3$ , so use the method your class or calculator requires consistently.
Finding the range instead of the interquartile range is wrong because the range is $\max - \min$ , while the IQR is $Q_3 - Q_1$ .
Treating the $75$ th percentile as $75\%$ of the maximum value is wrong because percentiles describe position in the ordered data, not a percent of the largest number.
Calling every value outside $Q_1$ and $Q_3$ an outlier is wrong because outliers are usually checked against the fences $Q_1 - 1.5(IQR)$ and $Q_3 + 1.5(IQR)$ .

Practice Questions

1 Find $Q_1$ , the median, $Q_3$ , and $IQR$ for the data set $4, 7, 9, 10, 12, 15, 18, 21$ .
2 For the data set $3, 5, 6, 8, 10, 13, 15, 17, 20$ , find the five-number summary.
3 A data set has $Q_1 = 24$ and $Q_3 = 40$ . Find the $IQR$ , the lower outlier fence, and the upper outlier fence.
4 Two classes have the same median test score, but Class A has a larger $IQR$ than Class B. Explain what this means about the spread of the middle $50\%$ of scores.

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