Lesson 4: Measures of Variation

Property

A measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes with a single number how its values vary.
The first, and easiest, measure is the range. This is given by the difference between the highest and lowest data values.
While easy to compute, it only tells us how far apart the extreme values are and gives no indication about the spread of the data values between the two extremes.

Examples

• A student's test scores are 85, 92, 78, 95, and 88. The highest score is 95 and the lowest is 78. The range is $95 - 78 = 17$.
• Two basketball players' points per game are recorded. Player A: {15, 17, 16, 18}. Player B: {5, 10, 25, 30}. Player A's range is $18 - 15 = 3$, while Player B's is $30 - 5 = 25$, showing Player B's scoring is less consistent.
• The data sets {2, 8, 8, 9, 12} and {2, 3, 4, 5, 12} both have a range of 10. However, the first set is clustered high while the second is more evenly spread, showing a limitation of using only the range.

Explanation

Measures of variability, like range, tell you about the spread of your data. While the mean or median tells you the center, variability describes if the data points are all clustered together or widely scattered apart.

Property

Quartiles divide an ordered data set into four equal parts. 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. The median of the entire data set is the second quartile ($Q_2$).

Examples

• For the data set $\{2, 5, 6, 8, 11, 12, 15, 18\}$, the median ($Q_2$) is $9.5$. The lower half is $\{2, 5, 6, 8\}$, so $Q_1 = \frac{5+6}{2} = 5.5$. The upper half is $\{11, 12, 15, 18\}$, so $Q_3 = \frac{12+15}{2} = 13.5$.
• For the data set $\{3, 7, 8, 10, 14, 16, 19\}$, the median ($Q_2$) is $10$. The lower half is $\{3, 7, 8\}$, so $Q_1 = 7$. The upper half is $\{14, 16, 19\}$, so $Q_3 = 16$.

Explanation

To find the quartiles, first order your data from least to greatest and find the median (the second quartile, $Q_2$). The first quartile, $Q_1$, is the median of the data points that are less than $Q_2$. The third quartile, $Q_3$, is the median of the data points that are greater than $Q_2$. These values are essential for understanding the spread of the data and for calculating the interquartile range.

Property

The interquartile range (IQR) is the difference between the third quartile ($Q_3$) and the first quartile ($Q_1$).

Examples

• For a data set with a first quartile of $Q_1 = 10$ and a third quartile of $Q_3 = 25$, the interquartile range is $25 - 10 = 15$.
• If the quartiles of a data set are $Q_1 = 38.5$ and $Q_3 = 52$, the interquartile range is $52 - 38.5 = 13.5$.

Explanation

The interquartile range, or IQR, measures the spread of the middle half of your data. To find the IQR, you first need to identify the first quartile ($Q_1$) and the third quartile ($Q_3$). The IQR is then calculated by simply subtracting the value of $Q_1$ from the value of $Q_3$. This value represents the range of the central 50% of the data and is less sensitive to extreme values, or outliers, than the total range.