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

The 5 number summary consists of five values: Minimum, Q1, Q2 (median), Q3, and Maximum. To find them:

1. Order the data from smallest to largest. The ends are the Minimum and Maximum.
2. Find the median of the entire data set (Q2).
3. Find the median of the lower half of the data (Q1).
4. Find the median of the upper half of the data (Q3).

The interquartile range (IQR) is the distance from Q1 to Q3, representing the middle 50% of the data. IQR = Q3 - Q1.

Examples

• For the data {2, 5, 6, 9, 11, 14, 17}, the 5 number summary is: Min=2, Q1=5, Q2=9, Q3=14, Max=17. The IQR is $14 - 5 = 9$.
• For the data {10, 20, 25, 35, 45, 50}, the median (Q2) is 30. The lower half is {10, 20, 25}, so Q1=20. The upper half is {35, 45, 50}, so Q3=45. The IQR is $45 - 20 = 25$.
• Given a 5 number summary of Min=5, Q1=12, Q2=18, Q3=22, Max=30, the range is $30-5=25$ and the interquartile range (IQR) is $22 - 12 = 10$.

Explanation

The 5 number summary provides a quick snapshot of your data's distribution. The IQR is a powerful measure of spread because it focuses on the middle half of the data, which means it isn't affected by unusually high or low outliers.