Range and Interquartile Range (IQR) - Sixth Grade Mathematics

Measures of variation quantify the spread, dispersion, and consistency of numerical data, describing how closely values cluster around the center or how widely they scatter.

The Concept of Statistical Variability

Two classes can have the exact same mean score of 80 on a test, yet have completely different score distributions: Class A: 78, 79, 80, 81, 82 (tightly clustered, highly consistent). Class B: 60, 70, 80, 90, 100 (widely scattered, highly variable). Measures of center describe the middle, but measures of variation describe the spread.

The Range

The range is the simplest measure of variation, calculated as the absolute difference between the largest value (maximum) and the smallest value (minimum).

Formula for Range:
Range = Maximum Value - Minimum Value

For Class A above: Range = 82 - 78 = 4 points. For Class B above: Range = 100 - 60 = 40 points. While easy to calculate, the range depends entirely upon the two most extreme values and provides no insight into how the remaining numbers are distributed between them.

Quartiles and the Five-Number Summary

To better understand spread without being deceived by extremes, statisticians divide an ordered data set into four equal quarters (quarters of 25% each) using three divider points called quartiles.

The Five-Number Summary:
1. Minimum (Min): The smallest value in the data set.
2. First Quartile (Q1): The median of the lower half of data (25th percentile).
3. Second Quartile (Q2 or Median): The overall median splitting data 50/50.
4. Third Quartile (Q3): The median of the upper half of data (75th percentile).
5. Maximum (Max): The largest value in the data set.

Dividing Data into Quarters (25% per quarter):
  [ Min ] ------ 25% ------ [ Q1 ] ------ 25% ------ [ Q2 ] ------ 25% ------ [ Q3 ] ------ 25% ------ [ Max ]

When calculating Q1 and Q3, if the overall data set has an odd number of items, do NOT include the overall median in either half!

Step-by-Step Calculation of Quartiles

Follow this exact procedure:

Data Set (11 students' quiz scores):
12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 29

Step 1: Locate the overall Median (Q2):
        Count = 11 (odd). The middle is the 6th value: 20.
        Q2 = 20.

Step 2: Identify the Lower Half (strictly below Q2):
        Lower Half: 12, 14, [15], 17, 18
        Find median of lower half: Q1 = 15.

Step 3: Identify the Upper Half (strictly above Q2):
        Upper Half: 21, 23, [24], 26, 29
        Find median of upper half: Q3 = 24.

Five-Number Summary:
Min = 12,  Q1 = 15,  Median (Q2) = 20,  Q3 = 24,  Max = 29.

The Interquartile Range (IQR)

The Interquartile Range (IQR) is the distance between the first quartile and the third quartile, representing the spread of the middle 50% of the data.

Formula for Interquartile Range:
IQR = Q3 - Q1

For the quiz scores above: IQR = 24 - 15 = 9 points. Because the IQR discards the lowest 25% and highest 25% of values, it is completely immune to extreme outliers!

The 1.5 x IQR Rule for Outliers

Statisticians use a mathematical boundary fence to formally classify outliers: Lower Boundary Fence = Q1 - (1.5 * IQR) Upper Boundary Fence = Q3 + (1.5 * IQR) Any data point lying below the lower fence or above the upper fence is mathematically designated as an outlier.

Exercises

Exercise 1. Find the Range, Q1, Median (Q2), Q3, and IQR for each data set: (a) 5, 8, 10, 12, 15, 18, 20; (b) 22, 25, 28, 30, 31, 35, 38, 42.

Exercise 2. A data set has Q1 = 30 and Q3 = 50. (a) What is the IQR? (b) Using the 1.5 x IQR rule, calculate the lower and upper fences for outliers; (c) State whether a data value of 85 would be considered an outlier.

Exercise 3. Two basketball players track points scored per game over 9 games: Player 1: 15, 16, 18, 19, 20, 21, 22, 24, 25 Player 2: 8, 10, 12, 15, 20, 25, 28, 30, 32 Compare their ranges and IQRs. Which player showed greater consistency?

Solutions

Solution to Exercise 1. For (a), Min = 5, Max = 20. Range = 20 - 5 = 15. Median (Q2) = 12. Lower half: 5, 8, 10 -> Q1 = 8. Upper half: 15, 18, 20 -> Q3 = 18. IQR = Q3 - Q1 = 18 - 8 = 10. For (b), Min = 22, Max = 42. Range = 42 - 22 = 20. Count = 8 (even). Median (Q2) = (30 + 31)/2 = 30.5. Lower half: 22, 25, 28, 30 -> Q1 = (25 + 28)/2 = 26.5. Upper half: 31, 35, 38, 42 -> Q3 = (35 + 38)/2 = 36.5. IQR = 36.5 - 26.5 = 10.

Solution to Exercise 2. For (a), IQR = Q3 - Q1 = 50 - 30 = 20. For (b), 1.5 * IQR = 1.5 * 20 = 30. Lower fence = Q1 - 30 = 30 - 30 = 0. Upper fence = Q3 + 30 = 50 + 30 = 80. For (c), since 85 is strictly greater than the upper fence of 80, the value 85 is an outlier.

Solution to Exercise 3. For Player 1: Range = 25 - 15 = 10; Median = 20; Q1 = (16 + 18)/2 = 17; Q3 = (22 + 24)/2 = 23; IQR = 23 - 17 = 6. For Player 2: Range = 32 - 8 = 24; Median = 20; Q1 = (10 + 12)/2 = 11; Q3 = (28 + 30)/2 = 29; IQR = 29 - 11 = 18. Both players had the same median (20 points), but Player 1 had a much smaller range (10 vs 24) and smaller IQR (6 vs 18), demonstrating far greater scoring consistency from game to game.