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Range and Interquartile Range

Measures of Spread (Dispersion)

Essential Understanding: While the mean, median, and mode tell us about the center of data, measures of spread tell us how spread out or varied the data is. Two data sets can have the same mean but very different spreads!

Range: Simplest measure
IQR: Middle 50% spread
Semi-IQR: Half of IQR

Understanding Quartiles

Quartiles Divide Data into Four Equal Parts

\(Q_1\)
Lower Quartile
25th percentile
\(Q_2\)
Median
50th percentile
\(Q_3\)
Upper Quartile
75th percentile

25% of data falls below \(Q_1\), 50% below \(Q_2\), and 75% below \(Q_3\)

Measures of Spread

Range

Definition: The difference between the largest and smallest values.

\[ \text{Range} = \text{Maximum} - \text{Minimum} \]

Pros: Easy to calculate

Cons: Affected by outliers

Interquartile Range (IQR)

Definition: The range of the middle 50% of the data.

\[ IQR = Q_3 - Q_1 \]

Pros: Not affected by outliers

Use: Better for skewed data

Semi-Interquartile Range

Definition: Half of the interquartile range.

\[ \text{Semi-IQR} = \frac{Q_3 - Q_1}{2} = \frac{IQR}{2} \]

Also called: Quartile deviation

Interactive Box Plot Explorer

Box and Whisker Plot Generator

Enter your data to see a box plot and calculate all measures of spread!

Lower Quartile (\(Q_1\))
Median (\(Q_2\))
Upper Quartile (\(Q_3\))
IQR (Box)
Minimum
12
\(Q_1\)
18
Median (\(Q_2\))
26
\(Q_3\)
32
Maximum
38
Range
26
IQR
14
Semi-IQR
7

Step-by-Step Example

Example: Finding Range, IQR, and Semi-IQR

Data: The test scores of 11 students are: 45, 52, 58, 62, 65, 70, 73, 78, 82, 88, 95

Visualizing the quartiles:

45
52
58
62
65
70
73
78
82
88
95
Lower 25% ← Middle 50% (IQR) → Upper 25%
1
Find the Range:
Range = Maximum - Minimum = 95 - 45 = 50
2
Find the Median (\(Q_2\)):
With 11 values, median is the 6th value = 70
3
Find \(Q_1\) (Lower Quartile):
Look at lower half: 45, 52, 58, 62, 65
Median of lower half = 3rd value = 58
4
Find \(Q_3\) (Upper Quartile):
Look at upper half: 73, 78, 82, 88, 95
Median of upper half = 3rd value = 82
5
Calculate IQR:
IQR = \(Q_3 - Q_1\) = 82 - 58 = 24
6
Calculate Semi-IQR:
Semi-IQR = \(\frac{IQR}{2} = \frac{24}{2}\) = 12

Comparing Data Sets

Using Spread to Compare

Question: Two classes took the same test:

  • Class A: Mean = 72, Range = 40, IQR = 15
  • Class B: Mean = 72, Range = 25, IQR = 8

Analysis:

  • Both classes have the same average (mean = 72)
  • Class A has a larger range (40 vs 25), suggesting more extreme scores
  • Class A has a larger IQR (15 vs 8), meaning the middle 50% of scores are more spread out
  • Conclusion: Class B's scores are more consistent/clustered together

CSEC Practice Questions

Test Your Understanding

1
Find the range of: 8, 15, 12, 20, 5, 18
5
12
15
20
Solution: Range = Maximum - Minimum = 20 - 5 = 15
2
For the data 3, 5, 7, 9, 11, 13, 15, find \(Q_1\).
5
7
9
11
Solution: The median is 9 (4th value). The lower half is 3, 5, 7. The median of the lower half is 5, so \(Q_1 = 5\).
3
If \(Q_1 = 24\) and \(Q_3 = 48\), find the interquartile range.
12
24
36
72
Solution: IQR = \(Q_3 - Q_1\) = 48 - 24 = 24
4
The IQR of a data set is 18. What is the semi-interquartile range?
9
18
36
6
Solution: Semi-IQR = \(\frac{IQR}{2} = \frac{18}{2}\) = 9

Key Points to Remember

  • Range = Maximum - Minimum (simple but affected by outliers)
  • Quartiles divide ordered data into four equal parts: \(Q_1, Q_2, Q_3\)
  • IQR = \(Q_3 - Q_1\) (spread of middle 50%, not affected by outliers)
  • Semi-IQR = \(\frac{IQR}{2}\) (also called quartile deviation)
  • A smaller IQR means data is more consistent/clustered
  • A larger IQR means data is more spread out
CSEC Examination Tips
  • Always sort the data first before finding quartiles
  • For quartiles: find the median first, then find the median of each half
  • If asked to compare data sets, discuss both central tendency (mean/median) AND spread (range/IQR)
  • When describing spread, use phrases like "more consistent", "less varied", "more spread out"
  • In box plots: the box represents the IQR, the line inside is the median
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