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Difficulty: MediumMeasures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

The table below shows the distribution of scores achieved by 20 students on a final exam:

ScoreFrequency
603
705
806
904
1002

What is the interquartile range (IQR) of the scores?

Answer: 20

Answer

The interquartile range (IQR) of the scores is 20.
For a set of 20 ordered scores, the first quartile Q1Q_1 is the median of the first 10 scores (average of the 5th and 6th values), and the third quartile Q3Q_3 is the median of the last 10 scores (average of the 15th and 16th values). Using the cumulative frequencies, the 5th and 6th scores are both 70 (Q1=70Q_1 = 70), and the 15th and 16th scores are both 90 (Q3=90Q_3 = 90). Thus, the interquartile range is IQR=Q3−Q1=90−70=20IQR = Q_3 - Q_1 = 90 - 70 = 20.

Step-by-Step Solution

1
Determine the cumulative frequency to locate quartile positions.
Score 60 occupies positions 1 to 3; Score 70 occupies positions 4 to 8; Score 80 occupies positions 9 to 14; Score 90 occupies positions 15 to 18; Score 100 occupies positions 19 to 20.
Tracking data positions in a frequency distribution allows efficient determination of medians and quartiles without expanding the raw list.
2
Calculate the first quartile (Q1Q_1).
Q1=70+702=70Q_1 = \frac{70 + 70}{2} = 70.
With 20 total values, the lower half consists of the first 10 values (positions 1 through 10). The median of these 10 values is the average of the 5th and 6th values, which are both 70.
3
Calculate the third quartile (Q3Q_3).
Q3=90+902=90Q_3 = \frac{90 + 90}{2} = 90.
The upper half consists of the last 10 values (positions 11 through 20). The median of these 10 values is the average of the 15th and 16th values, which are both 90.
4
Subtract Q1Q_1 from Q3Q_3 to find the interquartile range.
IQR=Q3−Q1=90−70=20IQR = Q_3 - Q_1 = 90 - 70 = 20.
The interquartile range represents the spread of the middle 50% of the dataset.

Key Concept

Interquartile Range (IQR) from a Frequency Distribution
Estimated Time:1m 30s
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Topics

Measures of Dispersion and Position (Range, IQR, Standard Deviation, Percentiles)

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