Interquartile Range (IQR) Statistics Example 4

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Example 4

hard
Daily temperatures (°C) for two weeks: 15, 16, 14, 18, 20, 22, 19, 35, 17, 16, 18, 21, 20, 19. Find the IQR and use the 1.5×IQR1.5 \times IQR rule to identify any outliers. Then recalculate the mean with and without the outlier(s).

Solution

  1. 1
    Step 1: Sort: 14,15,16,16,17,18,18,19,19,20,20,21,22,35. 14 values. Q2=18+192=18.5Q_2 = \frac{18+19}{2}=18.5. Lower half: {14,15,16,16,17,18,18}, Q1=16Q_1=16. Upper half: {19,19,20,20,21,22,35}, Q3=20Q_3=20. IQR=20−16=4IQR=20-16=4.
  2. 2
    Step 2: Fences: Lower=16−6=1016-6=10, Upper=20+6=2620+6=26. The value 35 > 26, so 35 is an outlier. Mean with outlier: 27014≈19.3\frac{270}{14}\approx 19.3. Mean without outlier: 23513≈18.1\frac{235}{13}\approx 18.1. The outlier raises the mean by about 1.2°C.

Answer

IQR=4IQR = 4. The value 35°C is an outlier (above the upper fence of 26°C). Mean with outlier ≈ 19.3°C; mean without ≈ 18.1°C.
The IQR-based outlier detection method is practical for identifying unusual values in real data. Removing outliers and comparing the mean before and after quantifies their influence. This helps decide whether outliers represent errors (remove) or genuine extreme observations (keep but note).

About Interquartile Range (IQR)

The interquartile range (IQR) is the range of the middle 50% of data, calculated as Q3−Q1Q_3 - Q_1. It measures spread while ignoring the top and bottom 25% of values, making it resistant to outliers.

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More Interquartile Range (IQR) Examples

Example 1 easy

Given the data set: 4, 7, 9, 12, 15, 18, 22, 25, 30, find the interquartile range (IQR).

Example 2 medium

Test scores: 55, 60, 65, 70, 72, 75, 78, 80, 85, 90, 95. Find the IQR and use it to determine the bo

Example 3 medium

Two classes took the same test. Class A: [formula], [formula]. Class B: [formula], [formula]. Which

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