Worksheet for calculating interquartile range (IQR) with examples and practice problems.

Let's solve the Interquartile Range (IQR) problems step by step. The IQR is calculated as:

$$
\text{IQR} = Q_3 - Q_1
$$

Where:
- $ Q_1 $ is the first quartile (median of the lower half of the data),
- $ Q_3 $ is the third quartile (median of the upper half of the data).

We’ll go through each data set, sort it (if not already sorted), find $ Q_1 $ and $ Q_3 $, and compute the IQR.

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Problem 1: 11, 14, 20, 23, 24, 30, 37

Step 1: Data is already in order.

Number of data points: 7 (odd)

- Median (Q₂): Middle value → 23
- Lower half: 11, 14, 20 → median is 14 → $ Q_1 = 14 $
- Upper half: 24, 30, 37 → median is 30 → $ Q_3 = 30 $

$$
\text{IQR} = Q_3 - Q_1 = 30 - 14 = \boxed{16}
$$

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Problem 2: 5, 2, 8, 15, 9, 10, 11

Step 1: Sort the data:
5, 2, 8, 15, 9, 10, 11 →
Sorted: 2, 5, 8, 9, 10, 11, 15

Number of data points: 7 (odd)

- Median (Q₂): Middle value → 9
- Lower half: 2, 5, 8 → median = 5 → $ Q_1 = 5 $
- Upper half: 10, 11, 15 → median = 11 → $ Q_3 = 11 $

$$
\text{IQR} = 11 - 5 = \boxed{6}
$$

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Problem 3: 7, 11, 12, 22, 25

Step 1: Already sorted.

Number of data points: 5 (odd)

- Median: 12
- Lower half: 7, 11 → median = $ \frac{7+11}{2} = 9 $ → $ Q_1 = 9 $
- Upper half: 22, 25 → median = $ \frac{22+25}{2} = 23.5 $ → $ Q_3 = 23.5 $

$$
\text{IQR} = 23.5 - 9 = \boxed{14.5}
$$

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Problem 4: 4, 6, 25, 15, 16, 18, 19, 7

Step 1: Sort the data:
4, 6, 7, 15, 16, 18, 19, 25

Number of data points: 8 (even)

- Median (Q₂): Average of 4th and 5th values: $ \frac{15 + 16}{2} = 15.5 $
- Lower half: 4, 6, 7, 15 → median = $ \frac{6+7}{2} = 6.5 $ → $ Q_1 = 6.5 $
- Upper half: 16, 18, 19, 25 → median = $ \frac{18+19}{2} = 18.5 $ → $ Q_3 = 18.5 $

$$
\text{IQR} = 18.5 - 6.5 = \boxed{12}
$$

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Problem 5: 16, 35, 14, 16, 16, 17, 34

Step 1: Sort the data:
14, 16, 16, 16, 17, 34, 35

Number of data points: 7 (odd)

- Median: 16
- Lower half: 14, 16, 16 → median = 16 → $ Q_1 = 16 $
- Upper half: 17, 34, 35 → median = 34 → $ Q_3 = 34 $

$$
\text{IQR} = 34 - 16 = \boxed{18}
$$

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Problem 6: 23, 28, 21, 41, 29, 37, 39, 42, 37

Step 1: Sort the data:
21, 23, 28, 29, 37, 37, 39, 41, 42

Number of data points: 9 (odd)

- Median: 37
- Lower half: 21, 23, 28, 29 → median = $ \frac{23+28}{2} = 25.5 $ → $ Q_1 = 25.5 $
- Upper half: 37, 39, 41, 42 → median = $ \frac{39+41}{2} = 40 $ → $ Q_3 = 40 $

$$
\text{IQR} = 40 - 25.5 = \boxed{14.5}
$$

---

Problem 7: 3, 28, 47, 11, 25, 19, 28, 45

Step 1: Sort the data:
3, 11, 19, 25, 28, 28, 45, 47

Number of data points: 8 (even)

- Median: average of 4th and 5th = $ \frac{25 + 28}{2} = 26.5 $
- Lower half: 3, 11, 19, 25 → median = $ \frac{11+19}{2} = 15 $ → $ Q_1 = 15 $
- Upper half: 28, 28, 45, 47 → median = $ \frac{28+45}{2} = 36.5 $ → $ Q_3 = 36.5 $

$$
\text{IQR} = 36.5 - 15 = \boxed{21.5}
$$

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Problem 8: 24, 26, 41, 45, 50, 39, 22, 25, 44

Step 1: Sort the data:
22, 24, 25, 26, 39, 41, 44, 45, 50

Number of data points: 9 (odd)

- Median: 39
- Lower half: 22, 24, 25, 26 → median = $ \frac{24+25}{2} = 24.5 $ → $ Q_1 = 24.5 $
- Upper half: 41, 44, 45, 50 → median = $ \frac{44+45}{2} = 44.5 $ → $ Q_3 = 44.5 $

$$
\text{IQR} = 44.5 - 24.5 = \boxed{20}
$$

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✔ Final Answers:

| Problem | IQR |
|--------|-----|
| 1 | 16 |
| 2 | 6 |
| 3 | 14.5|
| 4 | 12 |
| 5 | 18 |
| 6 | 14.5|
| 7 | 21.5|
| 8 | 20 |

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🔍 Summary:

To find the IQR:
1. Sort the data.
2. Find the median (Q₂).
3. Split the data into lower and upper halves.
4. Find Q₁ (median of lower half) and Q₃ (median of upper half).
5. Subtract: $ \text{IQR} = Q_3 - Q_1 $

This measures the spread of the middle 50% of the data — useful for identifying outliers.

Let me know if you'd like a visual explanation or box plot!