Find the lower quartile, upper quartile, and interquartile range for each set of data.

Let's solve each part of the Alpha Exercise step by step.

We are asked to find:
- Lower Quartile (Q₁): The median of the lower half of the data.
- Upper Quartile (Q₃): The median of the upper half of the data.
- Interquartile Range (IQR): $ \text{IQR} = Q₃ - Q₁ $

We will follow these steps for each dataset:
1. Sort the data in ascending order (if not already).
2. Find the median (middle value) — this is the second quartile (Q₂).
3. Split the data into two halves around the median.
4. Find Q₁ as the median of the lower half.
5. Find Q₃ as the median of the upper half.
6. Compute IQR = Q₃ − Q₁.

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a) 4, 5, 7, 8, 8, 8, 9, 10, 10

Already sorted.

- Number of values: 9 (odd)
- Median (Q₂): 5th value → 8
- Lower half: 4, 5, 7, 8 → middle two: 5 and 7 → average = $ \frac{5+7}{2} = 6 $
→ Q₁ = 6
- Upper half: 8, 9, 10, 10 → middle two: 9 and 10 → average = $ \frac{9+10}{2} = 9.5 $
→ Q₃ = 9.5
- IQR = 9.5 − 6 = 3.5

✔ Answer a): Q₁ = 6, Q₃ = 9.5, IQR = 3.5

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b) 8, 9, 11, 12, 12, 12, 14, 14, 15, 17

Sorted: yes

- Number of values: 10 (even)
- Median (Q₂): average of 5th and 6th values → $ \frac{12 + 12}{2} = 12 $
- Lower half: first 5 values → 8, 9, 11, 12, 12
- Median = 3rd value → 11 → Q₁ = 11
- Upper half: last 5 values → 12, 14, 14, 15, 17
- Median = 3rd value → 14 → Q₃ = 14
- IQR = 14 − 11 = 3

✔ Answer b): Q₁ = 11, Q₃ = 14, IQR = 3

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c) 7.1, 7.2, 7.4, 7.4, 7.5

Sorted: yes

- Number: 5 (odd)
- Median (Q₂): 3rd value → 7.4
- Lower half: 7.1, 7.2 → median = $ \frac{7.1 + 7.2}{2} = 7.15 $ → Q₁ = 7.15
- Upper half: 7.4, 7.5 → median = $ \frac{7.4 + 7.5}{2} = 7.45 $ → Q₃ = 7.45
- IQR = 7.45 − 7.15 = 0.3

✔ Answer c): Q₁ = 7.15, Q₃ = 7.45, IQR = 0.3

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d) 19, 29, 31, 21, 28, 27, 24

First, sort:
→ 19, 21, 24, 27, 28, 29, 31

- Number: 7 (odd)
- Median (Q₂): 4th value → 27
- Lower half: 19, 21, 24 → median = 21 → Q₁ = 21
- Upper half: 28, 29, 31 → median = 29 → Q₃ = 29
- IQR = 29 − 21 = 8

✔ Answer d): Q₁ = 21, Q₃ = 29, IQR = 8

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e) 0.6, 0.23, 0.2, 0.7, 0.14, 0.1, 0.68

Sort:
→ 0.1, 0.14, 0.2, 0.23, 0.6, 0.68, 0.7

- Number: 7 (odd)
- Median (Q₂): 4th value → 0.23
- Lower half: 0.1, 0.14, 0.2 → median = 0.14 → Q₁ = 0.14
- Upper half: 0.6, 0.68, 0.7 → median = 0.68 → Q₃ = 0.68
- IQR = 0.68 − 0.14 = 0.54

✔ Answer e): Q₁ = 0.14, Q₃ = 0.68, IQR = 0.54

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f) 20, 31, 25, 45, 46, 20, 34, 31

Sort:
→ 20, 20, 25, 31, 31, 34, 45, 46

- Number: 8 (even)
- Median (Q₂): average of 4th and 5th → $ \frac{31 + 31}{2} = 31 $
- Lower half: 20, 20, 25, 31 → median = average of 2nd and 3rd → $ \frac{20 + 25}{2} = 22.5 $ → Q₁ = 22.5
- Upper half: 31, 34, 45, 46 → median = average of 2nd and 3rd → $ \frac{34 + 45}{2} = 39.5 $ → Q₃ = 39.5
- IQR = 39.5 − 22.5 = 17

✔ Answer f): Q₁ = 22.5, Q₃ = 39.5, IQR = 17

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✔ Final Answers:

| Part | Lower Quartile (Q₁) | Upper Quartile (Q₃) | Interquartile Range (IQR) |
|------|---------------------|---------------------|----------------------------|
| a) | 6 | 9.5 | 3.5 |
| b) | 11 | 14 | 3 |
| c) | 7.15 | 7.45 | 0.3 |
| d) | 21 | 29 | 8 |
| e) | 0.14 | 0.68 | 0.54 |
| f) | 22.5 | 39.5 | 17 |

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