8. Given the following data, (A) Find the median, quartiles and semi-interquartile range, (B) Find the 8th deciles and 85th percentiles. $$ 1976,2162,1502,1782,1523,2033,1564,1320,1951 $$, 1714,1841 $$ 1648,1345,1837,1718,2047,1954,2000,1479,2571 $$, 1739,1781

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Sure, let's break down the problem into its two main parts:

### **Part A: Median, Quartiles, and Semi-Interquartile Range**

**1. Organize the Data:**

First, arrange the given data in ascending order:

```
1320, 1345, 1479, 1502, 1523, 1564, 1648, 1714, 1718, 1739, 
1781, 1782, 1837, 1841, 1951, 1954, 1976, 2000, 2033, 2047, 
2162, 2571
```

**Total Data Points (N):** 22

**2. Calculate the Median:**

- **Position of Median:** For an even number of data points, the median is the average of the 11th and 12th values.
  
  $$
  \text{Median} = \frac{1781 + 1782}{2} = 1781.5
  $$

**3. Determine the Quartiles:**

- **First Quartile (Q₁):** This is the median of the lower half (first 11 data points).

$$
  Q₁ = \text{6th value} = 1564
  $$

- **Third Quartile (Q₃):** This is the median of the upper half (last 11 data points).

$$
  Q₃ = \text{6th value in upper half} = 1976
  $$

**4. Calculate the Semi-Interquartile Range:**

$$
\text{Semi-Interquartile Range} = \frac{Q₃ - Q₁}{2} = \frac{1976 - 1564}{2} = 206
$$

---

### **Part B: 8th Decile and 85th Percentile**

**Understanding Deciles and Percentiles:**

- **8th Decile:** Represents the 80th percentile.
- **85th Percentile:** Directly given.

**1. Calculate the 80th Percentile (8th Decile):**

- **Position Formula:**

$$
  \text{Position} = \left(\frac{80}{100}\right) \times (N + 1) = 0.8 \times 23 = 18.4
  $$

- **Interpolation:**
  
  - **18th Value:** 2000
  - **19th Value:** 2033
  
  $$
  \text{80th Percentile} = 2000 + 0.4 \times (2033 - 2000) = 2013.2 \approx 2013
  $$

**2. Calculate the 85th Percentile:**

- **Position Formula:**

$$
  \text{Position} = \left(\frac{85}{100}\right) \times (N + 1) = 0.85 \times 23 = 19.55
  $$

- **Interpolation:**
  
  - **19th Value:** 2033
  - **20th Value:** 2047
  
  $$
  \text{85th Percentile} = 2033 + 0.55 \times (2047 - 2033) = 2040.7 \approx 2041
  $$

---

### **Summary of Results**

- **Median:** 1781.5
- **First Quartile (Q₁):** 1564
- **Third Quartile (Q₃):** 1976
- **Semi-Interquartile Range:** 206
- **8th Decile (80th Percentile):** 2013
- **85th Percentile:** 2041
Median: 1781.5 First Quartile (Q₁): 1564 Third Quartile (Q₃): 1976 Semi-Interquartile Range: 206 8th Decile (80th Percentile): 2013 85th Percentile: 2041

8. Given the following data, (A) Find the median, quartiles and semi-interquartile range, (B) Find the 8th deciles and 85th percentiles. \( 1976,2162,1502,1782,1523,2033,1564,1320,1951 \), 1714,1841 \( 1648,1345,1837,1718,2047,1954,2000,1479,2571 \), 1739,1781

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