8. Given the following data, (A) Find the median, quartiles and semi-interquartile range, (B) Find the 8th deciles and 85 th 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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Certainly! Let's analyze the given dataset step-by-step to find the required statistical measures.

## **Given Data**

The combined dataset consists of 22 observations:

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

## **Step 1: Sort the Data in Ascending Order**

First, arrange the data from smallest to largest:

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

Number of observations (**n**) = 22

---

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

### **1. Median**

For an even number of observations, the median is the average of the two middle numbers.

- **Position of Median**: $$(n/2)^{th}$$ and $$(n/2 + 1)^{th}$$ terms
- **Calculation**:
  - 11th value = 1781
  - 12th value = 1782
  - **Median = (1781 + 1782) / 2 = 1781.5**

### **2. Quartiles**

**Quartiles** divide the data into four equal parts.

- **First Quartile (Q₁)**: Median of the lower half (first 11 data points)
  - Lower Half: 1320, 1345, 1479, 1502, 1523, 1564, 1648, 1714, 1718, 1739, **1781**
  - **Q₁ = 1564** (6th data point)

- **Third Quartile (Q₃)**: Median of the upper half (last 11 data points)
  - Upper Half: 1782, 1837, 1841, 1951, 1954, **1976**, 2000, 2033, 2047, 2162, 2571
  - **Q₃ = 1976** (6th data point)

### **3. Semi-Interquartile Range**

The **Semi-Interquartile Range** measures the spread of the middle 50% of the data.

- **Formula**: $$(Q₃ - Q₁) / 2$$
- **Calculation**: $$(1976 - 1564) / 2 = 412 / 2 = 206$$

---

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

### **1. 8th Decile (80th Percentile) and 85th Percentile**

To find percentiles and deciles, we'll use the **Linear Interpolation Method**.

#### **a. Calculating Positions**

- **Formula**: $$i = \frac{P}{100} 	imes (n + 1)$$

##### **For the 80th Percentile (8th Decile):**

- $$i = 0.80 	imes 23 = 18.4$$
- **Interpretation**:
  - 18th data point = 2000
  - 19th data point = 2033
  - **80th Percentile = 2000 + 0.4 	imes (2033 - 2000) = 2000 + 13.2 = 2013.2**

##### **For the 85th Percentile:**

- $$i = 0.85 	imes 23 = 19.55$$
- **Interpretation**:
  - 19th data point = 2033
  - 20th data point = 2047
  - **85th Percentile = 2033 + 0.55 	imes (2047 - 2033) = 2033 + 7.7 = 2040.7**

---

## **Summary of Results**

### **A. Median, Quartiles, and Semi-Interquartile Range**

- **Median**: 1781.5
- **First Quartile (Q₁)**: 1564
- **Third Quartile (Q₃)**: 1976
- **Semi-Interquartile Range**: 206

### **B. 8th Decile and 85th Percentile**

- **8th Decile (80th Percentile)**: **2013.2**
- **85th Percentile**: **2040.7**

---

These calculations provide a comprehensive statistical overview of the given dataset.
**Median:** 1781.5 **First Quartile (Q₁):** 1564 **Third Quartile (Q₃):** 1976 **Semi-Interquartile Range:** 206 **8th Decile (80th Percentile):** 2013.2 **85th Percentile:** 2040.7

Given the following data, (A) Find the median,
quartiles and semi-interquartile range, (B) Find the 8th
deciles and 85 th percentiles.
,
1714,1841
,
1739,1781

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Given the following data, (A) Find the median, quartiles and semi-interquartile range, (B) Find the 8th deciles and 85 th percentiles. , 1714,1841 , 1739,1781