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

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**: \(i = \frac{P}{100} \times (n + 1)\) ##### **For the 80th Percentile (8th Decile):** - \(i = 0.80 \times 23 = 18.4\) - **Interpretation**: - 18th data point = 2000 - 19th data point = 2033 - **80th Percentile = 2000 + 0.4 \times (2033 - 2000) = 2000 + 13.2 = 2013.2** ##### **For the 85th Percentile:** - \(i = 0.85 \times 23 = 19.55\) - **Interpretation**: - 19th data point = 2033 - 20th data point = 2047 - **85th Percentile = 2033 + 0.55 \times (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.