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

To find the median, quartiles, and semi-interquartile range, we first need to organize the data in ascending order: \( 1320, 1345, 1479, 1502, 1523, 1564, 1600, 1714, 1718, 1739, 1781, 1782, 1837, 1841, 1951, 1954, 1976, 2000, 2033, 2047, 2162, 2571 \) A) **Median**: The median (middle value) is the average of the 10th and 11th values in this 21-number dataset, which are 1718 and 1739. Median = \( (1718 + 1739) / 2 = 1728.5 \) **Quartiles**: - **Q1** (1st quartile) is the median of the first half (first 10 values): \( (1479 + 1502) / 2 = 1490.5 \) - **Q3** (3rd quartile) is the median of the second half (the remaining 11 values): \( (2000 + 2033) / 2 = 2016.5 \) **Semi-interquartile range (SIQR)**: SIQR = \( Q3 - Q1 / 2 = (2016.5 - 1490.5) / 2 = 263 \) B) To find the 8th decile (D8) and the 85th percentile (P85), we can use the following method: **D8**: D8 is the value at the 80% position in the ordered data. Using \( N = 21 \), the position is \( 0.8 \times 21 = 16.8 \), which means D8 is slightly above the 16th value, which is \( 1954 \). So it can be approximated as slightly higher than this. **P85**: P85 is the value at the 85% position, calculated as \( 0.85 \times 21 = 17.85 \), putting us between the 17th and 18th values, \( 2000 \) and \( 2033 \). We can estimate it as a weighted average of those values. By following these steps, you'll keep your statistics sharper than a freshly sharpened pencil!