Question - Calculating Quartiles for a Data Set
Solution:
To find the lower quartile, median, and upper quartile for the given data set, we follow these steps:1. Median: This is the middle value when the data set is ordered from smallest to largest. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.The given data set is already ordered:1, 2, 2, 2, 3, 3, 6, 8, 8There are 9 values, so the middle one will be the 5th value (since (9 + 1) / 2 = 5):1, 2, 2, 2, 3, 3, 6, 8, 8Hence, the median is 3.2. Lower Quartile (Q1): This is the median of the lower half of the data (excluding the median if the number of values is odd).For our data set, the lower half (before the median) is:1, 2, 2, 2In this set of 4 values, the lower quartile is the average of the 2nd and 3rd value (because (4 + 1) / 2 = 2.5, which means we take the average of 2nd and 3rd):(2 + 2) / 2 = 4 / 2 = 2Hence, the lower quartile is 2.3. Upper Quartile (Q3): This is the median of the upper half of the data (excluding the median if the number of values is odd).For our data set, the upper half (after the median) is:3, 6, 8, 8In this set of 4 values, the upper quartile is the average of the 6 and 8:(6 + 8) / 2 = 14 / 2 = 7Hence, the upper quartile is 7.In summary:- Lower Quartile (Q1) = 2- Median = 3- Upper Quartile (Q3) = 7
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