Question

1. the box plot represents the distribution of the number of children in 30 different families. after further examination, the value of 12 is removed for having been recorded in error. the box plot represents the distribution of the same data set, but with the maximum, 12, removed. the median is 2 children for both plots. a. explain why the median remains the same when 12 was removed from the data set. b. when 12 is removed from the data set, does mean remain the same? explain your reasoning.

Explanation:

Step1: Understanding the median

The median is the middle value when the data is ordered. For \(n = 30\) data points (original data set), the median is the average of the \(15^{th}\) and \(16^{th}\) ordered values. When \(n=29\) (after removing 12), the median is the \(15^{th}\) ordered value. Since 12 is the maximum value, removing it does not affect the position of the middle - most values (the \(15^{th}\) and \(16^{th}\) values in the original ordered data and the \(15^{th}\) value in the new ordered data are the same).

Step2: Understanding the mean

The mean is calculated as \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). Let the sum of the original 30 data points be \(S=\sum_{i = 1}^{30}x_{i}\), and the mean of the original data set is \(\bar{x}_{1}=\frac{S}{30}\). After removing 12 (so the new sum is \(S - 12\) and \(n = 29\)), the new mean is \(\bar{x}_{2}=\frac{S - 12}{29}\). Since \(\frac{S}{30}
eq\frac{S - 12}{29}\) (cross - multiply: \(29S
eq30(S - 12)\), \(29S
eq30S-360\), \(S
eq360\) in general).

Answer:

a. The median is based on the position of values. Since 12 is the maximum value, removing it does not change the position of the middle - most values (the \(15^{th}\) and \(16^{th}\) values in the original ordered data and the \(15^{th}\) value in the new ordered data are the same).
b. No. The mean is \(\bar{x}=\frac{\sum_{i=1}^{n}x_{i}}{n}\). The original mean is \(\frac{\sum_{i = 1}^{30}x_{i}}{30}\), and the new mean is \(\frac{\sum_{i = 1}^{30}x_{i}-12}{29}\). Since \(\frac{\sum_{i = 1}^{30}x_{i}}{30}
eq\frac{\sum_{i = 1}^{30}x_{i}-12}{29}\) (by the formula for the mean and the change in the sum and the number of data points), the mean changes.