Question

from unit 1, lesson 10
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: Recall median concept

Median is middle - value. For 30 data points (even number of data), median is average of 15th and 16th ordered values. Removing 12 (the maximum) doesn't change relative position of these middle values.

Step2: Recall mean concept

Mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Let sum of original 30 values be $S$. Original mean $\bar{x}_1=\frac{S}{30}$. New sum is $S - 12$ and new number of data is $n=29$. New mean $\bar{x}_2=\frac{S - 12}{29}$. Since $\frac{S}{30}
eq\frac{S - 12}{29}$, mean changes.

Answer:

a. The median is the middle - value. For 30 data points (an even number of data), it is the average of the 15th and 16th ordered values. Removing the maximum value (12) does not change the relative position of these middle - ranked values, so the median remains the same.
b. The mean does not remain the same. The mean is calculated as the sum of all values divided by the number of values. Let the sum of the original 30 values be $S$. The original mean is $\frac{S}{30}$. After removing 12, the new sum is $S - 12$ and the new number of values is 29. The new mean is $\frac{S - 12}{29}$. Since $\frac{S}{30}
eq\frac{S - 12}{29}$, the mean changes.