Question

what effect does eliminating the highest value, 180, from the data set have on the mean and median?
25 50 50 60 70 85 85 90 90 180

Explanation:

Step1: Calculate the original mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here $n = 10$, and $\sum_{i=1}^{10}x_{i}=25 + 50+50 + 60+70+85+85+90+90+180=785$. So the original mean $\bar{x}_1=\frac{785}{10}=78.5$.

Step2: Calculate the original median

Since $n = 10$ (an even - numbered data set), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values. The ordered data set is $25,50,50,60,70,85,85,90,90,180$. The $\frac{n}{2}=5$th value is $70$ and the $(\frac{n}{2}+1)=6$th value is $85$. So the original median $M_1=\frac{70 + 85}{2}=77.5$.

Step3: Calculate the new mean after eliminating 180

Now $n = 9$, and $\sum_{i = 1}^{9}x_{i}=785-180 = 605$. So the new mean $\bar{x}_2=\frac{605}{9}\approx67.22$.

Step4: Calculate the new median after eliminating 180

The new ordered data set is $25,50,50,60,70,85,85,90,90$. Since $n = 9$ (an odd - numbered data set), the median is the $(\frac{n + 1}{2})$th value. $\frac{n+1}{2}=5$th value, so the new median $M_2 = 70$.

Step5: Analyze the changes

The mean changes from $78.5$ to approximately $67.22$, so the mean decreases. The median changes from $77.5$ to $70$, so the median decreases.

Answer:

The mean and median both decrease.