Question

practice 4 (from unit 1, lesson 9)
a student has these scores on ten assignments.
0 40 60 70 75 80 85 95 95 100
the teacher is considering dropping a lowest score.
what effect does eliminating the lowest value, 0, from the data set have on the mean and median?

Explanation:

Step1: Calculate the original mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. The original data set is $0,40,60,70,75,80,85,95,95,100$ with $n = 10$. $\sum_{i=1}^{10}x_{i}=0 + 40+60 + 70+75+80+85+95+95+100=700$. So the original mean $\bar{x}_1=\frac{700}{10}=70$.

Step2: Calculate the new mean

After dropping the lowest - value $0$, the new data set is $40,60,70,75,80,85,95,95,100$ with $n = 9$. $\sum_{i = 1}^{9}x_{i}=40+60+70+75+80+85+95+95+100=690$. So the new mean $\bar{x}_2=\frac{690}{9}\approx76.67$. The mean increases.

Step3: Calculate the original median

For a data set with $n = 10$ (an even number of data points), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered data points. The $\frac{10}{2}=5$th and $6$th ordered data points are $75$ and $80$. So the original median $M_1=\frac{75 + 80}{2}=77.5$.

Step4: Calculate the new median

For a data set with $n = 9$ (an odd number of data points), the median is the $(\frac{n + 1}{2})$th ordered data point. $\frac{9+1}{2}=5$th ordered data point. The new data set in ascending order is $40,60,70,75,80,85,95,95,100$, and the $5$th value is $80$. The median increases.

Answer:

Both the mean and the median increase.