Question

1. a student has these scores on their assignments. 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? 0, 40, 60, 70, 75, 80, 85, 95, 95, 100

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 $\sum_{i=1}^{10}x_{i}=0 + 40+60 + 70+75+80+85+95+95+100=690$. So the original mean $\bar{x}_{1}=\frac{690}{10}=69$.

Step2: Calculate the new mean

After removing $0$, $n = 9$, and $\sum_{i = 1}^{9}x_{i}=690 - 0=690$. The new mean $\bar{x}_{2}=\frac{690}{9}\approx76.67$.

Step3: Calculate the original median

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

Step4: Calculate the new median

After removing $0$, $n = 9$ (odd number of data points). The median is the $(\frac{n + 1}{2})$-th ordered value. $\frac{9+1}{2}=5$-th value. The ordered data set (after removing $0$) is $40,60,70,75,80,85,95,95,100$. The $5$-th value is $80$. So the new median $M_{2}=80$.

Answer:

The mean increases from $69$ to approximately $76.67$ and the median increases from $77.5$ to $80$.