Question

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

Step2: Calculate the new mean

After dropping the lowest score (\(0\)), the new data - set is \(40,60,70,75,80,85,95,95,100\). Here \(n = 9\) and \(\sum_{i = 1}^{9}x_{i}=40+60 + 70+75+80+85+95+95+100=700\). So the new mean \(\bar{x}_{2}=\frac{700}{9}\approx77.78\). The change in the mean is \(\bar{x}_{2}-\bar{x}_{1}=77.78 - 70 = 7.78\).

Step3: Calculate the original median

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

Step4: Calculate the new median

For \(n = 9\) (an odd - numbered data - set), the median is the \(\frac{n + 1}{2}\)th ordered value. \(\frac{n+1}{2}=5\). The 5th value in the new ordered data - set \(40,60,70,75,80,85,95,95,100\) is \(80\). The change in the median is \(M_{2}-M_{1}=80 - 77.5 = 2.5\).

The mean increases by approximately \(7.78\) and the median increases by \(2.5\).

Answer:

The mean increases and the median increases. The mean increases by approximately \(7.78\) and the median increases by \(2.5\).