Question

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? (lesson 1 - 9) 0, 40, 60, 70, 75, 80, 85, 95, 95, 100

Explanation:

Step1: Calculate original mean

The original data - set is \(0,40,60,70,75,80,85,95,95,100\). The sum of the data - set \(S_1=0 + 40+60 + 70+75+80+85+95+95+100 = 700\), and the number of data points \(n_1 = 10\). The original mean \(\bar{x}_1=\frac{S_1}{n_1}=\frac{700}{10}=70\).

Step2: Calculate original median

Since \(n_1 = 10\) (an even - numbered data - set), the median is the average of the \(\frac{n_1}{2}\)th and \((\frac{n_1}{2}+1)\)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 \(M_1=\frac{75 + 80}{2}=77.5\).

Step3: Calculate new mean after removing 0

The new data - set is \(40,60,70,75,80,85,95,95,100\). The sum of the new data - set \(S_2=40+60 + 70+75+80+85+95+95+100 = 700\), and the number of data points \(n_2 = 9\). The new mean \(\bar{x}_2=\frac{S_2}{n_2}=\frac{700}{9}\approx77.78\).

Step4: Calculate new median after removing 0

Since \(n_2 = 9\) (an odd - numbered data - set), the median is the \(\frac{n_2 + 1}{2}\)th ordered value. \(\frac{9+1}{2}=5\)th value, and the new median \(M_2 = 80\).

Step5: Analyze the changes

The mean increases from \(70\) to approximately \(77.78\) because the sum of the data remains the same while the number of data points decreases, and the removed value was a very low outlier. The median increases from \(77.5\) to \(80\) because the middle value of the ordered data - set shifts to a higher value when the lowest value is removed.

Answer:

The mean increases and the median increases.