Question

if a new data point at 12 is added to the graph, which will be true? the mean will increase, and the median will stay the same. the median will increase, and the mean will stay the same. the mean will increase more than the median, but both will increase. the median will increase more than the mean, but both will increase.

Explanation:

Step1: Analyze original data

Count data points: At 1:1, 2:3, 3:1, 4:3, 5:1. Total points: \(1 + 3 + 1 + 3 + 1 = 9\). Median (5th term) is 3 (sorted: 1,2,2,2,3,4,4,4,5). Mean: \(\frac{1\times1 + 2\times3 + 3\times1 + 4\times3 + 5\times1}{9}=\frac{1 + 6 + 3 + 12 + 5}{9}=\frac{27}{9}=3\).

Step2: Add data point 12

New total points: \(9 + 1 = 10\). Sorted data: 1,2,2,2,3,4,4,4,5,12. Median (average of 5th and 6th terms): \(\frac{3 + 4}{2}=3.5\) (so median increases from 3 to 3.5). Mean: \(\frac{27 + 12}{10}=\frac{39}{10}=3.9\) (mean increases from 3 to 3.9). Mean increase: \(3.9 - 3 = 0.9\), median increase: \(3.5 - 3 = 0.5\). So mean increases more than median, both increase.

Answer:

The mean will increase more than the median, but both will increase.