Question

the data represent the number of cans collected by different classes for a service project. 12 14 22 14 18 23 42 13 9 19 22 14. a. find the mean. b. find the median. c. eliminate the greatest value, 42, from the data set. explain how the measures of center change.

Explanation:

Step1: Calculate sum of data

$12 + 14+22 + 14+18+23+42+13+9+19+22+14=222$

Step2: Calculate number of data points

There are 12 data - points.

Step3: Calculate the mean

The mean $\bar{x}=\frac{222}{12}=18.5$

Step4: Arrange data in ascending order

$9, 12, 13, 14, 14, 14, 18, 19, 22, 22, 23, 42$

Step5: Calculate the median

Since $n = 12$ (even), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values. $\frac{14 + 18}{2}=16$

Step6: Remove 42 from data set

New data set: $9, 12, 13, 14, 14, 14, 18, 19, 22, 22, 23$

Step7: Calculate new sum

$9+12+13+14+14+14+18+19+22+22+23 = 170$

Step8: Calculate new number of data points

There are 11 data - points.

Step9: Calculate new mean

New mean $\bar{x}=\frac{170}{11}\approx15.45$

Step10: Calculate new median

Since $n = 11$ (odd), the median is the $\frac{n + 1}{2}=6$th ordered value, which is 14.

The mean decreased from 18.5 to approximately 15.45 because the large value of 42 was removed, reducing the total sum while the number of data points decreased by 1. The median decreased from 16 to 14 because the removal of 42 changed the position of the middle - value in the ordered data set.

Answer:

a. Mean: 18.5
b. Median: 16
c. Mean decreased from 18.5 to approximately 15.45, median decreased from 16 to 14.