Question

1. the data represent the number of cans collected by different classes for a service project. (lesson 1-9)

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 the mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
Here, $n = 12$ and $\sum_{i=1}^{12}x_{i}=12 + 14+22 + 14+18+23+42+13+9+19+22+14=222$.
So, $\bar{x}=\frac{222}{12}=18.5$.

Step2: Find the median

First, order the data set: $9,12,13,14,14,14,18,19,22,22,23,42$.
Since $n = 12$ (even), the median $M=\frac{x_{\frac{n}{2}}+x_{\frac{n}{2}+1}}{2}$.
$x_{6}=14$ and $x_{7}=18$.
So, $M=\frac{14 + 18}{2}=16$.

Step3: Analyze the effect of eliminating the greatest value

The new data set (after eliminating $42$) has $n = 11$ and $\sum_{i = 1}^{11}x_{i}=222-42 = 180$.
The new mean $\bar{x}_{new}=\frac{180}{11}\approx16.36$.
The new ordered data set: $9,12,13,14,14,14,18,19,22,22,23$.
The new median (since $n = 11$, odd) is $x_{\frac{n + 1}{2}}=x_{6}=14$.

Answer:

a. The mean is $18.5$.
b. The median is $16$.
c. When eliminating the greatest value ($42$):

• The mean changes from $18.5$ to approximately $16.36$ (the mean is affected by extreme values).
• The median changes from $16$ to $14$. The median is less affected by a single - extreme value compared to the mean.