Question

find a. the mean. b. the deviation from the mean for each data item, and c. the sum of the deviations in part (b) for the following group of data items 159, 161, 166, 170, 174

a. the mean is (simplify your answer)

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a data set $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$.
Here, $n = 5$, $x_1=159$, $x_2 = 161$, $x_3=166$, $x_4=170$, $x_5=174$.
$\sum_{i=1}^{5}x_i=159 + 161+166+170+174=830$.
$\bar{x}=\frac{830}{5}=166$.

Step2: Calculate the deviation from the mean for each data item

The deviation of a data item $x_i$ from the mean $\bar{x}$ is $x_i-\bar{x}$.
For $x_1 = 159$: $159 - 166=-7$.
For $x_2 = 161$: $161-166=-5$.
For $x_3 = 166$: $166 - 166=0$.
For $x_4 = 170$: $170-166 = 4$.
For $x_5 = 174$: $174-166=8$.

Step3: Calculate the sum of the deviations

Sum of deviations $=(-7)+(-5)+0 + 4+8$.
$(-7-5)+(4 + 8)=-12 + 12=0$.

Answer:

a. The mean is $166$.
b. The deviations are $-7,-5,0,4,8$.
c. The sum of the deviations is $0$.