Question

find the mean, standard deviation, and five - number summary for the data set. assume population data are given: 8, 13, 19, 21, 22, 33, 35, 39, 46, 51. the mean is 28.7 (round to one decimal place as needed). the standard deviation is (round to one decimal place as needed).

Explanation:

Step1: Confirm population size

The data set has $n=10$ values: 8, 13, 19, 21, 22, 33, 35, 39, 46, 51.

Step2: Recall population std dev formula

Population standard deviation:
$$\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \mu)^2}{n}}$$
where $\mu=26.7$ (given mean).

Step3: Calculate squared deviations

Compute each $(x_i - \mu)^2$:

• $(8-26.7)^2=(-18.7)^2=349.69$
• $(13-26.7)^2=(-13.7)^2=187.69$
• $(19-26.7)^2=(-7.7)^2=59.29$
• $(21-26.7)^2=(-5.7)^2=32.49$
• $(22-26.7)^2=(-4.7)^2=22.09$
• $(33-26.7)^2=(6.3)^2=39.69$
• $(35-26.7)^2=(8.3)^2=68.89$
• $(39-26.7)^2=(12.3)^2=151.29$
• $(46-26.7)^2=(19.3)^2=372.49$
• $(51-26.7)^2=(24.3)^2=590.49$

Step4: Sum squared deviations

$$\sum(x_i - \mu)^2=349.69+187.69+59.29+32.49+22.09+39.69+68.89+151.29+372.49+590.49=1874.1$$

Step5: Compute variance and std dev

Variance: $\frac{1874.1}{10}=187.41$
Standard deviation: $\sqrt{187.41}\approx13.7$

Step6: Calculate five-number summary

• Minimum: 8
• First Quartile ($Q_1$): Median of lower half (8,13,19,21,22) = 19
• Median ($Q_2$): Average of 5th/6th values: $\frac{22+33}{2}=27.5$
• Third Quartile ($Q_3$): Median of upper half (33,35,39,46,51) = 39
• Maximum: 51

Answer:

The standard deviation is 13.7
Five-number summary: 8, 19, 27.5, 39, 51