Question

chapter 3.2 homework
score: 12/102 answered: 8/33
question 9
3.2 measures of spread: standard deviation and variance
for the sample data shown, answer the questions. round to 2 decimal
x
5.1
5.3
6
6.4
10.1
17.3
18.1
21.9
24.2
25.9
find the mean:
find the median:
find the sample standard deviation:
question help: message instructor post to forum

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a sample $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here $n = 10$, and $\sum_{i=1}^{10}x_i=5.1 + 5.3+6+6.4+10.1+17.3+18.1+21.9+24.2+25.9 = 140.3$. So $\bar{x}=\frac{140.3}{10}=14.03$.

Step2: Calculate the median

Arrange the data in ascending - order: $5.1,5.3,6,6.4,10.1,17.3,18.1,21.9,24.2,25.9$. Since $n = 10$ (an even number), the median is the average of the $\frac{n}{2}$ - th and $(\frac{n}{2}+1)$ - th ordered values. The 5 - th value is $10.1$ and the 6 - th value is $17.3$. So the median $M=\frac{10.1 + 17.3}{2}=13.70$.

Step3: Calculate the sample standard deviation

The formula for the sample standard deviation $s$ is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}$.
First, calculate $(x_1-\bar{x})^2,(x_2-\bar{x})^2,\cdots,(x_{10}-\bar{x})^2$:
$(5.1 - 14.03)^2=(-8.93)^2 = 79.7449$;
$(5.3 - 14.03)^2=(-8.73)^2 = 76.2129$;
$(6 - 14.03)^2=(-8.03)^2 = 64.4809$;
$(6.4 - 14.03)^2=(-7.63)^2 = 58.2169$;
$(10.1 - 14.03)^2=(-3.93)^2 = 15.4449$;
$(17.3 - 14.03)^2=(3.27)^2 = 10.6929$;
$(18.1 - 14.03)^2=(4.07)^2 = 16.5649$;
$(21.9 - 14.03)^2=(7.87)^2 = 61.9369$;
$(24.2 - 14.03)^2=(10.17)^2 = 103.4289$;
$(25.9 - 14.03)^2=(11.87)^2 = 141.0069$.
$\sum_{i = 1}^{10}(x_i-\bar{x})^2=79.7449+76.2129+64.4809+58.2169+15.4449+10.6929+16.5649+61.9369+103.4289+141.0069 = 627.731$.
Then $s=\sqrt{\frac{627.731}{9}}\approx\sqrt{69.7479}\approx8.35$.

Answer:

Mean: $14.03$
Median: $13.70$
Sample standard deviation: $8.35$