Question

a statistics student gave a survey to students which asked how many first cousins they have. the data from the first nine responses are shown below. 6, 2, 7, 8, 8, 8, 5, 10, 10 what is the standard deviation of the data? 2.38 first cousins 2.52 first cousins 5.65 first cousins 6.36 first cousins

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data set $x_1,x_2,\cdots,x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$.
Here, $n = 9$, and $\sum_{i=1}^{9}x_i=6 + 2+7+8+8+8+5+10+10=64$.
So, $\bar{x}=\frac{64}{9}\approx7.11$.

Step2: Calculate the squared - differences

The formula for the squared - difference $(x_i-\bar{x})^2$:
For $x_1 = 6$: $(6 - 7.11)^2=(-1.11)^2 = 1.2321$
For $x_2 = 2$: $(2 - 7.11)^2=(-5.11)^2=26.1121$
For $x_3 = 7$: $(7 - 7.11)^2=(-0.11)^2 = 0.0121$
For $x_4 = 8$: $(8 - 7.11)^2=(0.89)^2 = 0.7921$
For $x_5 = 8$: $(8 - 7.11)^2=(0.89)^2 = 0.7921$
For $x_6 = 8$: $(8 - 7.11)^2=(0.89)^2 = 0.7921$
For $x_7 = 5$: $(5 - 7.11)^2=(-2.11)^2 = 4.4521$
For $x_8 = 10$: $(10 - 7.11)^2=(2.89)^2 = 8.3521$
For $x_9 = 10$: $(10 - 7.11)^2=(2.89)^2 = 8.3521$
The sum of squared - differences $\sum_{i = 1}^{n}(x_i-\bar{x})^2=1.2321+26.1121 + 0.0121+0.7921+0.7921+0.7921+4.4521+8.3521+8.3521=50.8909$

Step3: Calculate the variance

The variance $s^2$ (for a sample) is given by $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$.
Since $n = 9$, $s^2=\frac{50.8909}{9-1}=\frac{50.8909}{8}=6.3613625$

Step4: Calculate the standard deviation

The standard deviation $s=\sqrt{s^2}$.
So, $s=\sqrt{6.3613625}\approx2.52$

Answer:

2.52 first cousins