Question

compute the standard deviation for the set of data.
2, 5, 6, 8, 14
a. 16
b. 4\t\t\tc. 80\td. 2

please select the best answer from the choices provided
○ a
○ b
○ c
○ d

Explanation:

Step1: Calculate the mean

The mean $\bar{x}$ of a data set $x_1, x_2, \dots, x_n$ is given by $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. For the data set $2,5,6,8,14$, $n = 5$.
$\sum_{i=1}^{5}x_i=2 + 5+6 + 8+14=35$
$\bar{x}=\frac{35}{5}=7$

Step2: Calculate the squared differences from the mean

For each data point $x_i$, calculate $(x_i-\bar{x})^2$:

• For $x_1 = 2$: $(2 - 7)^2=(-5)^2 = 25$
• For $x_2 = 5$: $(5 - 7)^2=(-2)^2 = 4$
• For $x_3 = 6$: $(6 - 7)^2=(-1)^2 = 1$
• For $x_4 = 8$: $(8 - 7)^2=(1)^2 = 1$
• For $x_5 = 14$: $(14 - 7)^2=(7)^2 = 49$

Step3: Calculate the variance

The variance $s^2$ is given by $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}$ (using sample variance formula, since it's a set of data, not the entire population).
$\sum_{i = 1}^{5}(x_i-\bar{x})^2=25 + 4+1 + 1+49 = 80$
$n-1=5 - 1 = 4$
$s^2=\frac{80}{4}=20$ (Wait, but maybe the question is using population standard deviation? Let's check with population variance: $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}=\frac{80}{5} = 16$. Then standard deviation is $\sqrt{16}=4$)

Step4: Calculate the standard deviation

If we use population standard deviation (since the data might be considered as the entire population here), standard deviation $\sigma=\sqrt{s^2}$, where $s^2$ is population variance.
We found population variance $s^2 = 16$, so $\sigma=\sqrt{16}=4$

Answer:

B. 4