Question

ex: determine the sample variance and sample standard deviation {5,8,12,15} n = , 𝑥̅ =

Explanation:

Step1: Calculate the sample mean $\bar{x}$

The sample data is $\{5,8,12,15\}$, and $n = 4$. The formula for the sample mean is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. So, $\bar{x}=\frac{5 + 8+12 + 15}{4}=\frac{40}{4}=10$.

Step2: Calculate $(x-\bar{x})$ and $(x - \bar{x})^2$ for each data - point

For $x = 5$: $x-\bar{x}=5 - 10=-5$, $(x - \bar{x})^2=(-5)^2 = 25$.
For $x = 8$: $x-\bar{x}=8 - 10=-2$, $(x - \bar{x})^2=(-2)^2 = 4$.
For $x = 12$: $x-\bar{x}=12 - 10 = 2$, $(x - \bar{x})^2=2^2 = 4$.
For $x = 15$: $x-\bar{x}=15 - 10 = 5$, $(x - \bar{x})^2=5^2 = 25$.

Step3: Calculate the sum of $(x - \bar{x})^2$

$\sum_{i = 1}^{n}(x_{i}-\bar{x})^2=25 + 4+4 + 25=58$.

Step4: Calculate the sample variance $s^{2}$

The formula for the sample variance is $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2}{n - 1}$. Here, $n=4$, so $s^{2}=\frac{58}{4 - 1}=\frac{58}{3}\approx19.33$.

Step5: Calculate the sample standard deviation $s$

The sample standard deviation $s=\sqrt{s^{2}}=\sqrt{\frac{58}{3}}\approx4.39$.

Answer:

The sample variance $s^{2}=\frac{58}{3}\approx19.33$, and the sample standard deviation $s=\sqrt{\frac{58}{3}}\approx4.39$.