Question

1. for the following set of scores: 6, 2, 3, 0, 4

a. if the scores are a population, what are the variance and standard deviation?
b. if the scores are a sample, what are the variance and standard deviation?

Explanation:

Step1: Calculate the mean for population

The population mean $\mu$ for the scores $x = \{6,2,3,0,4\}$ is $\mu=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 5$ and $\sum_{i=1}^{5}x_{i}=6 + 2+3+0 + 4=15$. So, $\mu=\frac{15}{5}=3$.

Step2: Calculate the population variance

The population variance $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}$.
$(6 - 3)^{2}=9$, $(2 - 3)^{2}=1$, $(3 - 3)^{2}=0$, $(0 - 3)^{2}=9$, $(4 - 3)^{2}=1$.
$\sum_{i = 1}^{5}(x_{i}-\mu)^{2}=9 + 1+0+9+1 = 20$.
So, $\sigma^{2}=\frac{20}{5}=4$.

Step3: Calculate the population standard - deviation

The population standard deviation $\sigma=\sqrt{\sigma^{2}}=\sqrt{4}=2$.

Step4: Calculate the sample variance

The sample variance $s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}$ (where $\bar{x}$ is the sample mean, which is the same as the population mean $\mu$ in this case since the data set is the same).
We already know $\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=20$ and $n = 5$.
So, $s^{2}=\frac{20}{4}=5$.

Step5: Calculate the sample standard - deviation

The sample standard deviation $s=\sqrt{s^{2}}=\sqrt{5}\approx2.24$.

Answer:

a. Variance: 4, Standard deviation: 2
b. Variance: 5, Standard deviation: $\sqrt{5}\approx2.24$