Question

1. for the following population of $n = 8$ scores: 1, 3, 1, 10, 1, 0, 1, 3

a. calculate $ss$, $sigma^{2}$, and $sigma$.
b. which formula should be used to calculate $ss$? explain.

Explanation:

Step1: Calculate the mean ($\mu$)

The scores are $1,3,1,10,1,0,1,3$. The sum of the scores $\sum X=1 + 3+1+10+1+0+1+3=20$. Since $N = 8$, $\mu=\frac{\sum X}{N}=\frac{20}{8}=2.5$.

Step2: Calculate $SS$ (Sum - of - Squares)

The formula for $SS=\sum(X - \mu)^2$.
$(1 - 2.5)^2=(-1.5)^2 = 2.25$
$(3 - 2.5)^2=(0.5)^2 = 0.25$
$(1 - 2.5)^2=(-1.5)^2 = 2.25$
$(10 - 2.5)^2=(7.5)^2 = 56.25$
$(1 - 2.5)^2=(-1.5)^2 = 2.25$
$(0 - 2.5)^2=(-2.5)^2 = 6.25$
$(1 - 2.5)^2=(-1.5)^2 = 2.25$
$(3 - 2.5)^2=(0.5)^2 = 0.25$
$SS=2.25+0.25+2.25+56.25+2.25+6.25+2.25+0.25 = 72$

Step3: Calculate $\sigma^{2}$ (Variance)

The formula for variance $\sigma^{2}=\frac{SS}{N}$. Since $SS = 72$ and $N = 8$, $\sigma^{2}=\frac{72}{8}=9$.

Step4: Calculate $\sigma$ (Standard Deviation)

The formula for standard deviation $\sigma=\sqrt{\sigma^{2}}$. Since $\sigma^{2}=9$, $\sigma=\sqrt{9}=3$.

For part b:
The definitional formula $SS=\sum(X - \mu)^2$ was used. This formula is appropriate when we know the population mean ($\mu$) and the individual scores ($X$). It directly measures the squared - deviation of each score from the mean and sums them up to get the total sum of squares.

Answer:

a. $SS = 72$, $\sigma^{2}=9$, $\sigma = 3$
b. The definitional formula $SS=\sum(X - \mu)^2$ should be used. It is appropriate when the population mean and individual scores are known as it directly calculates the sum of the squared - deviations of scores from the mean.