4.6 – Calculating Standard Deviation

Standard Deviation

Now that we have calculated the variance, calculating the standard deviation is a very simple step. We simply take the square root of the appropriate variance. As with variance, there is a population formula and a sample formula:

Let’s continue using the same set of scores with which we have been working:

[latex]\text{Sum of Squares (Computational Formula)}=SS=\Sigma X^2-\frac{(\Sigma X)^2}{N}[/latex]

[latex]=139 - \frac{(25)^2}{5} = 139 - \frac{625}{5} = 139 - 125 = 14[/latex]

Calculating Population Standard Deviation (σ)

If we were told that this set of scores was a population, we would then calculate population variance (σ²):

 [latex]\text{Population Variance}=\sigma^2=\frac{SS}{N}=\frac{14}{5}=2.8[/latex]

And then we would calculate the population standard deviation (σ):

 [latex]\text{Population Standard Deviation}=\sigma=\sqrt{\sigma^2}=\sqrt{2.8}=1.673[/latex]

This result tells us that the average deviation in this distribution of population scores is σ = 1.673. In other words, the scores in the distribution deviate from the mean by about 1.673 points on average.

Calculating Sampe Standard Deviation (s)

If we were told that this set of scores was a population, we would then calculate population variance (s²):

[latex]\text{Sample Variance}=s^2=\frac{SS}{n-1}=\frac{14}{5-1}=\frac{14}{4}=3.5[/latex]

And then we would calculate the sample standard deviation (s):

 [latex]\text{Population Standard Deviation}=\sigma=\sqrt{\sigma^2}=\sqrt{2.8}=1.673[/latex]

This result tells us that the average deviation in this distribution of sample scores is s = 1.673. In other words, the scores in the distribution deviate from the mean by about 1.673 points on average.

As you can see, the process and formulas between populations and sample are mostly the same (in fact, the Sum of Squares calculation is exactly the same. Just be sure to use the n – 1 in the denominator of the sample variance.