Calculating the Standard Deviation by the Raw Score Method

We can calculate the standard deviation for a population of scores using the raw score method by using the following formula:

We can calculate the standard deviation for a sample of scores using the raw score method by using the following formula:

The two formulas only differ by the term in the denominator. With the population the denominator consists of N, while for the sample we use n-1 in the denominator.

Lets use the scores in the following worksheet to find the standard deviation for a population and for a sample by the raw score method.

Worksheet for Calculating the Standard Deviation for 5 scores

2	4
4	16
6	36
8	64
10	100
---	---
30	220

At the bottom of the first column is the sum of the scores (30) and at the bottom of the second column is the sum of the squared scores (220). We need these quantities and the number of scores (5) to calculate the standard deviation.

First lets calculate the standard deviation for a population.

Now lets use the same data to calculate the standard deviation for a sample.

So for our five scores the standard deviation considering the scores as a population is 2.828 while the standard deviation for the same five scores if we consider them a sample is 3.162

We can also use the computer and a spreadsheet program to calculate the standard deviation and the variance for a population and for a sample. For our next topic in Lesson 6 let's look at how to do this.