How to calculate the standard deviation

Now we need to calculate what is in the brackets () the participant's raw score minus the mean

Column A Column B - = 5 - 6 -1 6 - 6 0 9 - 6 3 2 - 6 -4 8 - 6 2 NB: We do NOT total this column

We now need to square the scores in Column B

Column B Column C (The data squared) -12 = 1 02 = 0 32 = 9 -42 = 16 22 = 4

Now you need to add up the data in column C to give you the sum (∑)

Column C (The data squared) 1 0 9 16 4 Total 30

Then find the square root:

S = 2.75

Now you need to divide 30 by 4

(4 = the total number of scores minus one) = 7.5 (this gives you the variance in the scores)

Imagine this:

You have just done an ethics test. The score was out of 12. Your raw scores look like this (this is the actual score each participant got on their ethics test) Participants (pps) Raw scores Pp1 5 Pp2 6 Pp3 9 Pp4 2 Pp5 8

Now use the formula and tables provided to calculate the standard deviation on your data for the two groups. (Remember, you need to calculate each group separately).

Pps group Pp1 15 15 - 34.2 -19.2 368.64 Pp2 18 18 - 34.2 -16.2 262.44 Pp3 12 12 - 34.2 -22.2 492.84 Pp4 58 58- 34.2 23.8 566.44 Pp5 49 49 - 34.2 14.8 219.04 Pp6 53 53 - 34.2 18.8 353.44 Total 205 /6 = 2,262.84 Total Mean = 34.2

Square root:

= square root of 452.57 S = 21.27

What is a normal distribution? (Figure 2.)

A negative skew means the majority of participants score highly on a test (there were not enough challenging questions). A positive skew would mean the majority of participants got a low score (the test was too hard)

Practice: Imagine you have tested two groups of participants (based on two different classes) on a test of the understanding of the Behaviourist approach which was scored out of a possible total of 60. You found that the following scores:

Group 1 Group 2 Pp1 15 Pp1 33 Pp2 18 Pp2 32 Pp3 12 Pp3 42 Pp4 58 Pp4 28 Pp5 49 Pp5 30 Pp6 53 Pp6 40

What is a normal distribution? (Figure 1.)

It is often called a 'Bell curve' because of the shape it makes. It looks at how certain traits, such as IQ are distributed amongst a population. In a normal population, the bell curve would look like the image above with the majority of the population having an IQ falls within one standard deviation on either side of the mean (the mean in figure 1. is represented by '0'). So if an average IQ is 100, and the standard deviation is 15, then 68% of the population would have an IQ that ranged between 85 and 115 (one SD either side of the mean - 100 plus or minus 15). 99% of the population would fall within three standard deviations. That is between 55 and 145. 1% of the population would fall outside of this. So therefore only a tiny percentage of people would be considered to have genius IQ of 160 or above. However, if you had a population made up of mainly members of Mensa, your bell curve would probably look like figure 2.

What is a standard deviation?

The standard deviation provides some idea about the distribution of scores around the mean (average). The smaller the standard deviation, the more narrow the range between the lowest and highest scores or, more generally, that the scores cluster closely to the average score. The larger the population we have, the more likely we are to have a normal distribution. Assuming that we have a normal distribution, 68 % of scores should be within one standard deviation (sd) each side of the mean. 95% should be within two sd of the mean. 99% should be within three sd of each side of the mean. Therefore if our mean is 6, 68% of participants should have score between 3.25 (6 - 2.75) and 8.75 (6 + 2.75). Consider participant three who scored 9 on the ethics test. Did they do well compared with the other participants in the test? The answer, of course would be yes! As they scored just above one standard deviation, they performed better than 68% of the other participants.

First of all, to calculate the standard deviation of these scores you would need to calculate the mean of those scores

The sum of the raw scores divided by the total number of scores For example: 5 + 6 + 9 + 2 + 8 = 30 30 / 5 = 6 So our mean score is 6