More on Standard Deviation

The idea of SD is that you want to know how much variation there is in your data around the mean. The smaller the SD, the more closely the data are concentrated in the distribution; the larger the SD, the more widely or loosely distributed are the data in the distribution. In lecture, I referred to the SD as the “average freak.” Freaks are abnormal and by definition are not average and do not conform to the mean. So the standard deviation tells us how far from the mean value the “average freak” is.

Here’s an example. One distribution has a SD of 6 and a different distribution has a SD of 20. A SD of 6 is much smaller than an SD of 20. That means that the data in the first distribution are much closely grouped than the data in the second distribution.

Remember that the mean is simply the average score. For example, the average of 20 and 0 is 10 (twenty plus zero divided by two equals ten). The average of 11 and 9 is also 10 (eleven plus nine equals twenty divided by two equals 10).  But in the second case our data points range from 9 to 11 and are much more closely concentrated around the average (10) then in the first case where our range from 0 to 20. Therefore, the SD in the second case would be a much smaller number because the values are more closely concentrated around the mean.

In-class example:

Today we looked at test scores. I told you that both of my classes have the same average of 70%, but that they have different SDs. Tutorial 1 had a SD of 15% and Tutorial 2 had a SD of 10%. This means that Tutorial 1 scores ranged between 55-85% (or 15% plus and minus 70%). Tutorial 2 scores only ranged between 60-80%. From this we would conclude that test scores in Tutorial 2 are more closely grouped around the mean; therefore, the mean score of 70% is actually more representative of students in Tutorial 2.