Explaining Standard Deviation in Simple Terms

Robert Niles (2008) calls standard deviation the "mean of the mean." If we keep in mind that 'mean' in this case means 'average', then this definition is fairly accurate. Standard deviation measures how much - on average - individual scores of a given group vary (or deviate) from the average (or mean) score for this same group. In other words, the value of standard deviation helps show how many subjects in the group score within a certain range of variation from the mean score for the entire group. Still other way to explain it is that standard deviation measures the spread of individual results around a mean of all the results.

Formulating this explanation in a few sentences still does not offer complete clarity, which shows that the concept of standard deviation is rather complex. The existing formulas for calculating standard deviation, one of which Niles (2008) presents, are confusing for someone not trained in statistics and with poor math skills, which further proves how difficult it is to understand and master this concept. The best way to explain it is with examples - and in my opinion, the simpler the example, the better.

Imagine a class of 40 people taking an exam. Once the exam is graded, the instructor calculates the mean value of all the results. In order to do this, she must add all the results together and then divide the resulting figure by the number of people who took the exam. Considering that the highest possible score was 100 and the lowest was zero, and to best show how standard deviation works, let's assume that the mean score for this exam in this class was 50.

To determine standard deviation, we must split our total dataset, which is 100 points, into smaller, even values. It is up to an individual researcher how to split it, but it obviously makes sense in this case to do so in tens. Thus we have 10 value units, from 10 to 100. The mean score is 50, right in the middle of our dataset.

Let's say that 16 students scored between 40 and 60 on the exam, which means that they scored within 10 points, either higher or lower, of the average score. This means that 40% of the entire class (16 divided by 40 and multiplied by 100) scored within one value unit of the mean score.

Let's say that another 12 students scored between 30 and 70 on the exam, which means that they scored within 20 points, either higher or lower, of the average score. These 12 students account for 30% of the entire class (12 divided by 40 and multiplied by 100). Together with the students who scored within 10 points of the average score, they make up a group of 28 students - or 70% of the entire class - who scored within two value units of the mean score.

Practical statistics tell us that approximately 68% of scores in any group fall within one standard deviation from the mean score. This information can be found both in the article by Robert Niles (2008) and in the Glossary section of the National Center for Education Statistics (2008). Based on this, we can say that in our example, 20 points is the approximate value of one standard deviation. Since the deviation can be both higher and lower than the mean score, standard deviation would be displayed as +/- 20.

Check your understanding by answering the following question:
Practical statistics also tell us that approximately 95% of scores in any group fall within two standard deviations from the mean score. In our example, what is the range for two standard deviations and how many students will score within that range?

References
National Center for Education Statistics. (2008). Glossary - S. Online at: http://nces.ed.gov/programs/coe/glossary/s.asp
Niles, R. (2008). Standard deviation. RobertNiles.com. Online at: http://www.robertniles.com/stats/stdev.shtml