Standard Deviation
Standard deviation is the square roots of the sum of the squared differences between each score and the mean average of all scores.
For a population the formula is:
For a sample the formula is:
Where SD is the standard deviation of a population,
xi is each of the observations,
ma is the mean average, and
n is the number of scores.
This may sound intimidating, but luckily statistical calculators, Excel and other software automate the calculation of standard deviations.
There are other formulas that can be used to make the calculation in fewer steps. The following formula is for easier calculation of the standard deviation of a sample.
This formula requires the total of all the observations squared (a value multiplied by itself) ∑ X2 and the total of all values (∑ X) which is then squared (∑ X)2.
From our point of view two issues need to be reviewed. We need some level of understanding of what standard deviation represents, and the issue of the standard deviation of a population versus calculating an estimate of the population parameter when using samples drawn for a population.
STANDARD DEVIATION for a population versus a sample: When a set of data contains all the possible data points for a population the calculation of standard deviation provides a specific value that characterizes that population. Such a specific value for a statistic is called a parameter. However, it is much more likely that the data one is working with is not the complete population, but only a sample. When a sample is used we need to estimate the standard deviation from the sample. In this situation 1 is subtracted from the N number of cases. (See the formulas above) For small samples this N-1 tends to increase the standard deviation slightly, making it a conservative estimate of the population's parameter. As sample size increases the effect of N-1 declines and the results of the two formulas converge towards the population parameter.
WHAT DOES STANDARD DEVIATION MEAN? The standard deviation statistic is a number that marks a distance on the measurement scale. In very general terms it is the average difference between each score and the mean average. A standard deviation is central to many of the statistics used to make inferences and test hypotheses.
HOW IS STANDARD DEVIATION INTERPRETED? A calculated standard deviation is an estimate of how scores are distributed away from the mean average. If this distribution is approximately normal (bell shaped curve), then .34 (34%) of the cases will occur within one standard deviation. Also, if one adds one standard deviation to the mean and subtracts one standard deviation from the mean the between these two numbers it is estimated that 68% of the cases will occur. This range give a reasonable range to use when discussing where about 2/3rds of the cases will be found. In more advanced interpretations a researcher may use a fraction of the standard deviation and a normal distribution table to estimate a different proportion of the cases.
