How to Calculate and Interpret Standard Deviation

In statistics, the standard deviation measures dispersion about (or around) the mean for a set of data values. The standard deviation is usually denoted with the letter σ (lowercase sigma). It can apply to the following: a probability distribution, a random variable, and a population or a data set. Standard deviation still remains one of the most common measures of statistical dispersion, which basically translates into how widely spread are the values in a particular data set. This article will show you how to quickly and easily calculate and interpret standard deviation.

Difficulty: Easy

Instructions

Things You'll Need:

• spreadsheet software or statistical software program

1. 
   Standard Deviation Formula

   The formula for standard deviation is shown in the picture. To calculate standard deviation: subtract the mean from all of the numbers in the data set, then square the differences, then find the average of all of these squared differences, and finally take the square-root. To put it simply, this is calculating the ‘root-mean-square-deviation’ about the mean.

2. Standard deviation may also be expressed simply as the square root of the variance.

3. Interpretation: If many data points are far from the mean, then the standard deviation is large. On the other hand, if many data points are close to the mean, then the standard deviation is small. If all data values in the distribution are equal, then the standard deviation is zero. Additionally, standard deviation is expressed in the same units as the data itself.

Tips & Warnings

• Many formulas in statistics and statistical analysis use the standard deviation.
• For a normally distributed set of data, the empirical rule states that about 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ, about 95% of the values are within two standard deviations, and about 99.7% are within three standard deviations.