<< Chapter < Page Chapter >> Page >
  • In general, a value = mean + (#ofSTDEVs)(standard deviation)
  • where #ofSTDEVs = the number of standard deviations
  • 7 is one standard deviation more than the mean of 5 because: 7=5+ (1) (2)
  • 1 is two standard deviations less than the mean of 5 because: 1=5+ (−2) (2)

Calculating the standard deviation

If x is a data value, then the difference " x - mean" is called its deviation . In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the data is for a population, in symbols a deviation is x μ . For sample data, in symbols a deviation is x - x .

The procedure to calculate the standard deviation depends on whether the data is for the entire population or comes from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is a population or a sample. The lower case letter s represents the sample standard deviation and the Greek letter σ (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ .

To calculate the standard deviation, we need to calculate the variance first. The variance is an average of the squares of the deviations (the x - x values for a sample, or the x μ values for a population). The symbol σ 2 represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s 2 represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.

If the data is from a population, when we calculate the average of the squared deviations to find the variance, we divide by N, the number of items in the population. If the data is from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n-1, one less than the number of items in the sample. You can see that in the formulas below.

    Formulas for sample standard deviation

  • s = size 12{s={}} {} Σ ( x x ¯ ) 2 n 1 or s = size 12{s={}} {} Σ f · ( x x ¯ ) 2 n 1
  • For the sample standard deviation, the denominator is n-1, that is the sample size MINUS 1.

    Formulas for population standard deviation

  • σ = size 12{σ={}} {} Σ ( x μ ¯ ) 2 N or σ = size 12{σ={}} {} Σ f · ( x μ ¯ ) 2 N
  • For the population standard deviation, the denominator is N, the number of items in the population.

In these fomulas, f represents the frequency with which a value appears. For example, if a value appears once, f is 1. If a value appears three times in the data set, f is 3.

In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. If you are using a TI-83,83+,84+ calculator, you need to select the appropriate standard deviation σ or s from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean.
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics: custom version modified by r. bloom' conversation and receive update notifications?

Ask