Test Validity

Standard Deviation

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Explanation

We have seen that standard deviation is a measure of variation. Now we will be more specific. Standard deviation measures how far from the average (the mean) the data is spread. The standard deviation measures the average distance between each of the data points and the mean.

One thing that you should keep in mind when using or reading about standard deviation and variance:

  • The standard deviation of a sample of data (s) involves dividing by (n-1)
  • The standard deviation of a population of data (σ) involves dividing by n
When using software to calculate these values, check which version is used and make sure you use the version for samples unless your data represents the entire population of whatever you are measuring (which is pretty rare). For example, Excel has the function stdev for calculating the sample standard deviation and the function stdevP for calculating the population standard deviation. The sample (n-1) method is known as the unbiased method. In reality, the differences are small unless n is very small, and in such cases you probably have insufficient data anyway. The difference is worth knowing about, however.

Exploration

Calculating the Standard Deviation

When you look at the formula below, you will see that it is made up of the mean subtracted from each value, then squared. These values are added together and the final sum is divided by n-1. You might notice that this is similar to the formula for calculating the mean, and you'd be right.

The standard deviation is the average distance between each point and the mean

There are two common formulae for calculating standard deviation. We will show you both of them at here. The first one you will see highlights the fact that the standard deviation is the average distance between each point in your data and the mean.

Here is the formula for calculating the standard deviation of a sample of data. Click on any part of the formula to see a description of its role. There is help on squaring and square roots above if you need it.

It says that you subtract the mean, which is x, from each value in turn and square the result. You add all of these values together and divide the result by one less than the number of values in your data (n-1). Finally, you find the square root of the result of the division by n-1 and that is your standard deviation.

Note that any part of the formula in brackets is calculated first, for example ∑(x - x) uses the brackets to indicate that you do the subtractions first and then add up the results of all the subtractions.

In case you are wondering, we square the differences and then take square roots for two reasons:

  • Half the distances from the mean are positive and half are negative, so adding them up would produce a value of zero! Squaring makes numbers positive and removes that problem;
  • We could leave the value squared, but square rooting it brings it back into units that match the units of the original measurements. If we measured people's heights in cm, then we can report the standard deviation in cm too.
You might see the formula for standard deviation in text books written as below. This is the same formula, but written in a way that makes it easier to calculate.

It doesn't matter which formula you use as they both give the same result.

Application

Let's calculate the standard deviation for cog_ablty, which has a mean of 77.33
  1. First we calculate the differences, x - x, and the squared differences, (x-x)2.

    The values of cog_ablty are shown below with the differences and squared differences filled in for all except the first five.
    Subtract the mean from each value and enter the result in the first column (keep the minus sign if there is one).
    For example, the first value is 54 - 77.33 = -23.33
    Then square the number from the first column and enter its value in the second column. To square a number, multiply it by itself (or use the [x^2] button) on the calculator.
    Enter answers to 2 decimal places. You can use the [RD] button on the calculator to round down.
    Valuex - x(x-x)2
    54      
    74      
    97      
    98      
    74      
    58-19.33373.65
    780.670.45
    75-2.335.43
    74-3.3311.09
    60-17.33300.33
    60-17.33300.33
    9214.67215.21
    10022.67513.93
    62-15.33235.01
    9517.67312.23
    74-3.3311.09
    50-27.33746.93
    9315.67245.55
    9315.67245.55
    835.6732.15
    72-5.3328.41
    9012.67160.53
    9214.67215.21
    76-1.331.77
    780.670.45
    9416.67277.89
    72-5.3328.41
    51-26.33693.27
    9719.67386.91
    54-23.33544.29
    Calculator
    ( You need to enable Java to see this applet. )
    Help

  2. The formula for standard deviation requires you to add up the squared differences. That is the ∑ (x-x)2 part.
    Add them up now and enter the total here.   Help
  3. You have 30 values in your data. What is n-1?   Help
  4. What is 7266.7 divided by n-1?   Help
  5. Now find the square root of that number, that is the √ part: s=   Help
Now you have a value for the standard deviation, you just need to report that value. It can be useful to report the units of measurement, too. The standard deviation of cog_ablty is 15.83 Points.
Central Tendency | Confidence Intervals