Measures of Dispersion: Standard Deviation

What is the Standard Deviation?

The standard deviation of a data set is based on how much each data value deviates from the mean, and is equal to the square root of the variance. The greater the dispersion of values, the larger the standard deviation. Much of statistical theory is based on the standard deviation and the 'normal' distribution.

When is the Standard Deviation Useful?

The standard deviation is a widely used measure of variation. It is a useful measure when your data distribution is very close to a normal curve. In this situation, the mean is the best measure of central tendency, and the standard deviation is the best measure of dispersion.

In a normal distribution, if you measure 1 standard deviation to either side of the mean, you will find that 68.3% of the observations fall into this area; 95.5% of the observations fall within 2 standard deviations to either side of the mean; and 99.7% of observations fall within 3 standard deviations of the mean.

How is the Standard Deviation Calculated?

The standard deviation is calculated as follows:

Translated to English:

  1. Calculate the mean (average or) for the data set.

  2. Determine the deviation from the mean () for each value by subtracting the mean from the value. A negative deviation means that observation fell below the mean. A positive deviation indicates that the observation fell above the mean.

  3. Calculate the square of the deviation for each observation calculated in step 2. This will always be a positive number (a negative value times a negative value equals a positive value).

  4. Add up the squares calculated in step 3:  

  5. Subtract 1 from the number of observations:  -1

  6. Divide the total from step #4 by the result of step 5:  

  7. Calculate the square root of the value calculated in step 6:  

  8. The result is the standard deviation  

An Exercise in Calculating the Standard Deviation

Let's take a look at a simple data set. The mean of the values in the 'Data Value' column is 50 (see below). Use this figure for the 'Deviation from the Mean' calculation in the next column. To better understand the process, walk through the steps of the exercise yourself.

After you understand the concept of standard deviation, you will normally use the standard deviation function on a calculator for the tedious and complex calculations. The GraphTool applet in the Tables Graphs & Charts lesson will calculate mean and standard deviation, as well as draw a graph of your own data set.

There are two versions available for the Windows95 calculator: scientific and standard. The scientific version of the calculator provides a method to enter your data values, then use the 's' function to calculate the standard deviation. The calculator is located in the Windows® accessories program group. The help option provides the information you need to learn how to enter numbers and use the 's' function.

 

Data
Value

Deviation
from Mean

Square of
Deviation

1

1 - 50 =

-49

- 49 * - 49 =

2401

44

44 - 50 =

-6

-6 * -6 =

36

45

45 - 50 =

-5

-5 * -5 =

25

46

46 - 50 =

-4

- 4 * - 4 =

16

48

48 - 50 =

-2

- 2 * -2 =

4

48

48 - 50 =

-2

- 2 * - 2 =

4

49

49 - 50 =

-1

- 1 * - 1 =

1

50

50 - 50 =

0

0 * 0 =

0

50

50 - 50 =

0

0 * 0 =

0

51

51 - 50 =

1

1 * 1 =

1

52

52 - 50 =

2

2 * 2 =

4

52

52 - 50 =

2

2 * 2 =

4

54

54 - 50 =

4

4 * 4 =

16

55

55 - 50 =

5

5 * 5 =

25

55

55 - 50 =

5

5 * 5 =

25

100

100 - 50 =

50

50 * 50 =

2500

Mean

   50

Sum of Squares

 

5062

# of Observations - 1

(16 - 1) = 15

Standard
Deviation

Square root of (5062 / 15) = 18.4


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