Standard deviation is used as a measure of the degree of spread in a population. It is widely quoted along with a measure of central tendency, such as the mean or the median. The symbol for Standard Deviation is the Greek letter sigma, and this word is often used interchangeably with Standard deviation.
Why Standard Deviation is Used as a Summary Statistic
When a sample is taken from a population and measured, it is usually necessary to summarise what has been found. For example, to estimate the average salary of those working in New York, a sample of perhaps 2,000 people would be asked to provide salary details. Any subsequent report about the survey could not produce all 2,000 numbers, and the numbers need to be condensed into a more manageable number.
There are two main types of summary statistic. The first deals with central tendency, and includes the mean (average), median (middle value) and others. The second type of summary statistic is the degree of spread, and there are several measures of this, like the standard deviation and Inter-Quartile Range. The standard deviation is the most useful, and is the most widely used.
Definition of Standard Deviation
Standard deviation is defined as the square root of the average squared distance from the mean. It is always a positive number, and has the same units as the parameter being measured. For example, if a sample of distance measurements is taken in miles, the standard deviation will be in miles also.
Step by Step Calculation of Standard Deviation
1. Calculate the mean.
2. For each data point, subtract the mean to give the difference.
3. Square each difference (multiply it by itself) to give the squared differences.
4. Add all of the squared differences together, to give the sum of squared differences.
5. Divide the sum of squared differences by the number of points, to give the mean squared difference, or variance.
6. Take the square root of the variance to give the standard deviation.
Example Standard Deviation Calculation
To calculate the standard deviation of the numbers 1, 2, 3, 4, and 5:
1. The mean of the sample is (1 + 2 + 3 + 4 + 5) / 5 = 3
2. The differences are:
1 - 3 = -2
2 - 3 = -1
3 - 3 = 0
4 - 3 = 1
5 - 3 = 2
3, The differences squared are:
-2 x -2 = 4
-1 x -1 = 1
0 x 0 = 0
1 x 1 = 1
2 x 2 = 4
4. The sum of squared differences is:
4 + 1 + 0 + 1 + 4 = 10
5. The average of the squared differences is:
10 / 5 = 2
6. The standard deviation is square root of 2 = 1.414
Summary of Standard Deviation
Standard deviation is the most important summary statistic used to represent the spread of a data set. It is extremely useful in statistics generally. It is particularly useful when setting up SPC Chart limits, and when choosing SPC Chart rules.
References for Basic Statistics
For more information about standard deviation, the article How is Standard Deviation Used is useful. There are many good references available, but Statistics For Dummies is a good starting point for beginners.
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