How is Standard Deviation Used

More Properties of Sigma in Statistics

Feb 27, 2009 Martin Bell

The standard deviation function is widely used in statistics. The alternative formula for sigma is described, along with how to combine two or more standard deviations.

Standard deviation may be calculated for a sample without calculating the sample mean first. This is useful when calculating the standard deviation of a large sample. The difference between the sample standard deviation and the population standard deviation, and how to combine two or more standard deviation values are described.

Alternative Standard Deviation Calculation

The article What is Standard Deviation describes how to calculate the standard deviation of the numbers 1, 2, 3, 4, and 5 using the conventional equation. The standard deviation was calculated as 1.414

There is an alternative method that gives exactly the same result:

1. Square each number:

1

4

9

16

25

2. Sum the squares:

1 + 4 + 9 + 16 + 25 = 55. This is the sum of squares.

3. Sum the original numbers

1 + 2 + 3 + 4 + 5 = 15. This is the sum

4. Square the sum:

15 x 15 = 225. This is the square of the sum.

5. Divide the square of the sum by the number of data points:

225 / 5 = 45. This is the average square.

6. Subtract the average square from the sum of squares:

55 - 45 = 10.

7. Divide this number by the number of samples:

10 / 5 = 2

8. Take the square root:

square root of 2 = 1.414, as before.

Sample vs Population Standard Deviation

In the example shown, the entire population was only five numbers. When a sample is taken from a much larger population, and when the sample size is small, the population standard deviation is underestimated. The formula in step 7 then changes slightly. Instead of dividing by the number of the sample, the sum is divided by the number in the sample minus one:

7. Divide this number by the number in the sample minus 1:

10 / ( 5 - 1) = 2.5

The standard deviation is now the square root of 2.5 = 1.581, and it is called the sample standard deviation.

Pooled Standard DeviationSometimes it is necessary to combine two standard deviations. For example, if the standard deviation of the height of women in Minnesota is 5 cm, and the standard deviation of the height of men in Minnesota is 6 cm, what is the overall standard deviation of the population of Minnesota?

The key to this is that standard deviations should not be added directly. Variances are added, where variance is the standard deviation squared:

Overall variance = Variance for men + Variance for women

So, overall variance = SD(men) x SD (men) + SD(women) + SD(women)

= 6 x 6 + 5 x 5

= 36 + 25

= 61

So the overall standard deviation is the square root of 61 = 7.8 cm

Combining two standard deviations in this way is valid when the mean of the two samples is roughly the same. To be strictly accurate, the pooled variance is the variance between the samples added to the variance within the samples.

Summary of Standard DeviationThere is an alternative, easy to use formula for standard deviation. The standard deviation of samples underestimates the standard deviation of the population, and needs to be adjusted. Standard deviations are combined by adding the variances.

References for Basic Statistics

The article What is Standard Deviation describes how to calculate the basic standard deviation. There are many good references available, but Statistics For Dummies is a good starting point for beginners.

The copyright of the article How is Standard Deviation Used in Engineering is owned by Martin Bell. Permission to republish How is Standard Deviation Used in print or online must be granted by the author in writing.

Alternative Equation for Standard Deviation

Both Formulae for Standard Deviation

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What is 9+1?

Comments

Mar 17, 2009 10:20 PM

Guest :

Pooled standard deviation = square root of weighted average of the squares of standard deviations!!!
Var(people) = (n1*Var(men)+n2*Var(Women))/(n1+n2), where n1 and n2 is the degree of freedom of variance estimations (ie. sample size - 1).
You can use it to estimate variance of people when avg. height of man and woman are equal.

You should use the law of error propagation when avg. height of man and woman are NOT equal!
Var(people) = Var(women)+Var(men)

Tamas

Mar 18, 2009 2:59 AM

Martin Bell :

Your equation is correct, but as you pointed out, this is only valid when the the average heights are equal. Since average heights differ by a number that is large (10 cm) compared to the standard deviation (5 or 6 cm) , then it is correct to use the error propogation rule.

2 Comments