Lesson Standard Deviation

Source code of 'Standard Deviation'

This Lesson (Standard Deviation) was created by by Shruti_Mishra(0)

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About Shruti_Mishra: I am a maths graduate from India and am currently persuing masters in Operations Research.

<a href="Standard_deviation.wikipedia">Standard deviation</a> (S.D) or {{{sigma}}} is the measure of the spread of a series from its <a href="mean.lesson">mean</a> value. <a href="Standard_deviation.wikipedia">S.D.</a> indicates the differences and variability which a set of number has. <a href="Standard_deviation.wikipedia">S.D.</a> is mathematically the square root of <a href="variance.wikipedia">variance</a> which is defined as the <a href="mean.lesson">mean</a> of the square of the differences between the elements and their <a href="mean.lesson">mean</a>:

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{{{sigma=sqrt(Variance)=sqrt((sum((x[i] - mean)^2,1,n))/n)}}}
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The steps to calculate the standard deviation are:
1. Calculate the <a href="mean.lesson">mean</a> of the series.
2. Calculate the differences between the elements and the <a href="mean.lesson">mean</a>: {{{x[i]-mean}}} for all elements.
3. Calculate the squares of the differences.
4. Calculate the <a href="mean.lesson">mean</a> of the squares by adding all the squares and dividing by the number of elements.
5. Calculate the square root of the the <a href="mean.lesson">mean</a>. This is the <a href="Standard_deviation.wikipedia">standard deviation</a>.

Example: Calculate the standard deviation of the series {1,3,12,5,9}.
Solution: The standard deviation can be calculated as follows:
1. <a href="mean.lesson">Mean</a> of the series = (1+3+12+5+9)/5 = 30/5 = 6
2. The differences between the individual elements and <a href="mean.lesson">mean</a> would be: {1-6, 3-6, 12-6, 5-6, 9-6} = {-5,-3,6,-1,3}
3. The squares of the differences would be {25,9,36,1,9}.
4. The <a href="mean.lesson">mean</a> of the squares would be (25+9+36+1+9)/5 = 80/5 = 16
5. The square root of the <a href="mean.lesson">mean</a> of squares would be {{{sqrt(16)}}} = 4

Thus the <a href="Standard_deviation.wikipedia">standard deviation</a> would be 4.