Method

Finding the Standard Deviation of a Data Set

The standard deviation, σ, of a data set is calculated using the following formula. σ = sqrt((x_1 - x)^2 + (x_2 - x)^2 + ... + (x_n - x)^2/n) In this formula, n is the number of values in the data set and x is the mean of the data set. Performing this calculation in one step makes for a convoluted expression. Therefore, it is best divided into a few, smaller steps. Consider the following data set as an example. 1, 5, 3, 4, 5, 12 Follow these steps when finding the standard deviation.

Find the Mean, x

First, calculate the mean of the data set. The example data set has 6 values, so the denominator is 6.

x = 1+5+3+4+5+12/6

AddTerms

Add terms

x = 30/6

CalcQuot

Calculate quotient

x = 5

The mean of the data is 5.

Find the Deviation of Each Data Value, x_i- x

For each data value, x - x can now be calculated and added to a table. This shows how much each data point varies from the mean.

x	x - x
1	1 - 5 = - 4
5	5 - 5= 0
3	3 - 5 = - 2
4	4 - 5 = - 1
5	5 - 5 = 0
12	12 - 5 = 7

Square the Deviations, (x_i - x)^2

Square the deviations, and add them to a new column in the table.

x	x - x	(x - x)^2
1	- 4	(- 4)^2 = 16
5	0	0^2 = 0
3	-2	(- 2)^2 = 4
4	- 1	(- 1)^2 = 1
5	0	0^2 = 0
12	7	7^2 = 49

Find the Mean of the Squared Deviations, σ^2

The squared deviations should be added and divided by the number of data values. In other words, the mean of the squared deviations is found.

16 + 0 + 4 + 1 + 0 + 49/6

AddTerms

Add terms

70/6

CalcQuot

Calculate quotient

11.666666 ...

RoundDec

Round to 2 decimal place(s)

11.67

This value is called the variance of the data set.

Take the Square Root of the Mean of the Squared Deviations, σ

Finally, take the square root of the just found quotient to get the standard deviation. Here, the fraction is used instead of the quotient, to avoid rounding errors. σ = sqrt(70/6) ≈ 3.4 Thus, a randomly chosen value from this data set is expected to deviate roughly 3.4 units from the mean.

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