Method

Finding the Standard Deviation of a Data Set

σ,

of a data set is calculated using the following formula.

σ = \frac{( x _{1} - x ) ^{2} + ( x _{2} - x ) ^{2} + \dots + ( x _{n} - x ) ^{2}}{n}

In this formula,

n

is the number of values in the data set and

\overline{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,

\overline{x}

First, calculate the mean of the data set. The example data set has

6

values, so the denominator is

6 .

\overline{x} = \frac{1 + 5 + 3 + 4 + 5 + 1 2}{6}

AddTerms

Add terms

\overline{x} = \frac{3 0}{6}

CalcQuot

Calculate quotient

\overline{x} = 5

The mean of the data is

5 .

Find the Deviation of Each Data Value,

x_{i} - \overline{x}

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

$x$	$x - \overline{x}$
$1$	$1 - 5 = - 4$
$5$	$5 - 5 = 0$
$3$	$3 - 5 = - 2$
$4$	$4 - 5 = - 1$
$5$	$5 - 5 = 0$
$12$	$12 - 5 = 7$

Square the Deviations,

(x_{i} - \overline{x})^{2}

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

$x$	$x - \overset{x}{ˉ}$	$(x - \overset{x}{ˉ})^{2}$
$1$	$- 4$	$(- 4)^{2} = 16$
$5$	$0$	$0^{2} = 0$
$3$	$- 2$	$(- 2)^{2} = 4$
$4$	$- 1$	$(- 1)^{2} = 1$
$5$	$0$	$0^{2} = 0$
$12$	$7$	$7^{2} = 49$

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.

\frac{1 6 + 0 + 4 + 1 + 0 + 4 9}{6}

AddTerms

Add terms

\frac{7 0}{6}

CalcQuot

Calculate quotient

11.666666 \dots

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 = \frac{7 0}{6} \approx 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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