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Standard Deviation

Standard Deviation

You might want to look at Measures of Central Tendency before learning about standard deviation. If you are already familiar with the mean of a data set, let's go!

Standard deviation is a measure of dispersion, and it is used in statistics to see how spread out values are from the mean in a data set.

Standard deviation formula

The formula for standard deviation is:

\[ \sigma = \sqrt{\dfrac{\sum(x_i-\mu)^2}{N}}\]

Where:

\(\sigma\) is the standard deviation

\(\sum\) is the sum

\(x_i\) is an individual number in the data set

\( \mu\) is the mean of the data set

\(N\) is the total number of values in the data set

So, in words, the standard deviation is the square root of the sum of how far each data point is from the mean squared, divided by the total number of data points.

The variance of a set of data is equal to the standard deviation squared, \(\sigma^2\).

Standard deviation graph

The concept of standard deviation is pretty useful because it helps us predict how many of the values in a data set will be at a certain distance from the mean. When carrying out a standard deviation, we assume that the values in our data set follow a normal distribution. This means that they are distributed around the mean in a bell-shaped curve, as below.

 probability standard deviation graph studysmarterStandard deviation graph. Image: M W Toews, CC BY-2.5 i

The \(x\)-axis represents the standard deviations around the mean, which in this case is \(0\). The \(y\)-axis shows the probability density, which means how many of the values in the data set fall between the standard deviations of the mean. This graph, therefore, tells us that \(68.2\%\) of the points in a normally-distributed data set fall between \(-1\) standard deviation and \(+1\) standard deviation of the mean, \(\mu\).

How do you calculate standard deviation?

In this section, we will look at an example of how to calculate the standard deviation of a sample data set. Let's say you measured the height of your classmates in cm and recorded the results. Here's your data:

165, 187, 172, 166, 178, 175, 185, 163, 176, 183, 186, 179

From this data we can already determine \(N\), the number of data points. In this case, \(N = 12\). Now we need to calculate the mean, \(\mu\). To do that we simply add all the values together and divide by the total number of data points, \(N\).

\[ \begin{align} \mu &= \frac{165 + 187+172+166+178+175+185+163+176+183+186+179}{12} \\ &= 176.25. \end{align} \]

Now we have to find

\[ \sum(x_i-\mu)^2.\]

For this we can construct a table:

\(x_i\)

\(x_i - \mu\)

\((x_i-\mu)^2\)

165

-11.25

126.5625

187

10.75

115.5625

172

-4.25

18.0625

166

-10.25

105.0625

178

1.75

3.0625

175

-1.25

1.5625

185

8.75

76.5625

163

-13.25

175.5625

176

-0.25

0.0625

183

6.75

45.5625

186

9.75

95.0625

179

2.75

7.5625

For the standard deviation equation, we need the sum by adding all the values in the last column. This gives \(770.25\).

\[ \sum(x_i-\mu)^2 = 770.25.\]

We now have all the values we need to plug into the equation and get the standard deviation for this data set.

\[ \begin{align} \sigma &= \sqrt{\dfrac{\sum(x_i-\mu)^2}{N}} \\ &= \sqrt{\frac{770.25}{12}} \\ &= 8.012. \end{align}\]

This means that, on average, the values in the data set will be \(8.012\, cm\) away from the mean. As seen on the normal distribution graph above, we know that \(68.2\%\) of the data points are between \(-1\) standard deviation and \(+1\) standard deviation of the mean. In this case, the mean is \(176.25\, cm\) and the standard deviation \(8.012\, cm\). Therefore, \( \mu - \sigma = 168.24\, cm\) and \( \mu - \sigma = 184.26\, cm\), meaning that \(68.2\%\) of values are between \(168.24\, cm\) and \(184.26\, cm\) .

The age of five workers (in years) in an office was recorded. Find the standard deviation of the ages: 44, 35, 27, 56, 52.

We have 5 data points, so \(N=5\). Now we can find the mean, \(\mu\).

\[ \mu = \frac{44+35+27+56+52}{5} = 42.8\]

We now have to find

\[ \sum(x_i-\mu)^2.\]

For this, we can construct a table such as above.

\(x_i\)\(x_i - \mu\)

\((x_i-\mu)^2\)

441.21.44
35-7.860.84
27-15.8249.64
5613.2174.24
529.284.64

To find

\[ \sum(x_i-\mu)^2,\]

we can simply add all the numbers in the last column. This gives

\[ \sum(x_i-\mu)^2 = 570.8\]

We can now plug everything into the standard deviation equation.

\[ \begin{align} \sigma &= \sqrt{\dfrac{\sum(x_i-\mu)^2}{N}} \\ &= \sqrt{\frac{570.8}{5}} \\ &= 10.68. \end{align}\]

So the standard deviation is \(10.68\) years.

Standard Deviation - Key takeaways

  • Standard deviation is a measure of dispersion, or how far away the values in a data set are from the mean.
  • The symbol for standard deviation is sigma, \(\sigma\)
  • The equation for standard deviation is \[ \sigma = \sqrt{\dfrac{\sum(x_i-\mu)^2}{N}} \]
  • The variance is equal to \(\sigma^2\)
  • Standard deviation is used for data sets that follow a normal distribution.
  • The graph for a normal distribution is bell-shaped.
  • In a data set that follows a normal distribution, \(68.2\%\) of values fall within \(\pm \sigma\) the mean.


Images

Standard deviation graph: https://commons.wikimedia.org/wiki/File:Standard_deviation_diagram.svg

Frequently Asked Questions about Standard Deviation

Standard deviation is a measure of dispersion, used in statistics to find the dispersion of values in a data set around the mean.

No, standard deviation cannot be negative because it is the square root of a number.

By using the formula 𝝈=√ (∑(xi-𝜇)^2/N) where 𝝈 is the standard deviation, ∑ is the sum, xi is an individual number in the data set, 𝜇 is the mean of the data set and N is the total number of values in the data set.

Final Standard Deviation Quiz

Question

What does standard deviation mean?

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Answer

Standard deviation is a measure of dispersion used in statistics to tell us how spread out the values are from the mean in a data set.

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Question

What is the formula for standard deviation?

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Answer

\(\sigma = \dfrac{\sqrt{\sum (x_i - \mu)^2}}{N}\).

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Question

What is the formula for standard deviation in words?

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Answer

The square root of the sum of how far each data point is from the mean squared divided by the total number of data points.

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Question

 What do the symbols in the standard deviation formula stand for?

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Answer

\(\sigma\) is the standard deviation, \(\sum\) is the sum, \(x_i\) is an individual number in the data set, \(\mu\)  is the mean of the data set, and \(N\) is the total number of values in the data set.

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Question

What type of distribution do data sets for which we can do a standard deviation follow?  

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Answer

A normal distribution.

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Question

What percentage of data points fall within \(\pm 1\sigma\) of the mean?

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Answer

\(68.2\%\).

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Question

What is the standard deviation of the standard normal distribution?

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Answer

One.

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Question

Where is the mean of the standard normal distribution?

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Answer

At zero.

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Question

Which of these tools can help you find the standard deviation of a data set?

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Answer

A table of values.

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Question

What is another word for the mean of a data set?

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Answer

The average.

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Question

How do you find the mean of a set of data?

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Answer

Add up all the numbers in the data set, and then divide by how many numbers there are in the set.

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Question

Which of the following tell you about how far the data is spread out from the mean of the data?

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Answer

The standard deviation.

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Question

What shape does the normal distribution graph have?

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Answer

A bell curve.

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Question

What percentage of values fall between 2 standard deviations of the mean?

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Answer

\(91.4\%\)

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Question

How do you find the variance of a data set?  

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Answer

Square the standard deviation.

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