15.09 Standard deviation

Standard deviation is a measure of spread, which helps give us a meaningful estimate of the variability in a data set. It is a weighted average of the distance of each data point from the mean. A small standard deviation indicates that most scores are close to the mean, while a large standard deviation indicates that the scores are more spread out away from the mean value.

Larger standard deviation - more spread out:

Smaller standard deviation - closer to the mean:

The symbol used for the standard deviation of a population is:

In statistics mode on a calculator, the symbol \sigma_{n} or \sigma_{x} may be used.

Standard deviation can be calculated using a formula. However, as this process is time consuming we will be using our calculator to find the standard deviation.

Standard deviation is a weighted average of how far each piece of data varies from the mean.

The standard deviation is a more complex calculation but takes every data point into account. The standard deviation is significantly impacted by outliers.

There is a second type of standard deviation for if you are working with a sample and not a population. This is the sample standard deviation, with the symbol s_{x}. This will normally have a value very close to \sigma_{x}. In this course, when standard deviation is mentioned it will only be referring to population standard deviation and represented by \sigma_{x}. Be careful when using your calculator or online tools that you are finding the population standard deviation and not the sample standard deviation.

Examples

Example 1

The test results for four Geography classes are shown. The classes are labelled as \text{W},\,\text{X},\,\text{Y}, and \text{Z}.

Select the the option that correctly lists the classes in order from largest standard deviation to smallest standard deviation.

A

Y,\,W,\,Z,\,X

B

W,\,X,\,Y,\,Z

C

X,\,W,\,Z,\,Y

D

X,\,Z,\,W,\,Y

Worked Solution

Create a strategy

Compare how spread out the data is for each class.

Apply the idea

Class Y has all the data on the mean, so it will have the lowest standard deviation.

Class W has most of the data near the mean, so it will have the second lowest standard deviation.

Class X has all its data away from the mean so it will have the largest standard deviation.

So the order is: X,\,Z,\,W,\,Y or option D.

Reflect and check

Using a calculator, we can compute the standard deviation of each graph given the score and its frequency.

Watch question walkthrough

Example 2

The number of runs scored by Mario in each of his innings is listed below. 33,\,32,\,32,\,32,\,31,\,32,\,32,\,32,\,32,\,32

What was his standard deviation? Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the population deviation function, \sigma_{x} on your calculator.

Apply the idea

Using Statistics mode enter each data point into your calculator then press the \sigma_{x} button. \sigma_{x} \approx 0.45

Example 3

Use technology to determine the standard deviation for the data represented by the frequency table.

Score	Frequency
15	13
16	9
17	23
18	19
19	8
20	13

Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the population deviation function, \sigma_{x} on your calculator.

Apply the idea

Using Statistics mode enter each score along with its frequency into a data table on your calculator. Then find the standard deviation.\sigma_{x} \approx 1.58

Example 4

Use technology to determine the standard deviation for the data represented by the grouped frequency table, using the class centres.

Class	Class centre	Frequency
40 \leq x \lt 45	42.5	4
45 \leq x \lt 50	47.5	11
50 \leq x \lt 55	52.5	16
55 \leq x \lt 60	57.5	17
60 \leq x \lt 65	62.5	7
65 \leq x \lt 70	67.5	12
70 \leq x \lt 75	72.5	11
75 \leq x \lt 80	77.5	5

Round your answer to two decimal places.

Worked Solution

Create a strategy

Enter the class centres as the scores and frequencies in your calculator to find the standard deviation.

Apply the idea

Since we are given grouped data, we can only get an estimate of the standard deviation since we are using the class centres rather than the actual scores.

Using Statistics mode we can enter the class centres as the values alongside the frequencies, and then find the standard deviation.

\sigma_{x} \approx 9.76