# 11.04 Standard deviation

Lesson

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:

 $\text{Standard deviation}$Standard deviation $=$= $\sigma$σ (pronounced sigma)

In statistics mode on a calculator, the following symbol might be used:

 $\text{Standard deviation}$Standard deviation $=$= $\sigma_n$σn​

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

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.

• larger value indicates a wider spread (more variable) data set.
• A smaller value indicates a more tightly packed (less variable) data set.

#### Worked examples

##### Example 1

Calculate the variance and standard deviation for this data set: $75,75,75,80,80,80,80,92,107,107,107,107$75,75,75,80,80,80,80,92,107,107,107,107

Using Statistics mode enter values and calculate standard deviation

Standard deviation is given as $\sigma_x\approx13.59$σx13.59.

##### Example 2

Calculate the mean for the data represented in the frequency table

Value Frequency
$75$75 $3$3
$80$80 $4$4
$92$92 $1$1
$107$107 $4$4

Using Statistics mode enter the values and frequencies in your calculator

This data set is equivalent to the previous examples so, once again, the standard deviation is given as $\sigma_x\approx13.59$σx13.59.

##### Example 3

Estimate the standard deviation for the data represented in the grouped frequency table

Class Frequency
$30-<40$30<40 $12$12
$40-<50$40<50 $16$16
$50-<60$50<60 $25$25
$60-<70$60<70 $4$4

Since we are given grouped data, we can only get an estimate of the standard deviation. We first need to determine the class centres, which will be used to represent each class. For instance, the class centre for the first interval is $\frac{30+40}{2}=35$30+402=35.

Class Class Centre Frequency
$30-<40$30<40 $35$35 $12$12
$40-<50$40<50 $45$45 $16$16
$50-<60$50<60 $55$55 $25$25
$60-<70$60<70 $65$65 $4$4

Using Statistics mode can enter information similarly to example 2 using the class centres as the values alongside the frequencies.

For this data set, the standard deviation is given as $\sigma_x\approx8.91$σx8.91.

### Sample vs population standard deviation

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 $s_x$sx. This will normally have a value very close to $\sigma_x$σx. In this course, when standard deviation is mentioned it will only be referring to population standard deviation and represented by $\sigma_x$σx. Be careful when using your calculator or online tools that you are finding the population standard deviation and not the sample standard deviation.

Remember!

For this course you should always use the $\sigma_x$σx value for standard deviation.

#### Practice questions

##### Question 1

The test results for four Geography classes are shown. The classes are labelled as $W$W, $X$X, $Y$Y and $Z$Z.

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

$Y$Y, $W$W, $Z$Z, $X$X

A

$W$W, $X$X, $Y$Y, $Z$Z

B

$X$X, $W$W, $Z$Z, $Y$Y

C

$X$X, $Z$Z, $W$W, $Y$Y

D

$Y$Y, $W$W, $Z$Z, $X$X

A

$W$W, $X$X, $Y$Y, $Z$Z

B

$X$X, $W$W, $Z$Z, $Y$Y

C

$X$X, $Z$Z, $W$W, $Y$Y

D

##### Question 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$33,32,32,32,31,32,32,32,32,32

1. What was his batting average?

2. What was his standard deviation?

##### Question 3

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

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