New Zealand
Level 7 - NCEA Level 2

# Standard Deviation

Lesson

Standard deviation is a measure of spread, which helps give us a meaningful estimate of the variability in a data set. A small standard deviation means most scores are close to the mean. Conversely, a large standard deviation means the scores are very spread out.

The standard deviation is found by calculating the square root of the variance.

Variance is the average of the squared differences from the mean. Here is its formula.

$\sigma^2=\frac{1}{n}\Sigma\left(x_i-\mu\right)^2$σ2=1nΣ(xiμ)2

The following are the steps required. It is clear that for data sets of only moderate size, the amount of calculation needed is quite large and this makes calculations time-consuming and error-prone. For this reason, modern statistics depends heavily on automation by computer software and by hand-held calculators.

## To Calculate a Population Standard Deviation

This is the formula by which a calculator calculates the standard deviation of a data set from a full population. That is, it is the formula used for census data rather than sample data.

$\sigma=\sqrt{\frac{1}{n}\Sigma\left(x_i-\mu\right)^2}$σ=1nΣ(xiμ)2

In this formula, the numbers $x_i$xi are the values in the data set. There is one value for each subscript $i$i.
There are $n$n numbers $x_i$xi in the data set. So, $i$i goes from $1$1 to $n$n in the summation.
The symbol $\mu$μ (Greek letter 'mu') is the population mean.
The Greek letter $\sigma$σ (sigma) is used for the population standard deviation.
The symbol $\Sigma$Σ (upper case sigma) is the summation symbol.

### steps

1. Calculate the mean.  $\mu=\frac{1}{n}\Sigma_{i=1}^n\ x_i$μ=1nΣni=1 xi
2. Find the difference from the mean for each score. $x_i-\mu$xiμ
3. Square each of the differences.    $\left(x_i-\mu\right)^2$(xiμ)2
4. Sum the squared differences.  $\Sigma\left(x_i-\mu\right)^2$Σ(xiμ)2
5. Divide the sum by the number of scores. $\frac{1}{n}\Sigma\left(x_i-\mu\right)^2$1nΣ(xiμ)2
6. Take the square root.  $\sigma=\sqrt{\frac{1}{n}\Sigma\left(x_i-\mu\right)^2}$σ=1nΣ(xiμ)2

#### Worked Examples

##### Question 1

Find the following based on this set of scores:

$19,18,14,19,10$19,18,14,19,10

1. Find the mean.

2. Complete the following table.

Score($x$x) $(x-$(xmean$)$) $(x-$(xmean$)^2$)2
$19$19 $\editable{}$ $\editable{}$
$18$18 $\editable{}$ $\editable{}$
$14$14 $\editable{}$ $\editable{}$
$19$19 $\editable{}$ $\editable{}$
$10$10 $\editable{}$ $\editable{}$
3. Thus, find the standard deviation, correct to 2 decimal places.

4. Find the range of the set of scores.

##### Question 2

The mean income of people in Country A is $\$1906919069. This is the same as the mean income of people in Country B. The standard deviation of Country A is greater than the standard deviation of Country B. In which country is there likely to be the greatest difference between the incomes of the rich and poor?

1. Country A

A

Country B

B

Country A

A

Country B

B

### Outcomes

#### S7-4

S7-4 Investigate situations that involve elements of chance: A comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions B calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology.

#### 91267

Apply probability methods in solving problems