# Mean and Standard Deviation

## Mean and Standard Deviation Revision

**Mean and Standard Deviation**

**Mean**, **median** and **mode** are averages of data sets – known as **measures of central tendency**. **Range** is a measure of the spread of the data. We can also analyse the spread of the data in more sophisticated ways, such as **standard deviation**.

**Recap: Mean, Median, Mode and Range**

The **mean** is the sum of the data points divided by the total number of data points. The **median** is the middle value of the data points when they are ordered (or the midpoint between the two middle values if there are an even number of data points). The **mode** is the value that appears the most often. The **range** is the difference between the highest and the lowest value. The **interquartile range** is the range after we discard the top and bottom quarters of the data.

**Notation for the mean:** The **mean** is often written as \bar{x}=\dfrac{\sum{x}}{n} where x is each data point, or \bar{x}=\dfrac{\sum{fx}}{\sum{f}} where x is a data value and f is the frequency of that data value.

**Variance and Standard Deviation**

**Variance**, like **range**, is a measure of the **spread of the data**. However, variance is a more intelligent measure than range as it takes all of the data into account. The formula for variance looks a little scary:

\dfrac{\sum{(x-\bar{x})^{2}}}{n} or \dfrac{\sum{x^{2}}}{n}-\bar{x}^{2} or \dfrac{\sum{fx^{2}}}{\sum{f}}-\bar{x}^{2}

It is easier to remember a simple rule:

**Variance** is** (mean of the squares) – (square of the mean)**

**Standard deviation** is the **square root of the variance.**

\text{standard deviation}=\sqrt{\text{variance}}

**Notation for Mean and Variance**

The **mean** can be written as \mathbb{E}(x), which means the **expectation** or **expected value** of x.

**Note:** the **mean of the squares** is written as \mathbb{E}(x^{2}), so the **variance **is \mathbb{E}(x^{2})-(\mathbb{E}(x))^{2}

The **standard deviation** is often written as the Greek letter \sigma, which means that the **variance** can be written as \sigma^{2}. You may also see the variance written as \text{var}(x).

We can also use **S _{xy} notation:**

S_{xy}=\sum{(x-\bar{x})(y-\bar{y})}=\sum{xy}-\dfrac{\sum{x}\sum{y}}{n}

Using this notation, the variance of x is equal to \dfrac{S_{xx}}{n}.

**Example 1: Finding the Mean and Standard Deviation**

Find the mean and standard deviation of 6,10,4,7,24,13,9,15.

**[3 marks]**

\begin{aligned}\text{mean}&=\dfrac{1}{8}(6+10+4+7+24+13+9+15)\\[1.2em]&=11\end{aligned}

Then to find the standard deviation,

\begin{aligned}&\text{mean of squares}\\[1.2em]&=\dfrac{1}{8}(6^{2}+10^{2}+4^{2}+7^{2}+24^{2}+13^{2}+9^{2}+15^{2})\\[1.2em]&=\dfrac{1}{8}(36+100+16+49+576+169+81+225)\\[1.2em]&=156.5\end{aligned}

\begin{aligned}\text{square of mean}&=11^{2}\\[1.2em]&=121\end{aligned}

\begin{aligned}\text{variance}&=\text{mean of squares}-\text{square of mean}\\[1.2em]&=156.5-121\\[1.2em]&=35.5\end{aligned}

\begin{aligned}\text{standard deviation}&=\sqrt{\text{variance}}\\[1.2em]&=\sqrt{35.5}\\[1.2em]&=5.96\end{aligned}

**Example 2: Standard Deviation From Grouped Frequency Table**

The number of cars sold by 100 small used car dealerships was monitored over the course of a day. The results have been collated into a table.

What is the mean and standard deviation of the number of cars sold?

**[6 marks]**

To start, set up a table like this:

\begin{aligned}\text{mean}&=\bar{x}\\[1.2em]&=\dfrac{\sum{fx}}{\sum{f}}\\[1.2em]&=\dfrac{446}{100}\\[1.2em]&=4.46\end{aligned}

\begin{aligned}\text{variance}&=\dfrac{\sum{fx^{2}}}{\sum{f}}-\bar{x}^{2}\\[1.2em]&=\dfrac{2330}{100}-4.46^{2}\\[1.2em]&=23.3-19.8916\\[1.2em]&=3.4084\end{aligned}

\text{standard deviation} = \sqrt{3.4084} = 1.846 (3 dp)

## Mean and Standard Deviation Example Questions

**Question 1:**

Consider the data set 10,4,7,11,3; find its:

a) mean

b) median

c) variance

**[4 marks]**

a)

\begin{aligned}\text{mean}&=\dfrac{10+4+7+11+3}{5}\\[1.2em]&=7\end{aligned}

b) For the median we must order the data:

3,4,7,10,11We can see that the middle value is 7, so the median is 7

c)

\begin{aligned}\text{variance}&=\dfrac{10^{2}+4^{2}+7^{2}+11^{2}+3^{2}}{5}-7^{2}\\[1.2em]&=\dfrac{100+16+49+121+9}{5}-49\\[1.2em]&=\dfrac{295}{5}-49=59-49\\[1.2em]&=10\end{aligned}**Question 2:**

If the standard deviation of a data set is 9.2, what is its variance?

If it has a mean of 5, what is the mean of the squares?

**[3 marks]**

\sigma=9.2

\begin{aligned}\text{variance}&=\sigma^{2}\\[1.2em]&=9.2^{2}\\[1.2em]&=84.64\end{aligned}

\text{variance}=\text{mean of squares}-\text{square of mean}

84.64=\text{mean of squares}-5^{2}

84.64=\text{mean of squares}-25

\begin{aligned}\text{mean of squares}&=25+84.64\\[1.2em]&=109.64\end{aligned}

**Question 3:**

Mrs Chivers gives her class a spelling test, with ten words. Students receive a score out of 10 corresponding to the number of words they spelt correctly. Calculate the following for the results of the class:

a) the mean

b) the variance

She has collated the result in the table below.

**[6 marks]**

Set up the table as in the second example:

\begin{aligned}\text{mean}&=\bar{x}\\[1.2em]&=\dfrac{\sum{fx}}{\sum{f}}\\[1.2em]&=\dfrac{185}{30}\\[1.2em]&=6.17\end{aligned}

\begin{aligned}\text{variance}&=\dfrac{\sum{fx^{2}}}{\sum{f}}-\bar{x}^{2}\\[1.2em]&=\dfrac{1265}{30}-6.17^{2}\\[1.2em]&=42.2-38.0\\[1.2em]&=4.14\end{aligned}