The number chosen for representing a set of data is called its average. It must,

- • Represent all the values
- • Should not be an exaggerated one - not too small or not too big

There are three averages:

- Mean
- Mode
- Median

*Mean*

This is the sum of all values divided by the number of data - Σ,sigma, means, add

**Mean = ∑ x / n
**

*Mode*

This is the value that occurs most frequently.

The difference between the highest and lowest values is called the range of the data.

**E.g.** Data: 3, 4, 8, 6, 11, 21

Range = 21 - 3 = 18

*Median*

This is the middle value, when the data is arranged in order of size. Now, let's try some examples.

Data that exists in the exact form since its collection is usually considered raw data.

**E.g.1**

The heights of five plants in a garden are 3cm, 4cm, 7cm, 12cm and 9cm. Find the averages.

Mean = ∑ x / n = 3 + 4 + 7 + 12 + 9 / 5 = 7cm

There is no mode, as each value occurs only once.

To find the median, let's rearrange them in order of size:

3, 4, **7**, 9, 12

The middle value is 7. So, the median = 7cm.

**E.g.2**

The lengths of 6 carpets are 7m, 15m, 15m, 9m, 22m, 4m. Find the averages.

Mean = ∑ x / n = 7 + 15 + 15 + 9 + 22 + 4 / 6 = 12m

The mode = 15m

To find the median, let's rearrange them in order of size:

4, 7, 9, 15, 15, 22

The middle value = 9 + 15 /2 = 12, and so is the median.

Raw data, when arranged in a table for convenience, is in tabular form.

**E.g.**

The frequency of shoe sizes of students in a certain class is as follows:

shoe-size (x) | frequency (f) |

3 | 3 |

4 | 5 |

5 | 10 |

6 | 8 |

7 | 4 |

Here, we have a slightly different approach;

Mean = sum of fx/n ∑ fx / n = 3X3 + 4X5 + 5X10 + 6X8 + 7X4 /30 = 5.2

The**Median Class** is the class where (n/2)^{th} value lies in. In this case, 30/2 = 15th value lies in **shoe-size 5 class**. So, it is the median class.

The** Modal class** is the class with the highest frequency. So, the modal class is **shoe-size 5 class**.

Raw data, when arranged in classes for easy handling, form grouped data. It is usually called a group frequency table.

**E.g.**

The marks obtained by a group of students for maths are as follows:

Marks (x) | frequency (f) |

0 - 20 | 3 |

21 - 40 | 6 |

41 - 60 | 9 |

61 - 80 | 8 |

81 - 100 | 4 |

Mean = ∑ fx / n = 10X3 + 30X6 + 50X9 +70X8 + 90X4 /30 = 52.7 - x is the middle class value

The** Median Class** is the class where n/2 the value lies in. In this case, 30/2 = 15th value lies in **41 - 60 class**. So, it is the median class.

The** Modal class** is the class with the highest frequency. So, the modal class is **41 - 60 class**.

**The reliability of the Mean**

The mean can easily be influenced by the extremes values of data:

The heights of five plants are 2cm, 4cm, 7cm, 18cm, 19cm. Find the mean and comment on the result.

Mean = 2 + 4 + 7 + 18 + 19 / 5 = 10 cm

This value does not represent either the shortest plant - 2cm - or the tallest - 19cm. So, the mean in this case is not accurate; it may even mislead!

With the following applet, you can practise the averages of a randomly-generated set of raw data:

**Now, in order to complement what you have just learnt, work out the following questions:**

- Find the mean, median and mode of the following numbers - 1, 5, 3, 4, 3, 8, 2, 3, 4, 1.
- The marks scored by pupils in a certain class for an IQ test are as follows:

**Marks (x)****frequency (f)**3 2 4 3 5 6 6 4 7 3 8 2 Find the averages of the marks.

- The mean height of 4 boys is 1.2m and the mean height of 6 girls is 1.5m. Find the mean height of 10 pupils altogether.
- The marks for a certain test for a group of students are as follows:

**Marks (x)****frequency (f)**30 5 40 k 50 1 The mean mark for the group is 30. Find k.

- The median of five consecutive odd numbers is M. Find the mean of the numbers in terms of M. Hence, find the mean of the square of the same numbers.
- The numbers 4, 5, 9, 15 and k are arranged in ascending order so that the mean is the same as median. Find k. Without further calculation, determine the new mean if the numbers are doubled.
- A set of numbers are in the ratio 3: 5: 8: 12. The mean turns out to be 42. Find the range of the numbers.
- Hounslow United football club in West London managed to score 2, 4, 4, 4, 2, 1, 4 goals in their first 7 matches. Find the mean.The manager wants them to keep the average goal of the first 10 matches to be the same as the mean goal so far. How many goals should they score in the next three matches?
- The mean of 6 numbers is 8. The mean of two numbers is 5. Find the mean of the other four numbers.
- The list of numbers, 2, 3, 7, 8, x, 12, 16, is in ascending order. Its mode, median and mean are the same. Find x.

Move the mouse over, just below this, to see the answers:

- 3.4. 3, 3
- 5.5, 5, 5
- 1.53
- 7
- (M + 4), (M + 4)
^{2}+ 8 - 12, 18
- 54
- 3, 9
- 10
- 8

Now that you have read this tutorial, you will find the following tutorials very helpful too:

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7^{th} edition in print.