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

We call this special value, the **average.**

There are three averages:

- Mean
- Mode
- Median

*Mean*

This is the sum of all values divided by the number of data.

**Mean = ∑ x / n
**

*Mode*

This is the value that occurs most frequently.

*Median*

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

**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.

**E.g.3**

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

The**Median Class** is the class where n/2 the 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**.

**E.g.4**

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 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!

If you would like to practise more, please visit this page.

**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 2 40 k 50 1 The mean mark for the group is 30. Find k.

- The median of five consecutive odd numbers is T. Find the mean of the numbers in terms of T. 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.

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level.
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The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.

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