What does it make you think of when we say **grouped **and **ungrouped **data?

**Grouped data** is exactly what it sounds like - the data was grouped together into groups!

For example, this table shows the grouped data of masses of students in a class:

Mass (m kg) | Frequency |
---|---|

40 ≤ m < 45 | 2 |

45 ≤ m < 50 | 5 |

50 ≤ m < 55 | 9 |

55 ≤ m < 60 | 3 |

60 ≤ m < 65 | 1 |

The data was collected and organised into groups:

2 children are in the group 40 ≤ m < 45 so have a mass greater than or equal to 40 kg but less than 45 kg.

5 children are in the group 45 ≤ m < 50 so have a mass greater than or equal to 45 kg but less than 50 kg

etc.

**Ungrouped data** is when we have no groups and we would have e.g. 2 children with the exact weight 42.1 kg, then 3 children with 43 kg etc

Can you see that when we have grouped data, we don't know exactly what the values are?

Let's have a look at how to find our averages of **mean**, **median **and **mode **for grouped data:

**Mean**

We have to **estimate **the mean by using the midpoint value in each class.

For example, **42.5 **is the midpoint for 40 ≤ m < 45 since 42.5 is right in the middle between 40 and 45.

So the midpoints are** 42.5, 47.5, 52.5, 57.5** and** 62.5** kg

We multiply these by their respective frequencies to get the total mass of all the children:

(2 x 42.5) + (5 x 47.5) + (9 x 52.5) + (3 x 57.5) + (1 x 62.5) = 1,030 kg

There are 2 + 5 + 9 + 3 + 1 = 20 children in the class.

So, the estimated **mean mass** = 1,030 ÷ 20 =** 51.5 kg**

**Median**

There are 20 children, so the '**middle person**' will be the (20 + 1) ÷ 2 = **10.5th person**, i.e. the value between the 10th and 11th.

There are 2 people in the 40 ≤ m < 45 class.

Then the next 9 (i.e. the 3rd to 2 + 9=11th) are in the 50 ≤ m < 55 class.

So the 10th and 11th person both fall into the 50 ≤ m < 55 class.

Since we have grouped data and so don't know the exact values, we have the **median class**

__:__50 ≤ m < 55 kg

**Mode**

This is the class with the highest frequency.

We can see that the highest frequency here is 9 which corresponds to 50 ≤ m < 55 kg

So the **modal class** is 50 ≤ m < 55 kg

Let's grab a calculator and have a go at some questions!

If you get stuck, remember you can always click on the red button at the side of the screen and it will show you this introduction again.