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Frequency, Frequency Tables and Levels of Measurement

- Calculus
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- Accumulation Function
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- Frequency, Frequency Tables and Levels of Measurement
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- Statistics
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- Types of Data in Statistics
- Venn Diagrams

Have you ever wondered how your teacher may keep track of the results from a test? Or maybe how they may record the information to find out the most common results? When data is collected it can be useful to sort it and display it in a way that is easy to read. This is sometimes known as a way of representing data. One way of representing data is through a **frequency table**.

Frequency is a term that describes the number of times a value occurs in a collection of data.

A frequency table is a type of table used to represent the different frequencies from a collection of data.

Steve flips a coin 5 times to find out whether it will land showing heads or tails. He records the results of each flip in a table.

Flip | Heads or Tails |

1 | Heads |

2 | Tails |

3 | Heads |

4 | Tails |

5 | Tails |

Steve flips the coin another 5 times.

Flip | Heads or Tails |

1 | Heads |

2 | Tails |

3 | Heads |

4 | Heads |

5 | Heads |

We can now place the results from both tests into one frequency table:

Heads or Tails | Tally | Frequency |

Heads | 6 | |

Tails | 4 |

This tells us that out of the 10 coin tosses Steve made, 6 landed with heads facing up and 4 with tails facing up.

A frequency table can be useful when you have a large amount of data, this is because it groups the data into groups making it easier to understand.

A frequency table can look like this:

Variable | Tally | Frequency |

1 - 5 | 5 | |

6 - 10 | 3 | |

11 - 15 | 7 |

In order to create a frequency table, you must first look at your data and create suitable groups. Your group size may depend on the number of variables you have. Once you have grouped your data you can tally the number of times each variable occurs to find out the frequency. Let's have a look at an example:

A year 7 class has completed a maths test. The results, as a percentage, from the 25 students have been recorded below:

63 | 73 | 39 | 45 | 82 |

28 | 77 | 64 | 35 | 47 |

92 | 57 | 85 | 53 | 62 |

66 | 59 | 46 | 91 | 82 |

74 | 46 | 28 | 46 | 90 |

In order to put this into a frequency table, you first need to find the group sizes, the smallest variable is 28 and the biggest variable is 92. This data could be split into 7 groups:

Test Results (%) |

25 - 35 |

36 - 45 |

46 - 55 |

56 - 65 |

66 - 75 |

76 - 85 |

86 - 95 |

Now you can go through the data and mark a tally where the variables fall into each group:

Test Results (%) | Tally |

25 - 35 | |

36 - 45 | |

46 - 55 | |

56 - 65 | |

66 - 75 | |

76 - 85 | |

86 - 95 |

Finally, you can add the next column, the frequency for each of the groups:

Test Results (%) | Tally | Frequency |

25 - 35 | 3 | |

36 - 45 | 2 | |

46 - 55 | 5 | |

56 - 65 | 5 | |

66 - 75 | 3 | |

76 - 85 | 4 | |

86 - 95 | 3 |

From this frequency table, you can now understand the data much easier. Just by taking a quick look, you can easily see that 5 students scored between 46% and 55% on the maths test.

Sometimes, there may only be a small amount of variables, if this is the case there may not be a need to group them. For example, if you are looking at the number of siblings people have and the variables range from 0 - 4, you could just look at each variable separately.

A group of students records the number of cups of coffee they drink per day. The results are shown below:

2 | 3 | 6 | 1 | 0 |

3 | 4 | 2 | 2 | 3 |

5 | 1 | 1 | 4 | 2 |

7 | 3 | 2 | 5 | 1 |

To put this into a frequency table, you first need to find the group sizes, the smallest variable is 0 and the biggest variable is 6:

Number of coffees |

0-1 |

2-3 |

4-5 |

6-7 |

Number of coffees | Tally |

0-1 | |

2-3 | |

4-5 | |

6-7 |

Finally, you can add the next column, the frequency for each of the groups:

Number of Coffees | Tally | Frequency |

0-1 | 5 | |

2-3 | 9 | |

4-5 | 4 | |

6-7 | 2 |

By creating this frequency table you can now much easier read that the most common amount of coffees that a group of students drinks in a day is between 2 and 3.

The level of measurement tells you how data is recorded. There are four different levels of measurements used in statistics:

- Nominal
- Ordinal
- Interval
- Ratio

The nominal level of measurement is data that can be categorised but the data has no order.

Place of birth, eye colour, gender.

The ordinal level of measurement is data that can be categorised and ordered. Although there is an order between the data, you cannot see the intervals between each variable.

Satisfaction survey, top 5 goal scorers.

The interval level of measurement is data that can be categorised and ranked and there is a scale on which you know the difference between each of the variables.

Temperature, test scores.

The ratio level of measurement gives an order to the variables, where there is a difference between the variables, as well as a true zero point.

Height, age.

The true zero indicates that there is an absence of something on the scale.

Data is taken from a class to find out the most common hair colour. The results collected can be considered as the **nominal** level of measurement since the data can be categorised.

The weights of different piles of sugar are recorded. One result shows 0 kg. This can be considered the **ratio** level of measurement since there is a chance of a true zero within the results.

A group of people is asked what their highest level of education is. The results collected from this can be considered as the **ordinal** level of measurement since they can be categorised and ordered.

The temperature is recorded over a period of a week. When these results are collected the data can be considered as the **interval** level of measurement since the temperature has evenly spaced intervals.

- Frequency tables can be used to display data and help to easily understand data.
- To create a frequency table you must first group your data, before tallying up the amount of variables that fall into each group.
- There are four levels of measurement:
- Nominal
- Ordinal
- Interval
- Ratio

A frequency table is a way of representing a collection of data.

- Nominal
- Ordinal
- Interval
- Ratio

There are two types of frequency tables; grouped data and ungrouped data frequency tables.

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