February 18, 2020

Chapters

## Frequency Part 1

In this section, you will learn about one of the most important basic concepts in statistics: Frequency. To introduce this concept, we will go over the definition of frequency, how to calculate it and why it is important. In order to understand what a frequency is, you won’t need any prior knowledge about other statistical concepts

## What is Frequency?

If you’ve ever complained about the rising price of amusement park, concert, or movie tickets, you’re not alone. From 2018 to 2019, the inflation rate, or the change in price level, in the UK was 1.8% and an average of 2.3% in the last 10 years. An increase in the price level can explain why many expenses, such as concert or movie tickets, get more expensive.

Another possible explanation lies in statistics - specifically, in the concept of **frequency**. Have you ever wondered why many cinemas around the world raise prices for movies in the evening? Hiking up prices for evening shows reflects a high demand for later time slots, while lower demand for morning shows is reflected in the lower prices.

One way demand can be measured is by looking at the frequency of attendance at different times. The **definition** of frequency is simply how often something occurs and is measured by counting the number of times something happens. While there are a number of different ways you can capture frequency, the two most common ways are:

- Frequency
- Relative Frequency
- Cumulative Frequency

If we wanted to estimate the demand for movie times, we could look at the frequency of attendance at each time slot offered. Most probably, we would see a larger frequency for the evening than for the morning, meaning that the count, or amount, of people attending evening shows is greater than the count for morning shows.

## Calculating Frequency

One of the most useful and basic tools in statistics is the frequency of a variable. Telling us how often something occurs, frequency is also one of the easiest measures to calculate. To calculate the frequency, you simply **count how many times** a certain value or category occurs. This is also known in statistics as the “count” of a value or category.

### Frequency for Quantitative Variables

While the process of finding the frequency for all types of variables is the same, understanding the differences in the way these are presented and interpreted can be different.

Frequencies are often displayed in what is known as a **frequency chart**. A frequency chart displays each value or category and its frequency. Below, you’ll find an example of a frequency chart for the cinema goers between 11 and 20 years of age of cinema on a particular day.

Observation | Age | Frequency |

1 | 11 | 0 |

2 | 12 | 1 |

3 | 13 | 0 |

4 | 14 | 3 |

5 | 15 | 2 |

6 | 16 | 5 |

7 | 17 | 15 |

8 | 18 | 10 |

9 | 19 | 17 |

10 | 20 | 23 |

We arrived at each frequency by simply counting the number of people who arrived at the cinema and watched a movie for each age. For the first observation, for example, there were no 11-year olds who watched a movie the day we collected the data.

### Frequency for Qualitative Variables

Finding the frequency for a **qualitative variable** follows the same procedure as for quantitative variables. The only difference, however, is that we count how many times something occurs for each different category.

Let’s say, for example, that instead of wanting to collect and display data for each and every age, we simply wanted to know the frequencies for age groups. An age group would be an **ordinal variable** because, as we learned in previous sections, it can be ordered on a scale based on age.

For each age group, we would simply count how many people went to the cinema that day. Our frequency chart would then look something like the one below.

Ages | Age Group | Frequency |

10 and below | Children | 3 |

11-20 | Teenagers | 32 |

21-30 | Young Adults | 49 |

31-50 | Adults | 72 |

50 and above | Seniors | 35 |

For each age category, we simply counted the number of people that went to the cinema. While age is a quantitative variable, discrete because we are dealing with approximate ages, we can group each age into categories instead. This, apart from creating a new variable that is qualitative, results in a frequency chart that is easier to understand.

Above, we can see that the age group with the **highest frequency**, or the largest number of people, is adults.

## Calculating Relative Frequency

Relative frequency is a type of frequency. Instead of reflecting the total number of times something occurs, it tells us about the **proportion** of times something occurs. Meaning, it is the frequency of a value or category in relation to, or relative to, the total frequency.

Relative frequency is calculated by taking the frequency of a particular value or category and dividing it by the total count, which is mostly always equal to the sample size. Again, the process of calculating the frequency for quantitative variables is the same as for qualitative variables. The only** difference** is the length and appearance of the frequency chart.

Calculating relative frequency is best demonstrated by an example. Imagine you’re at a restaurant with five friends. Each of you have the option to buy a hamburger, stew or stir-fry. The frequency chart for this outing looks like:

Dish | Frequency |

Hamburger | 1 |

Stew | 2 |

Stir-fry | 2 |

Total | 5 |

We would then calculate the relative frequency as the frequency over the total sample size.

Dish | Frequency | Relative Frequency | Relative Frequency (%) |

Hamburger | 1 | ⅕ = 0.2 | 20% |

Stew | 2 | ⅖ = 0.4 | 40% |

Stir-fry | 2 | ⅖ = 0.4 | 40% |

Total | 5 | 1 | 100% |

As you can see, we can also express the relative frequency of a variable as a percentage. Often, the relative frequency as a percentage is preferred. Here, we can interpret the results of this outing as: 20% of the group ordered a hamburger, 40% of the group ordered stew and 40% ordered stir-fry.

## Calculating Cumulative Frequency

Cumulative frequency is easy to remember because it is the cumulation, or the addition, of each relative frequency. It tells us the cumulative occurrence of a variable and is calculated **by adding** each previous frequency. Below, you’ll find two different ways of expressing the same calculation of cumulative frequency using the same example as above.

Dish | Frequency | Cumulative Frequency | Relative Frequency | Cumulative Frequency (using relative frequency) | Cumulative Frequency (%) |

Hamburger | 1 | ⅕ = 0.2 | ⅕ = 0.2 | 0.2 | 20% |

Stew | 2 | 1+ ⅖ = 0.6 | ⅖ = 0.4 | 0.2+0.4 = 0.6 | 60% |

Stir-fry | 2 | (1+2+2)/5 = 1 | ⅖ = 0.4 | 0.6+0.4 = 1 | 100% |

Total | 5 | 1 |

Using the cumulative frequency can be important if we want to know how the proportion of frequency changes between variables. Here, for example, we can see that 60% of the group ordered stew and hamburgers. Naturally, **100% of the group** ordered one of the categories.

It’s important to note we are allowed to change the order of presentation, but we if we do, we must also change our interpretation. For example, we instead order our table like this:

Dish | Frequency | Relative Frequency | Cumulative Frequency (using relative frequency) | Cumulative Frequency (%) |

Stir-fry | 2 | 0.4 | 0.4 | 40% |

Stew | 2 | 0.4 | 0.4+0.4 = 0.8 | 80% |

Hamburger | 1 | 0.2 | 0.8+0.2 = 1 | 100% |

Total | 5 | 1 |

Now, the cumulative frequencies have changed from our previous results, although the last row should always be 100% because it encapsulates all counts. Our interpretation now is that 80% of the group ordered stir-fry and stew.