Frequency

In Statistics, Frequency can be defined as any number that indicates the times that “something” such as a value, word, or category exists in a set. It is denoted as (Frequency). There is the Simple Frequency, the Relative Frequency, and the Cumulative Frequency, as well its combinations e.g. Cumulative Relative Frequency.

Simple Frequency

Simple Frequency is the number that is produced by (simply) counting the times that “something” such as a value, word, or category exists in a set.

**Example**

In The previous picture, the word “Statistics” appears 7 times. Therefore, it can be said that its Simple Frequency is 7: .

In the next picture, we can count:

i) 6 Breaks, that is, its Simple Frequency is: .

ii) 4 Stars, that is, its Simple Frequency is: .

iii) 1 Sun, that is, its Simple Frequency is: .

iv) One “point”, that is, its Simple Frequency is: .

This is the Simple Frequency

Cumulative Frequency

The Cumulative Frequency is the summation of Simple Frequencies. In each step, the Cumulative Frequency is this number that is resulted when the “previous” Simple Frequencies are added. The word “previous” suggests that Simple Frequencies have been “somehow” ordered, -in a random way or not. The Cumulative Frequency can be denoted as: or as: .

**Example**

We have randomly ordered our Simple Frequencies, one below the other. Therefore, we can now calculate the Cumulative Frequency in each step, which :

i) In 1st case is 6. Here, we do not have previous Simple Frequency values, therefore:

ii) In 2nd case is 10. Here, we add the “current” Simple Frequency (4) to the previous result (6):

iii) In 3rd case is 11. Here, we add the “current” Simple Frequency (1) to the previous result (10):

iii) In 4th case is 12. Here, we add the “current” Simple Frequency (1) to the previous result (11):

The final Cumulative Frequency expresses the Total number of Observations that exist in this picture, which is 12.

Relative Frequency

The Relative Frequency is the Simple Frequency expressed as a Percentage, based on the total number of observations described by the Suimple Frequencies in a set. It can be denoted as: .

**Example**

The Relative Frequency can be calculated for the:

i) 1st case as: . Indeed, these 6 observations consist the half part of the total observations (12).

ii) 2nd case as: .

iii) 3rd case as: .

iii) 4th case as: .

Cumulative Relative Frequency

The Cumulative Relative Frequency is the summation of the Simple Frequencies. In each step, the Cumulative Relative Frequency is this number that is resulted if we add all the “previous” Relative Frequencies. The word “previous” suggests that Simple Frequencies have been “somehow” ordered, -in a random way or not. The Cumulative Relative Frequency can be denoted as: or as: .

**Example**

We have randomly ordered our Simple Frequencies, one below the other. Therefore, we can now calculate the Cumulative Relative Frequency in each step, which :

i) In 1st case is 50%. Here, we do not have previous Relative Frequency values, therefore:

ii) In 2nd case is 83.3%. Here, we add the “current” Relative Frequency (33.3%) to the previous result (50%):

iii) In 3rd case is 91.6%. Here, we add the “current” Relative Frequency (8.3%) to the previous result (83.3%):

iii) In 4th case is 100%. Here, we add the “current” Relative Frequency (8.3%) to the previous result (91.6%):

The final value of Cumulative Relative Frequency represents the Total Percentage of all observations of this picture. Therefore, it must be equal to 100%. *Subnote: Because values has been rounded, some decimal were “lost” in the way. This had as a result, the addition of all Relative Frequencies to be equal to 99.9% and not to 100%. *