# The Relative Frequency And Cumulative Frequencies

Relative frequency refers to the percentage of observations falling within a given class. It reveals the popularity of certain classes of data based on a sample. In other words, relative frequency tells us the number of times an event occurs relative to the total number of events. With reference to the example we used earlier in the introduction of the frequency distribution table, we could come up with the relative frequency for each interval using the formula:

Relative frequency = Absolute frequency / Total frequency

Total frequency is just the total number of observations.

## Example 1: Relative Frequency

$$\begin{array}{c|c|c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} & \textbf{Relative Frequency} \\ \hline -30\% \leq R_t \leq -20\% & \text{II} & \text{2} & \text{= 2/25 = 8%} \\ -20\% \leq R_t \leq -10\% & \text{I} & \text{1} & \text{= 1/25 = 4%} \\ -10\% \leq R_t \leq 0\% & \text{III} & \text{3} & \text{= 3/25 = 12%} \\ 0\% \leq R_t \leq 10\% & \text{IIIIII} & \text{6} & \text{= 6/25 = 24%} \\ 10\% \leq R_t \leq 20\% & \text{IIIIIII} & \text{7} & \text{= 7/25 = 28%} \\ 20\% \leq R_t \leq 30\% & \text{IIIII} & \text{5} & \text{= 20/25 = 8%} \\ 30\% \leq R_t \leq 40\% & \text{I} & \text{1} & \text{= 1/25 = 4%} \\ \textbf{Total} & \text{} & \textbf{25} & \text{= 25/25 = 100%} \\ \end{array}$$

Furthermore, we could come up with cumulative frequencies. The cumulative absolute frequency is the sum of the absolute frequencies up to and including the given interval. The cumulative relative frequency similarly sums up the relative frequencies up to and including the given relative frequency.

## Example 2: Cumulative Frequencies

$$\begin{array}{c|c|c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} & \textbf{Relative Frequency} & \begin{array}{c} \textbf{Cumulative absolute} \\ \textbf{frequency} \end{array} & \begin{array}{c} \textbf{Cumulative relative} \\ \textbf{frequency} \end{array} \\ \hline -30\% \leq R_t \leq -20\% & \text{II} & \text{2} & \text{= 2/25 = 8%} & 2 & \text{8%} \\ -20\% \leq R_t \leq -10\% & \text{I} & \text{1} & \text{= 1/25 = 4%} & 3 & \text{12%} \\ -10\% \leq R_t \leq 0\% & \text{III} & \text{3} & \text{= 3/25 = 12%} & 6 & \text{24%} \\ 0\% \leq R_t \leq 10\% & \text{IIIIII} & \text{6} & \text{= 6/25 = 24%} & 12 & \text{48%} \\ 10\% \leq R_t \leq 20\% & \text{IIIIIII} & \text{7} & \text{= 7/25 = 28%} & 19 & \text{76%} \\ 20\% \leq R_t \leq 30\% & \text{IIIII} & \text{5} & \text{= 20/25 = 8%} & 24 & \text{96%} \\ 30\% \leq R_t \leq 40\% & \text{I} & \text{1} & \text{= 1/25 = 4%} & 25 & \text{100%} \\ \textbf{Total} & \text{} & \textbf{25} & \text{= 25/25 = 100%} & \text{} & \text{} \\ \end{array}$$

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