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A frequency distribution refers to the presentation of statistical data in a tabular format to simplify data analysis. In a frequency distribution, data is subdivided into groups or intervals.
The standard procedure for constructing a frequency distribution involves the following steps:
Points to note:
You have been given the following data showing the percentage returns that certain classes of investment offer in a year. Use the data to construct a frequency distribution table.
$$ \begin{array}{c|c|c|c|c} \text{-10%} & \text{2%} & \text{32%} & \text{-28%} & \text{25%} \\ \hline \text{-25.60%} & \text{4%} & \text{11%} & \text{-14%} & \text{15%} \\ \hline \text{23%} & \text{13%} & \text{6%} & \text{-2.70%} & \text{8%} \\ \hline \text{12%} & \text{28%} & \text{17.50%} & \text{5.80%} & \text{20%} \\ \hline \text{4.60%} & \text{17%} & \text{-3.90%} & \text{22.40%} & \text{15%} \\ \end{array} $$
Solution
$$ \begin{array}{c|c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} \\ \hline -30\% \leq R_t < -20\% & \text{II} & \text{2} \\ -20\% \leq R_t < -10\% & \text{I} & \text{1} \\ -10\% \leq R_t < 0\% & \text{III} & \text{3} \\ 0\% \leq R_t <10\% & \text{IIIIII} & \text{6} \\ 10\% \leq R_t < 20\% & \text{IIIIIII} & \text{7} \\ 20\% \leq R_t < 30\% & \text{IIIII} & \text{5} \\ 30\% \leq R_t < 40\% & \text{I} & \text{1} \\ \textbf{Total} & \text{} & \textbf{25} \\ \end{array} $$
Absolute frequency is the actual number of observations in a given interval.
Relative frequency refers to the percentage of observations falling within a given class. It is calculated by dividing the absolute frequency of each return interval by the total number of observations. Using our earlier example when we introduced the frequency distribution table, we could come up with the relative frequency for each interval using the formula below:
$$\text{Relative Frequency}=\frac{\text{Absolute frequency}}{\text{Total frequency}}$$
Where \(\text{Total frequency}\) is the total number of observations.
$$
\begin{array}{c|c|c|c}
\textbf { Interval } & \textbf { Tally } & \textbf { Frequency } & \textbf { Relative Frequency } \\
\hline-30 \% \leq \mathrm{R}_{\mathrm{t}} \leq-20 \% & \text { II } & 2 & \frac{2}{25}=8 \% \\
-20 \% \leq \mathrm{R}_{\mathrm{t}} \leq-10 \% & \text { I } & 1 & \frac{1}{25}=4 \% \\
-10 \% \leq \mathrm{R}_{t} \leq 0 \% & \text { III } & 3 & \frac{3}{25}=12 \% \\
0 \% \leq \mathrm{R}_{t} \leq 10 \% & \text { IIIII } & 6 & \frac{6}{25}=24 \% \\
10 \% \leq \mathrm{R}_{t} \leq 20 \% & \text { IIIIII } & 7 & \frac{7}{25}=28 \% \\
20 \% \leq \mathrm{R}_{t} \leq 30 \% & \text { IIII } & 5 & \frac{5}{25}=20 \% \\
30 \% \leq \mathrm{R}_{t} \leq 40 \% & \text { I } & 1 &\frac{1}{25}=4 \% \\
\text { Total } & & 25 & \frac{25}{25}=100 \%
\end{array}
$$
In the above table, the absolute frequency of the 1st interval is 2. Similarly, the relative frequency of the 1st interval is 8%. The same applies to other intervals.
Cumulative absolute frequency is the sum of the absolute frequencies, including the given interval.
Cumulative relative frequency similarly sums up the relative frequencies up to and including the given relative frequency.
$$
\begin{array}{c|c|c|c|c|c}
\textbf { Interval } & \textbf { Tally } & \textbf { Frequency } & \textbf { Relative Frequency } & \begin{array}{c}
\textbf { Cumulative Absolute } \\
\textbf { Frequency }
\end{array} & \begin{array}{c}
\textbf { Cumulative Relative } \\
\textbf { Frequency }
\end{array} \\
\hline-30 \% \leq R \leq-20 \% & \text { II } & 2 & \frac{2 }{25}=8 \% & 2 & 8 \% \\
-20 \% \leq R_{4} \leq-10 \% & \text { I } & 1 & \frac{1 }{25}=4 \% & 3 & 12 \% \\
-10 \% \leq R_{1} \leq 0 \% & \text { III } & 3 & \frac{3}{ 25}=12 \% & 6 & 24 \% \\
0 \% \leq R_{1} \leq 10 \% & \text { IIIII } & 6 & \frac{6}{25}=24 \% & 12 & 48 \% \\
10 \% \leq R \leq 20 \% & \text { IIIII } & 7 & \frac{7}{25}=28 \% & 19 & 76 \% \\
20 \% \leq R \leq 30 \% & \text { IIII } & 5 & \frac{5}{25}=8 \% & 24 & 96 \% \\
30 \% \leq R \leq 40 \% & \text { I } & 1 & \frac{1}{25}=4 \% & 25 & 100 \% \\
\text { Total } & & 25 & \frac{25}{25}=100 \% & &
\end{array}
$$
In the above table, 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.
Question 1
The class frequency divided by the total number of observations is most likely called:
- Relative frequency.
- Percentage frequency.
- Cumulative relative frequency.
Solution
The correct answer is A.
Relative frequency refers to the percentage of observations falling within a given class. It is calculated by dividing the absolute frequency of each return interval by the total number of observations.
C is incorrect. Cumulative relative frequency sums up the relative frequencies up to and including the given relative frequency.
B is incorrect. There is no such term as percentage frequency.
Question 2
The number of tally sheet count for each value or a group is most likely known as:
- Class limit.
- Frequency.
- Class width.
Solution
The correct answer is B.
The frequency of a class is the number of data entries in the class.
A is incorrect. Each class will have a “lower-class limit” and an “upper-class limit”, which are the lowest and highest numbers in each class.
C is incorrect. The “class width” is the distance between the lower limits of consecutive classes.