Understanding a Parameter, a Statistic, and Frequency Distribution Tables

A parameter refers to a measure that is used to describe the characteristic of a population.  It is a numerical quantity that describes a given aspect of the population as a whole. You should note that we are referring to the population as a whole, not a sample. Many parameters could be used to measure different characteristics of the population, but just a few of them are used regularly by investment analysts. These include the mean and the standard deviation of investment returns. (Examiners often set questions on these parameters.)

A statistic, on the other hand, is a measure that describes a characteristic of a sample. For example, this could be the average value or the sample standard deviation of the sampled items.

Note: researchers use sample statistics to estimate the unknown population parameters. For instance, we use the sample mean to estimate the actual population mean.

Frequency Distribution

A frequency distribution refers to the presentation of statistical data in a tabular format to simplify the analysis. The data is subdivided into groups or intervals. The standard procedure for constructing a frequency distribution involves the following steps:

1. determine the number of classes you wish to have. For example, 5 – 20 is always a good number;

2. determine the interval size. To do this, you should be guided by the range and the number of classes. The range is the difference between the smallest and the largest observations. In the case of a fractional result, you should take the next higher whole number as the size of the interval;

3. determine the starting point – it could be the lower limit of the lowest observation or a convenient value just below the lower limit;

4. add the class interval to the starting point to get the second lower limit and repeat this process;

5. list the lower limits in a vertical column alongside the upper-class limits; and

6. lastly, you can now complete the table by counting the number of observations that fall under each class.

Points to note: classes should be mutually exclusive and have similar widths. Also, you must tabulate all class intervals even if they have zero observations. The sum of the frequencies should be equal to the number of observations. Tally bars offer a convenient tool for visual presentation of the number of observations in each class.

Question

You have been given the following data that show the percentage returns offered by certain classes of investment in a certain 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

We have 25 observations in total. The range is 60% (- 28% to 32%). If we were to choose an interval of just 1%, we would end up with 60 intervals which would be too many. So, we can use an interval of 10%. The lowest return intervals will be -30% ≤ R_t < -20% and the highest one will be 30% ≤ R_t < 40%.

$$ \begin{array}{c|c|c} \textbf{Interval} & \textbf{Tally} & \textbf{Frequency} \\ \hline -30\% \leq R_t < -20\% & \text{II} & \text{2} \\ -20\% \leq R_t < -10\% & \text{I} & \text{1} \\ -10\% \leq R_t < 0\% & \text{III} & \text{3} \\ 0\% \leq R_t <10\% & \text{IIIIII} & \text{6} \\ 10\% \leq R_t < 20\% & \text{IIIIIII} & \text{7} \\ 20\% \leq R_t < 30\% & \text{IIIII} & \text{5} \\ 30\% \leq R_t < 40\% & \text{I} & \text{1} \\ \textbf{Total} & \text{} & \textbf{25} \\ \end{array} $$