Frequency Distribution Formula

Frequency distribution formula and the table is usually a strong tool in statistical maths in order to determine the correct number of observations and their distribution within the given integrals by the graphical or tabular format of display. Thud this entire article is based on the frequency distribution formula and determining group data formula.

What is a Frequency Distribution?

In statistics, the frequency distribution is a tool to display the outcomes of the various samples in form of lists, graphs or tables where the tabular format is vastly applied and is the most common method of displaying numerous observations and their distribution in a given integral. Within a particular group or intervals, each entry from the sample in the table reflects the no. of counts or the frequency of the occurrence of the values in the intervals or the group mentioned. 

This tool is majorly used for the assessment of the similarities and the differences between the data frequencies using the statistical hypothesis testing that calculates the averages or the central tendency of a complete algorithm like mean, mode and median. The frequency distribution can be more clearly explained with the following table that elaborates the frequency of each value of the variants which is the number of students and their measurement of heights. The frequency could be organized in the following manner.

Example of Organized Frequency

Frequency Distribution Table 

A frequency distribution table is the easiest method to organize the frequency or the number of counts of the data in a given event in particular groups or intervals so that it can be more relatable and meaningful in nature. Thus it becomes easy to draw an outcome out of the organized data. It can be elaborated with an example.

As Ray is randomly throwing the dice multiple times and getting different numbers each time, he wants to now find out the frequency of getting each no. in the no. of throws he initiated. Thus a tabular format for the same will be as follows:-

Tabular Format of Frequency Distribution

With this tabular formula now it became very easy for Ray to understand the probability or frequency of him getting a six or a four with the help of three tabular categories. So it becomes an orderly format of the summarization of all the values. 

Construction of Frequency Distribution Formula

In the first tabular format, it was easy to distribute the students into groups in order to extract the exact frequency as it was for a single class but when there are multiple classes involved that could be a larger no. or a smaller number but to find out its distribution with unorganized data will be difficult. In such cases, the ideal number of classes can be determined by the following frequency distribution formula which is as follows:-

C (no. of classes) = 1 + 3.3 logn where(log is base 10) or alternatively the square root of frequency distribution formula is written as:

C = √n, where n is the total no. of observations of the data that has been distributed.

In order to calculate a range of data that is the minimum to the maximum range of heights of students in a different class, a different frequency distribution formula is used to determine the minimum to the maximum range of values that is also known as the width of the value. That can be expressed as follows:-

h = range/number of class (assuming that the class interval for all the classes are same)

Variance Formula For Grouped Data

The variance formula for group data of a population is given as follows:-

\[\sigma ^{2}=\frac{\Sigma f(m-\overline{x})^{2}}{n}\]

The sample variance formula is represented as follows:-

\[S^{2}=\frac{\Sigma f(m-\overline{x})^{2}}{n-1}\]

Where σ2 is variance

            f = frequency of the class

           m = midpoint of the class

           \[\overline{x}\] = mean of the data

             n = no. of items given 

This variance formula for grouped data formula.

The variance for the grouped data is expressed as follows:-can be further explained with the calculation of a grouped data given as the height of the trees as 3, 12, 94, 48, 72, 15.

1. The first step is to add all the numbers given as data

3 + 12 + 94 + 48 + 72+15 = 244

2. Squaring of the summed answer.

244 x 244 = 59536

And then divide the square value with the no. of items which is 6 in this case

59536 / 6 = 9922.67

3. Take all the set of numbers from the first step and square them all and add the squares of all the values

3 x 3 + 12 x 12 + 94 x 94 + 48 x 48 + 72 x 72+15 x15

= 9 + 144 + 8836 + 2304 + 5184 + 225 = 16702

4. Subtract the result of step 2 from step 3

16702 - 9922.67 = 6779.33

5. Just subtract 1 from the total no. of items, that is 6 in this case

6 - 1 = 5

6. Now to get the variance, divide the value of step 4 with step 5

6779.33 / 5 = 1355.86

7. Now the square root of the value of step 6 will provide the value of the deviation

\[\sqrt{1355.86}\]=  36.82

Therefore the answer of the grouped data is 36.82 derived with the help of the group data formula.