Harmonic Mean for Grouped Data

Harmonic mean can be defined as the reciprocal of the arithmetic mean of the reciprocal of the observations. When data is collected in its original form, it is called raw data or ungrouped data. A frequency distribution is the organisation of raw data in table form using classes and frequencies. This process of organising the data makes the data more meaningful for its users. 

An ungrouped frequency distribution can be used for data that can be enumerated and when the range of values in the data set is not large. A grouped frequency distribution can be used when the range of the values in the given data set is very large. The data must be grouped into classes that are more than one unit in width.

Harmonic Mean for Grouped Data

Harmonic mean for grouped data can be calculated by dividing the sum of observation (∑f) with the sum of reciprocal of given observations multiplied by their respective frequencies (∑f/x).

Formula:

H.M. = ∑f / ∑f/x

Where ∑f = total number of observations(also called n)

and ∑f/x = sum of reciprocal of observations

and f = frequency of observations

Steps to Calculate Harmonic Mean for Grouped Data

We can find out the Harmonic Mean for Grouped Data in five simple steps as given below:

• Step 1:

In the first step, we need to calculate the sum of frequencies.

∑f = Total number of observations

• Step 2:

Now, we need to find the class mark in step two, which is represented by x. Class mark is also called the mid-value of the class interval.

Formula to calculate x = (lower value of class interval + upper value of class interval)/ 2

• Step 3:

In the next step, we are supposed to find the value of f/x. We are finding out the reciprocal of observations basically here.

• Step 4:

After finding out all the values of f/x, we will add all the values of f/x to get ∑f/x

• Step 5:

Put the values of ∑f from step 1 and ∑f/x from step 4 above to get the resulting value of Harmonic Value for the grouped data.

H.M. = ∑f / ∑f/x

Grouped and Ungrouped Data

Grouped Data: Data organised in the form of class intervals is called the grouped data. Each group represents a frequency corresponding to that particular group.

Ungrouped Data: Raw data or data lying unorganised is called the Ungrouped data.

Further, it is important to note that we use class boundaries to separate the classes, ensuring no gaps in the frequency distribution. The class width or interval for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class from the lower (or upper) class limit of the previous class.

Example:

In a class test, nine students obtained marks out of total marks 50 were as follows:

 25, 30, 35, 40, 26, 29, 50, 45

We can see that the given data here is not grouped in any way and is arranged randomly. Data here represents individual students’ information about marks obtained by them. Hence, this kind of data is called ungrouped data.

Now, if we write given data in the form of class intervals such as:

The number of students who got marks between 25-35 is = 5

So, here we have grouped the given data.

25-35 is the first group or the class interval, and the class interval represents the information specific to that group. Here group 25-35 shows that five students are there in the class who obtained marks between 25 and 35.

Say, we make one more class interval such as number of students who got marks between 36 and 50 are = 4

So, 36-50 is our next class interval, and four is the frequency for this class interval.

This kind of organisation of data is called the grouping of data.

25, 30, 35, 40, 26, 29, 50, 45

Also, remember, the formula to calculate the harmonic mean for ungrouped data is:

H.M. = ∑n / ∑1/x

Conclusion

The harmonic mean is one of the measures of central tendency and dispersion. Means, which are generally known as mathematical averages, can be subdivided into Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). The harmonic mean is generally used to calculate the average speed. For e.g., If we want to find out the average speed of a vehicle moving from one direction to the other, we use harmonic mean. The harmonic mean is given by the reciprocal of the arithmetic mean of the reciprocals of the observations. In this article, we discussed in detail the calculation of harmonic mean for grouped data. We hope this study material will be helpful for students seeking basic concept clarity on the given topic.