Mode of Grouped Data

Mode of grouped data can be calculated using the mode formula. In the case of grouped frequency distribution, mode can't be obtained just by looking into the frequency, we first need to find out the modal class, in which lies the mode of the given data.

Let us learn about the mode of grouped data and how to find the mode of grouped data using the formula with solved examples and practice questions.

Mode is one of the measures of the central tendency of a given dataset which demands the identification of the central position in the data set as a single value. In the case of ungrouped data, the mode is simply the item having the greatest frequency. For grouped data, the mode is calculated using the formula,

How to find the Mode of Grouped Data?

First, assess the given data for grouped or ungrouped frequency distribution. In the case of grouped distribution, follow the steps as given below to find the mode:

• Step 1: Find the modal class, that is class interval with the maximum frequency.
• Step 2: Find the size of the modal class. (upper limit – lower limit.)
• Step 3: Calculate the mode using the mode formula, Mode = L + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right)\)h

Now, for any given data range, the mode can be calculated by the formula:

Mode = L + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right)\)h

where,

• L is the lower limit of the modal class
• h is the size of the class interval
• \(\mathrm{f}_{\mathrm{1}}\) is the frequency of the modal class
• \(\mathrm{f}_{\mathrm{0}}\) is the frequency of the class preceding the modal class
• \(\mathrm{f}_{\mathrm{2}}\) is the frequency of the class succeeding the modal class

In the case of the grouped data represented on the histogram, no individual values are there to be considered as mode and thus the modal class is taken up. Based on that, one finds the modal value.

Consider the graph given below.

• The frequency of the modal class = \(\mathrm{f}_{\mathrm{m}}\) or \(\mathrm{f}_{\mathrm{1}}\).
• The frequency of the preceding modal class = \(\mathrm{f}_{\mathrm{0}}\)
• The frequency of the class succeeding the modal class = \(\mathrm{f}_{\mathrm{2}}\)
• The lower limit of the modal class = \(I_{0}\).
• Thus, the mode is given by \(I_{0}\)+x.
• △AEB ~ △DEC. Thus, AB/CD = BE/DE = \(\frac{f_{1}-f_{0}}{f_{1}-f_{2}}\)
• △BEF ~ △BDC. Thus, FE/BC = BE/BD = \(\frac{f_{1}-f_{0}}{\left(f_{1}-f_{0}\right)+\left(f_{1}-f_{2}\right)}=\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\)
• FE = \(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}} \times B C=\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) f_{1}\)
• x = \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) f_{1}\)
• Thus, mode = \(I_{0}+x=I_{0}\) + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right) f_{1}\)

Related Topics

Given below is the list of topics related to the concept Mode of Grouped Data.

• Median
• Categorical Data
• Range in Statistics
• Discrete Data

Important Notes

Listed below are a few important points that help to summarize our learning on Mode of Grouped Data.

• For data that does not have any repeating numbers, modal value cannot be defined.
• In the case of ungrouped data, the mode can be found by observation, whereas for grouped data, the mode can be found using the formula.

What Is the Mode of Grouped Data in Statistics?

The mode is the value of the measure of central tendency giving us an idea about the most frequently occurring items in a data set.

What Is the Formula for Mode of Grouped Data?

For grouped data, the mode formula is given as, Mode = L + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right)\)h where, L is the lower limit of the modal class, h is the size of the class interval, \(\mathrm{f}_{\mathrm{1}}\) is the frequency of the modal class, \(\mathrm{f}_{\mathrm{0}}\) is the frequency of the class preceding the modal class, \(\mathrm{f}_{\mathrm{2}}\) is the frequency of the class succeeding the modal class.

How To Determine the Mode of Grouped Data?

To determine the mode of grouped data we assess the given data and look for the modal class. Accordingly, we find the other parameters to be used for the application of the mode formula, Mode = L + \(\left(\frac{f_{1}-f_{0}}{2 f_{1}-f_{0}-f_{2}}\right)\)h

How Do You Find the Mode of Grouped Data Graphically?

We can find the mode of a frequency distribution graphically using the histogram.

• Represent the given data in the form of a histogram where the height of the rectangles corresponds to the frequencies of the class interval.
• Identify the highest rectangle as it corresponds to the modal class.
• Join the top corners of the rectangle, representing the modal class with the immediate next corners of the adjacent rectangles such that the two lines cut each other.
• Draw a perpendicular line from that point of intersection onto the x-axis. The point where the perpendicular meets the x-axis gives the mode or the modal value.

How To Find Mode of Grouped Data with Unequal Classes?

To find the mode of the grouped data with unequal classes

• Calculate the heights of the respective class intervals by dividing the frequency of each interval by its height.
• Check out for the parameters as per the formula, Mode = L + H2 * H / (H1 + H2)
• Put the values in the formula.