Mode for the grouped data

This article will be about how to calculate the mode for grouped data, with solved examples with the formula.

The mode in a data group is the number or variable that is the most repeated. when it comes to ungrouped data, we just have to see the frequency of each number or variable, and the variable that has the greater frequency is the mode, this changes when we work with grouped data, because when we work with grouped data there are no numbers to count how many times each number is repeated, instead the data is organized in intervals and the way the mode is found changes.

Mode for grouped data

An inconvenient when we find the mode for grouped data, just as the median, is that the mode is the number, variable, or answer that repeats the most, but we do not have any specific number, instead of that we have intervals, and we cannot count the number one by one, so, it is possible that the mode we calculate is not a number in the compiled data, it is also possible that the mode is indeed the most repeated number, but this is not going to be all the time because the mode in grouped data is just an estimation of the mode.

Another thing that changes in the mode for grouped data is that the mode can be found in ungrouped data with both qualitative(variables expressed by text) and quantitative data (variables expressed by numbers), but is not the same case with grouped data. This happens because in ungrouped data, we only have to count the frequency of a variable, so we do not need any mathematical process, this changes with the grouped data, because we are going to use a formula to define the mode, so we can only find the mode in quantitative variables.

Formula for mode for grouped data

  • Mode
  • Mode = Li +
    Δ1/Δ1 + Δ2
    * A
  • where
  • Δ1 = fi - fi-1
  • Δ2 = fi - fi+1

Now we are going to explain every variable of the formula of the mode for grouped data, the mode will always be in the interval with the highest frequency.

Fi is the frequency of the interval that contains the mode, fi-1 is the frequency of the previous interval, fi+1 is the frequency of the next interval and with this two variables we can find both Δ1 and Δ2 by using the previous formula.

Li is the lower limit of the interval that contains the mode, it is the lowest number of the interval, for example, for the interval ]30,40] the value of Li would be 30.

A is the width of an interval and it is calculated by subtracting the upper limit minus the lower limit, for example, in the interval ]3,9] the width would be 9-3=6.

Examples for mode for grouped data

Example 1: A poll was made to some people about how many times they ate fast food during last month, calculate the mode of the results.

As we said before the mode is in the interval that has the highest frequency.

(With * the mode interval)

Fast food / month Frequency
]5-10] 22
]10 - 15] *41
]15 - 20] 14
]20 - 25] 8

Now knowing the interval where the mode is located, we are going to find every variable of the formula, so we can solve the problem.

  • We find "A" of ]10 - 15]
  • A = 15 - 10
  • A = 5

Then, having the value of A, and also knowing that Li is 10 we would only have to find the deltas (Δ1 and Δ2)

  • we find Δ1
  • Δ1 = fi - fi-1
  • Δ1 = 41 - 22
  • Δ1 = 19
  • we find Δ2
  • Δ2 = fi - fi+1
  • Δ2 = 41 - 14
  • Δ2 = 27

And last we find the mode.

  • Mode
  • Mode = Li +
    Δ1/Δ1 + Δ2
    * A
  • Mode = 10 +
    19/19 + 27
    * 5
  • Mode = 10 +
    19/46
    * 5
  • Mode = 10 + 0.41 * 5
  • Mode = 12.1

Example 2: Calculate the mode of the following data that was compiled after someone asked a group of people how many hours do they sleep a day.

First we find the mode interval.

(With * the mode interval)

Sleeping time Frequency
]2-4] 4
]4 - 6] 87
]6 - 8] *271
]8 - 10] 29
  • we find "A" of ]6 - 8]
  • A = 8 - 6
  • A = 2

And now we calculate delta 1 and delta 2.

  • Finding Δ1
  • Δ1 = fi - fi-1
  • Δ1 = 271 - 87
  • Δ1 = 184
  • Finding Δ2
  • Δ2 = fi - fi+1
  • Δ2 = 271 - 29
  • Δ2 = 242

And lastly we calculate the mode.

  • Mode
  • Mode = Li +
    Δ1/Δ1 + Δ2
    * A
  • Mode = 6 +
    184/184 + 242
    * 2
  • Mode = 6 +
    184/426
    * 2
  • Mode = 6 + 0.43 * 2
  • Mode = 6.86

Example 3: Some people were asked about how many times they went to the movies in the last 6 months, calculate the mode of the results.

As the first step we are going to find the mode interval.

(With * The mode interval)

Times they wont to the movies Frequency
]0-2] 8
]2 - 4] 12
]4 - 6] *13
]6 - 8] 6
  • we find "A" of the interval ]4 - 6]
  • A = 6 - 4
  • A = 2

And then we calculate the values of delta 1 and delta 2

  • finding Δ1
  • Δ1 = fi - fi-1
  • Δ1 = 13 - 12
  • Δ1 = 1
  • finding Δ2
  • Δ2 = fi - fi+1
  • Δ2 = 13 - 6
  • Δ2 = 7

And for the last step we use the formula to calculate the mode

  • Mode
  • Mode = Li +
    Δ1/Δ1 + Δ2
    * A
  • Mode = 4 +
    1/1 + 7
    * 2
  • Mode = 4 +
    1/8
    * 2
  • Mode = 4 + 0.125* 2
  • Mode = 4.25

Related articles

Mean median and mode Mean for grouped data Median for grouped data Quartiles Standard deviation Frequency table