Mode Formula

Introduction

• The mode indicates the most frequently occurring value or number in the record. You may need to find values that occur more often in your dataset.

• In such cases, find the mode for the given dataset. A particular record may or may not have a modal value. For data with no-repeat values, there may be no mode at all.

• A mode is relevant when studying a list of numbers (such as the heights of all the children in a class).

• It is the numerical quantity that comes most frequently on the list.

• There can be a tie if multiple numbers appear with equal frequency. In situations of interest, you need a mode to get the average grade that most students in your class get.

• There are many real applications and it is important to use the mode value.

• There are many aspects where just finding the average (or average) does not work. Therefore, in such cases, you tend to use the mode. In this tutorial, we will discuss the mode formula.

Central Tendencies

• Central tendency measurements describe a data set by identifying the centre position in the data set as a single representative value

• The average is the most common measurement of the central tendency used to describe a dataset.

• Statistical measures that represent the entire distribution or individual values in the dataset are shown as central trends.

• The goal is to provide an accurate description of all the data in the distribution.

The central tendency of a dataset can be determined using three important indicators: mean, median, and mode.

Mode

• The mode indicates the most frequently occurring value or number in the record.

• You may need to find values that occur more often in your dataset.

• A particular record may or may not have a modal value.

• For data with no-repeat values, there may be no mode at all.

• Modes are a very convenient way to find categorical data.

• The mode can be easily determined even for datasets that do not contain numbers.

Mode of Ungrouped Data

• Ungrouped data is the data that is obtained in its original form, it is just a list of numbers.

• Finding an ungrouped data mode is as easy as arranging the data values in ascending or descending order and then finding the values that repeat and how often.

• The most frequent observation is the mode of the given data, which is referred to here as the mode.

• The most common value in a series of observations is the mode for ungrouped data.

• Calculation of Modes for ungroup Data − Find the most frequently occurring observations.

Mode of a Grouped Data

Grouped data is the data that is organized into particular groups, called classes.

With a grouped distribution, it is not possible to calculate modes based solely on frequency.

Calculate the modal class to determine the data mode in such cases. The mode is in the modal class.

Mode for grouped data −

$$\mathrm{mode= l+h×\frac{f_m-f_1}{(f_m-f_1)+(f_m-f_2)}}$$

l indicates the lower limit of the given modal class.

h indicates the size of the given class interval.

f_m indicates the frequency of the given modal class.

f₁ indicates the frequency of the given modal class that precedes it.

f₂ indicates the frequency of the given modal class that succeeds it.

To calculate the mode of grouped data −

Step1. First of all, find the highest frequency of the given class interval.

Step2. Check the size of the class.

Step3. Now we have to use the formula of mode

$$\mathrm{mode= l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h}$$

Solved Examples

Example1: Which number is the mode from 1,2,3,4,5,1,6?

Solution: 1 is mode from these numbers because this is the only number from here that repeats the maximum time.

Example 2: What will be the mode of the data given below.

Bike colours	Black	Blue	Green	White	Red
Number of bikes	11	13	21	6	12

Solution: This is ungrouped data, it is clearly visible that green bikes have the highest frequency as shown in the table. So, the mode is 21.

Example 3: An Organisation maintains its records as the find the modal class whose frequency is maximum.

Age of the patients	0-20	20-40	40-60	60-80
Number of patients	35	315	120	50

Solution: Maximum frequency = 315

Modal class = 20-40

Example 4: What will be the mode of the following data.

Age of patients (in years)	0-20	20-40	40-60	60-80
Number of patients	35 = f0	315 = f1	120 = f2	50

Solution: Therefore, l=20,f₁=315, f₀=35,f₂=120,And h=20

Now,

$$\mathrm{mode= l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h}$$

Now putting the values in the formula,

$$\mathrm{mode= 20+\frac{315-35}{2\times 315-35-120}\times 20}$$

$$\mathrm{mode= 20+\frac{280}{475}\times 20}$$

$$\mathrm{Mode= 31.79}$$

Example 5: What is the mode of the given data set? {2,6,8,4,9,10,16,2,18,2}

Mode is 2 because the 2 has the highest frequency.

Example 6: Find the mode in the following data.

class interval	0-9	10-19	20-29	30-39	40-49	50-59
frequency	12	15	21	17	19	6

Solution: As it is clearly visible that frequency distribution is not continuous.

So, our first step is to make frequency distribution continuous. This is done by adding 0.5 in the upper limit and subtracting 0.5 in the lower limit.

Class interval	Continuous class interval	Frequency
0-9	-0.5-9.5	12
10-19	9.5-19.5	15
20-29	19.5-29.5	21
30-39	29.5-39.5	17
40-49	39.5-49.5	19
50-59	49.5-59.5	6

$$\mathrm{l=19.5}$$

$$\mathrm{f= 21}$$

$$\mathrm{f_0= 15}$$

$$\mathrm{f_2= 17}$$

$$\mathrm{h= 10}$$

$$\mathrm{Now, mode= l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h}$$

$$\mathrm{mode= 19.5+\frac{21-15}{42-15-17}\times 10}$$

$$\mathrm{mode = 19.5+6=25.5}$$

Conclusion

The mode of statistics or the mode of probability usually indicates the most frequently occurring value or number in the record. Therefore, as an example, when acquiring a mode, the mode has the highest number of occurrences after counting the number of occurrences of each number. If there are multiple numbers, then the number with the highest frequency will be the mode. But in the case of grouped data, we cannot find the mode simply by looking. So, here we use a formula that is a mode formula.

To calculate the mode of ungrouping data: Find the most common observations. And For the grouped data, the mode is

$$\mathrm{mode= l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h}$$

FAQs

1.What does the mode indicate?

The mode indicates the most frequently occurring value or number in the record.

2.How can we calculate the mode of ungrouped data?

Ungrouped data modes can be found by selecting the most common item in the data.

3.What do you mean by central tendency?

Central tendency measurements describe a data set by identifying the centre position in the data set as a single representative value. In general, the central tendency commonly used in statistics has three indicators: mean, median, and mode.

4.what do you mean by grouped data?

Grouped data is the data that has been organized into several groups, called classes.

5.what do you mean by ungrouped data?

Ungrouped data is the data that is obtained in its original form, it is just a list of numbers.