Finding Mode: There are many places where people use the mode to ensure that there should not be any loss due to less or excess number of products or services. But how do they use the mode? They monitor the frequency of a particular product or service as per the data collected and provide the product or service accordingly. For doing this, they take help from mode. Have you ever wondered how you always get your favourite juice at the local shop?

By observing the frequency of sales, the shopkeeper refills the stock with him, and for doing this, the shopkeeper uses the mode. Companies and any other service provider, not just the shopkeeper, are also use mode to get to know the most frequent product or service they need to provide to the customer.

What is Central Tendency?

The central tendency is a measure that tells us where the middle value of a dataset lies. There are different measures of central tendencies as listed below.

• Mean is the average of the dataset. It is calculated by adding the value of each item in a group and dividing it by the total number of observations. 
• Median is the mid-point value of the dataset dividing the data into two halves with items in the group above and below it.
• Mode is one of the most useful measures of central tendency. We use mode to examine categorical data, such as models of mobile phones or flavours of drink, for which a mathematical average median value based on order cannot be calculated.

Definition of Mode

Mode is the value of the data that appears most frequently in a dataset. In other words, the data having the highest frequency in the dataset is known as the mode.

Any set of data may have one mode, two modes, three modes, more than three modes, or no mode (zero modes) at all.

• A set of data having two modes is known as bimodal.
• A set of data having three modes is known as trimodal.
• If all the numbers in a set occur exactly once, that set has no mode.

Note:
Any dataset with more than one mode is known as multimodal.

Mode of Ungrouped Data

Data given as individual points such as \(10,\,13,\,04,\,20,\,25,……\) etc. is called ungrouped data.

For ungrouped data, the mode is the value of the observation, which occurs the maximum number of times in the dataset.

We can also say that the data with the maximum frequency is known as the mode. 

Example: Mode of the dataset \(3,\,5,\,7,\,3,\,8,9,3,\,1,\,3\) is \(3\) as the frequency of \(3\) is \(3\) here.

Steps to Calculate Mode of Ungrouped Data

The steps to calculate the mode of ungrouped data are as follows:

• Step 1: Arrange the data in ascending or descending order.
• Step 2: Write the frequency of each data. This is done by counting the number of times a specific observation is repeated in the given dataset.
• Step 3: Identify the data with the maximum frequency as the mode of the dataset.

Mode of Grouped Data

The data that is given in the form of class intervals such as \(0 – 10,\,10 – 20,…..\) etc. is called grouped data.

We cannot determine the mode for such grouped data by looking at the frequencies. Here we have a class with the maximum frequency known as the modal class. Mode is the value of the modal class.

The formula of mode for grouped data is given below:
\({\text{Mode}}\,{\text{ = }}\,l + \left( {\frac{{{f_1} – {f_0}}}{{2{f_1} – {f_0} – {f_2}}}} \right)\, \times \,h\)
where
\(l\)= lower limit of the modal class,
\(h\)= size of the class interval (assuming all class sizes to be equal),
\({f_1}\)= frequency of modal class,
\({f_0}\)= frequency of the class preceding the modal class,
\({f_2}\)= frequency of the class succeeding the modal class.

Modal Class

The class with the maximum number of frequencies in grouped data is called the modal class.

Steps to Calculate Mode

The steps to calculate Mode are as follows:

• Step 1: Find the modal class of the given data.
• Step 2: Identify the values of \(l,\,{f_1},\,{f_0},\,{f_2}\) and \(h\) from the data.
• Step 3: Now substitute the values in the formula given above.
• Step 4: Simplify it to get the mode of the data.

Advantages of Mode

The advantages of Mode are as follows:

• Mode is one of the central tendencies that is easy to understand and calculate.
• Mode is not affected by extreme values.
• Mode is easy to identify in a dataset as it depends on the frequency of the observations.
• Mode is useful for qualitative data.
• Mode can be computed for the frequency table, which does to have a boundary.
• Mode can be located graphically.

Disadvantages of Mode

The disadvantages of Mode are as follows:

• Mode is not defined if there are no repeats in a dataset.
• Mode is not based on all values.
• For a small number of datasets, the mode is unstable.
• The number of modes is not fixed for all types of datasets. Sometimes it may have one mode, two or more modes, or no mode at all.

Real-life Uses of Mode in Different Situations

To better understand datasets, individuals and companies use these central tendencies all the time in different fields.

Here are some situations where mode was used in different fields.

1. Healthcare: Many health insurance companies use mode to calculate the age of their customers so that they can know which age group of people is using their insurance the most.
2. Real Estate: Real estate agents use mode to calculate the requirement of their customer in a particular area, i.e. how many bedrooms a customer wants in that area, so that they can construct flats or buildings as per requirements.
3. Marketing: Marketing people use mode to understand how much advertisement is required for a particular product and which mode of advertisement is best (i.e., TV, radio, newspaper, or digital).

Mode is a beneficial central tendency, and it has many uses in real life as it makes our work easier.

Remarks

• The mode is the most commonly observed value of a data set, and it is easy to calculate for ungrouped data as we only need to see the frequency of datasets.
• For many cases, the modal value of the data differs from the average value in the data.
• There is a relation between central tendencies, mean, median, and mode \(3\,{\text{Median}} – 2{\text{Mean}} = {\text{Mode}}\)

Solved Examples – Finding Mode

Q.1. The marks obtained out of \(100\) by \(50\) students in an English test are given in the frequency table below:

Marks	\(36\)	\(50\)	\(60\)	\(66\)	\(70\)	\(76\)	\(80\)
Remarks	\(10\)	\(8\)	\(12\)	\(6\)	\(7\)	\(5\)	\(2\)

Find the modal marks of the students.
Sol:
We know that the mode in a dataset is defined as the number that occurs most frequently in a dataset.
Here, \(36\) occurs \(10\) times, \(50\) occurs \(8\) times, \(60\) occurs \(12\) times, \(66\) occurs \(6\) times, \(70\) occurs \(7\) times, \(76\) occurs \(5\) times, and \(80\) occurs \(2\)times.
Therefore, \(60\) is the most repeated number.
Hence, according to the definition of mode, the modal marks of the students are \(60\).

Q.2. What is mode of \(30,\,11,\,32,\,7,\,25,\,13,\,22,\,28,\,40\,\)?
Sol:
We know that mode is the value in a dataset that appears the highest number of times.
In the given data all the numbers occur only once.
Hence, no number is repeated more than the other.
So, there is no mode for the given data.

Q.3. What is the mode of \(8,\,11,\,12,\,17,\,11,\,16,\,9,\,8\)?
Sol:
Mode is the value that appears the highest number of times.
In the above dataset, \(8\) and \(11\) repeated twice, and \(12,\,17,\,16,\,9\) appear once.
So, the mode of the data are \(8\) and \(11\).

Q.4. Which of the following datasets have the least mode?
Dataset A: \(15,\,35,\,25,\,18,\,28,\,32,\,25,\,35,\,25\)
Dataset B: \(35,\,75,\,43,\,49,\,73,\,18,\,46,\,25,\,18,\)
Sol:
Since \(25\) has occurred three times in the dataset \(A\) and other numbers less than that.
The mode of dataset \(A\) is \(25\).
Since \(18\) has occurred three times in the dataset \(B\) and other numbers less than that.
The mode of dataset \(B\) is \(18\).
So, dataset \(B\) has the least mode value i.e., \(18\).

Q.5. Find the mode of the frequency distribution:

Class interval	Frequency
\(1000 – 1500\)	\(20\)
\(1500 – 2000\)	\(36\)
\(2000 – 2500\)	\(29\)
\(2500 – 3000\)	\(24\)
\(3000 – 3500\)	\(26\)
\(3500 – 4000\)	\(18\)
\(4000 – 4500\)	\(12\)

Sol:
Calculation of mode:
Here, the class \((1500 – 2000)\) has maximum frequency.
Therefore, \((1500 – 2000)\) is the modal class.
Lower limit of the modal class \(\left( l \right)\, = 1500\)
Class size of the modal class \(\left( h \right)\, = 500\)
Frequency of the modal class \(\left( {{f_1}} \right) = \,36\)
Frequency of class preceding modal class \(\left( {{f_0}} \right) = \,20\)
Frequency of class succeeding modal class \(\left( {{f_2}} \right) = \,29\)
We now that,
\({\text{Mode}}\,{\text{ = }}\,l + \frac{{{f_1} – {f_0}}}{{2{f_1} – {f_0} – {f_2}}}\, \times \,h\)
\( = 1500 + \frac{{36 – 20}}{{2 \times 36 – 20 – 29}} \times 500\)
\( = 1500 + \frac{{16}}{{23}} \times 500\)
\( = 1500 + \frac{{8000}}{{23}}\)
\( = \frac{{34500 + 8000}}{{23}}\)
\( = 1847.83\)
Hence, mode of the given data is \(1847.83\)

Q.6. The data \(18,\,18,\,30,\,40,\,20,\,10,\,5,\,20\) and \(x\) has mode value \(20\). Find the missing value \(x\).
Sol:
We can write the given data in frequency distribution as:

Data	Frequency
\(18\)	\(2\)
\(30\)	\(1\)
\(40\)	\(1\)
\(20\)	\(2\)
\(10\)	\(1\)
\(5\)	\(1\)
\(x\)	\(1\)

For \(20\) to be a mode, frequency of \(20\) should be the highest.
Here, the frequency of \(18\) and \(20\) are same i.e.\(2\) .
Therefore, \(20\) will be the mode only if \(x=20\).

Summary

The central tendency is a measure that tells us where the middle of a dataset lies. There are three main measures of central tendency such as mean, median, and mode. Mode is the observation with the maximum frequency in the dataset. A set of data having two modes is known as bimodal, that with three modes is known as trimodal. If all the numbers in a set occur exactly once, then that dataset has no mode.

For ungrouped datasets, the mode is the value of the observation, which occurs maximum time in the dataset. For grouped datasets, mode \(=\) \(l + \frac{{{f_1} – {f_0}}}{{2{f_1} – {f_0} – {f_2}}}\, \times \,h\). Mode is used in healthcare, real estate, and marketing. The relation between mean, median, and mode is given by \(3\,{\text{Median}} – 2{\text{Mean}} = {\text{Mode}}\)

Frequently Asked Questions (FAQs)

Q.1. How do I calculate the mode?
Ans: For ungrouped data, write the data in ascending or descending order and then write their frequency with respect to their occurrence. Data with the maximum frequencies is a mode of the datasets.
For grouped datasets, use the formula given below:
\({\text{Mode}}\,{\text{ = }}\,l + \left( {\frac{{{f_1} – {f_0}}}{{2{f_1} – {f_0} – {f_2}}}} \right)\, \times \,h\)
where \(l\)= lower limit of the modal class,
\(h\)= size of the class interval (assuming all class sizes to be equal),
\({f_1}\)= frequency of modal class,
\({f_0}\)= frequency of the class preceding the modal class,
\({f_2}\)= frequency of the class succeeding the modal class.

Q.2. What is the multimodal mode?
Ans: A dataset having more than one mode is known as multimodal.

Q.3. What if there is no mode?
Ans: If all the numbers in a set occur exactly once, that set has no mode. In this case, we cannot use mode as a measure of central tendency.

Q.4. What is a modal class?
Ans: Modal class is one of the class intervals with the maximum number of frequencies.

Q.5. How do you find the mode if there are two?
Ans: If datasets have two observations that have the same highest frequency, then both the observations are the mode. A dataset can have more than one mode, and that with two modes is known as bimodal.

Learn About Mode of Ungrouped Data Here

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