In the previous section we completed the discussion on Median. In this section we will see Mode.
It can be explained as follows:
• In a data set, all the values may not be different. Some values occur more than once. That is.,
♦ Some values may occur twice.
♦ Some values may occur thrice.
♦ Some other values may occur four times . . . and so on.
We can not specify a limit. It depends on the situation
• So, from the data set, we take out the value that occur the most number of times. This value is the mode.
■ This can be put in another way:
We just mentioned that:
♦ Some values may occur twice.
♦ Some values may occur thrice.
♦ Some other values may occur four times . . . and so on.
• When such multiple occurrences happen, we know that, we must convert the raw data into a frequency table. Such a frequency table will directly give us the number of times each value occurs. So the frequency table makes our task of finding the mode easy. All we have to do is:
• Take out that value which has the largest frequency.
■ From the above discussion, we get another information:
Unlike mean and median, mode need not be a 'number'. It can be any item like chocolate ice cream, table, chair, cricket, soccer etc.,
■ Also note that, the 'number of times' or 'frequency' is not the mode. An observation in the data set is the mode.
Solved example 25.24
Find the mode of the following marks (out of 10) obtained by 20 students:
Solution:
The first step is to arrange the raw data in ascending or descending order. The ascending order is shown below:
From the above sorted data, we can easily form the frequency table. It is shown below:
From the above table, we find that the score '9' has the maximum frequency 4. So 9 is the mode
Solved example 25.25
In a small unit of a factory, there are 5 employees : a supervisor and four labourers. The labourers draw a salary of Rs. 5,000 per month each while the supervisor gets Rs. 15,000 per month. Calculate the mean, median and mode of the salaries of this unit of the factory.
Solution:
In this problem, there are only 5 observations. So we do not need to form a frequency table. The data set is:
5000, 5000, 5000, 5000, 15000
Calculation of mean:
• Sum of all the values = 5000 + 5000 + 5000 + 5000 + 15000 = 35000
• No. of observations = 5
• Mean = 35000/5 = Rs. 7000
Calculation of median:
There are 5 values. So n = 5. It is an odd number. We can use the procedure that we wrote earlier:
• So the 3rd value 5000 is the median
Calculation of mode:
The following number of goals were scored by a team in a series of 10 matches:
Find the mean, median and mode of these scores.
Solution:
Calculation of mean:
• Sum of all the values = 2 + 3 + 4 + 5 + 0 + 1 + 3 + 3 + 4 + 3 = 28
• No. of observations = 10
• Mean = 28/10 = 2.8
Calculation of median:
There are 10 values. So n = 10. It is an even number. We can use the procedure that we wrote earlier:
We have completed this discussion on Mean, Median and Mode. Part III of this discussion can be seen in chapter 37.
In the next Chapter we will see Arithmetic progressions.
Mode
Mode is that value in the data that occurs the most.It can be explained as follows:
• In a data set, all the values may not be different. Some values occur more than once. That is.,
♦ Some values may occur twice.
♦ Some values may occur thrice.
♦ Some other values may occur four times . . . and so on.
We can not specify a limit. It depends on the situation
• So, from the data set, we take out the value that occur the most number of times. This value is the mode.
■ This can be put in another way:
We just mentioned that:
♦ Some values may occur twice.
♦ Some values may occur thrice.
♦ Some other values may occur four times . . . and so on.
• When such multiple occurrences happen, we know that, we must convert the raw data into a frequency table. Such a frequency table will directly give us the number of times each value occurs. So the frequency table makes our task of finding the mode easy. All we have to do is:
• Take out that value which has the largest frequency.
■ From the above discussion, we get another information:
Unlike mean and median, mode need not be a 'number'. It can be any item like chocolate ice cream, table, chair, cricket, soccer etc.,
■ Also note that, the 'number of times' or 'frequency' is not the mode. An observation in the data set is the mode.
The ready made garment and shoe industries make great use of this measure of central tendency. Using the knowledge of mode, these industries decide which size of the product should be produced in large numbers.
Let us illustrate this with the help of an example:Solved example 25.24
Find the mode of the following marks (out of 10) obtained by 20 students:
The first step is to arrange the raw data in ascending or descending order. The ascending order is shown below:
Solved example 25.25
In a small unit of a factory, there are 5 employees : a supervisor and four labourers. The labourers draw a salary of Rs. 5,000 per month each while the supervisor gets Rs. 15,000 per month. Calculate the mean, median and mode of the salaries of this unit of the factory.
Solution:
In this problem, there are only 5 observations. So we do not need to form a frequency table. The data set is:
5000, 5000, 5000, 5000, 15000
Calculation of mean:
• Sum of all the values = 5000 + 5000 + 5000 + 5000 + 15000 = 35000
• No. of observations = 5
• Mean = 35000/5 = Rs. 7000
Calculation of median:
There are 5 values. So n = 5. It is an odd number. We can use the procedure that we wrote earlier:
1. Sort the list in ascending or descending order
2. Take out the value whose i = (n+1)⁄2
3. This value is the median
• The list sorted in ascending order is:
5000, 5000, 5000, 5000, 15000
• i = (n+1)⁄2 = (5+1)⁄2 = 6⁄2 = 3• The list sorted in ascending order is:
5000, 5000, 5000, 5000, 15000
• So the 3rd value 5000 is the median
Calculation of mode:
From the data, we find that the salary '5000' has the maximum frequency 4. So Rs. 5000 is the mode.
We have completed a basic discussion about mean, median and mode. But the details that we have seen so far is not sufficient to get an accurate result on the central tendency of a data set. We need to gain more knowledge. Then only we will be able to do many problems encountered in science, engineering, business administration, social sciences etc., We will learn more in higher classes.
Now we will see some solved examples
Solved example 25.26The following number of goals were scored by a team in a series of 10 matches:
Solution:
Calculation of mean:
• Sum of all the values = 2 + 3 + 4 + 5 + 0 + 1 + 3 + 3 + 4 + 3 = 28
• No. of observations = 10
• Mean = 28/10 = 2.8
Calculation of median:
There are 10 values. So n = 10. It is an even number. We can use the procedure that we wrote earlier:
1. Sort the list in ascending or descending order
2. Take out the value whose i = n⁄2
3. Take out the value whose i = (n+2)⁄2
4. Calculate the mean of the values in (2) and (3). This mean is the median of the whole list
• The list sorted in ascending order is:
The numbers in yellow colour shows the sequence 'i'
• i = n⁄2 = 10⁄2 = 5. So the first member of the middle pair = 3
• (n⁄2 + 1) = 6. So the second member of the middle pair = 3
• Median of the list = Mean of 3 and 3 = (3+3)⁄2 = 6⁄2 = 3
Calculation of mode:
Calculation of mean:
• Sum of all the values = 41 +39 +48 +52 +46 +62 +54 +40 +96 +52 +98 +40 +42 +52 +60 = 822
• No. of observations = 15
• Mean = 822/15 = 54.8
Calculation of median:
There are 15 values. So n = 15. It is an odd number. We can use the procedure that we wrote earlier:
• So the 8th value 52 is the median
Calculation of mode:
• (n⁄2 + 1) = 6. So the second member of the middle pair = x+2
• Median of the list = Mean of x and (x+2) = (x+x+2)⁄2 = (2x+2)⁄2 = x+1
• But the median is given as 63. So we can write:
x+1 = 63 ⇒ x = 63-1 = 62
Solved example 25.29
Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.
Solution:
Arrange the given data in ascending order:
2. Table below is prepared by expanding the given table
• The numerator in (1) is calculated at the bottom end of third column in the above table. It's value is 305000
• The denominator is calculated at the bottom end of second column in the above table. It's value is 60
3. So we get x = 305000/60 = 5083.33
• The list sorted in ascending order is:
• i = n⁄2 = 10⁄2 = 5. So the first member of the middle pair = 3
• (n⁄2 + 1) = 6. So the second member of the middle pair = 3
• Median of the list = Mean of 3 and 3 = (3+3)⁄2 = 6⁄2 = 3
Calculation of mode:
From the data, we find that the value '3' has the maximum frequency 4. So '3' is the mode.
Solved example 25.27
In a mathematics test given to 15 students, the following marks (out of 100) are recorded:
Find the mean, median and mode of this data.
Solution:Calculation of mean:
• Sum of all the values = 41 +39 +48 +52 +46 +62 +54 +40 +96 +52 +98 +40 +42 +52 +60 = 822
• No. of observations = 15
• Mean = 822/15 = 54.8
Calculation of median:
There are 15 values. So n = 15. It is an odd number. We can use the procedure that we wrote earlier:
1. Sort the list in ascending or descending order
2. Take out the value whose i = (n+1)⁄2
3. This value is the median
• The list sorted in ascending order is:
• i = (n+1)⁄2 = (15+1)⁄2 = 16⁄2 = 8
• The list sorted in ascending order is:
• So the 8th value 52 is the median
Calculation of mode:
From the data, we find that the value '52' has the maximum frequency 3. So '52' is the mode.
Solved example 25.28
The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x.
Solution:
The given data is already arranged in ascending order.
There are 10 values. So n = 10. It is an even number. We can use the procedure that we wrote earlier:
• i = n⁄2 = 10⁄2 = 5. So the first member of the middle pair = x• (n⁄2 + 1) = 6. So the second member of the middle pair = x+2
• Median of the list = Mean of x and (x+2) = (x+x+2)⁄2 = (2x+2)⁄2 = x+1
• But the median is given as 63. So we can write:
x+1 = 63 ⇒ x = 63-1 = 62
Solved example 25.29
Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18.
Solution:
Arrange the given data in ascending order:
From the data, we find that the value '14' has the maximum frequency 4. So '14' is the mode.
Solved example 25.30
Find the mean salary of 60 workers of a factory from the following table:
Solution:
1. We have:
• The denominator is calculated at the bottom end of second column in the above table. It's value is 60
3. So we get x = 305000/60 = 5083.33
In the next Chapter we will see Arithmetic progressions.
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