Chapter 25.10 - Mode of a Data

In the previous section we completed the discussion on Median. In this section we will see Mode.

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.

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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:

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:

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
• So the 3^rd 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.

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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.

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Now we will see some solved examples

Solved example 25.26
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:

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:

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

• So the 8^th 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:

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

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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.

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Chapter 25.9 - Median of a Data

In the previous section we completed the discussion on Mean. In this section we will see Median.

Median

We are going to learn about median and it's significance. But first we will learn how to calculate the median:
■ Any list of numbers will have a median. It is the 'middle number'.
■ If the list contains an odd number of numbers, one of the number already in the list will be the median. 
• This number will be situated at the exact middle, 
      ♦ when the whole list is sorted in ascending or descending order.
■ If the list contains an even number of numbers, the median will not be already present in the list. We will have to do some calculations to find it.
• This is because, there will be two numbers at the exact middle,
    ♦ when the whole list is sorted in ascending or descending order
• The median is the mean of those two numbers at the middle

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Let us elaborate on the above two cases with the help of examples: fig.25.22

Case 1: When the list has an odd number of numbers

1. Consider fig. 25.22(a) below. It has 13 (an odd number) values sorted in ascending order. 

Fig.25.22

2. Each can be denoted as x_i. So we have x₁, x₂, x₃, . . . , upto x_n. where n = 13

3. 'i' denotes the position of each value in the sorted list

4. We want the value of 'i' of the median. For calculating that value of 'i', we adopt the following procedure:

(a) There are equal number of values on either side of the median. Let this equal numbers be 'x'

(b) So we get: 

x + 1 ( the median) + x = n 

⇒ 2x + 1 = n. ⇒ 2x = n-1 ⇒ x = ^(n-1)⁄₂

(c) So there are ^(n-1)⁄₂ numbers on the left side of the median

(d) That means, the 'i' value of the number just to the left of the median = ^(n-1)⁄₂

(e) So the 'i' value of the median = ^(n-1)⁄₂  + 1 = ^(n-1+2)⁄₂  = ⁽ⁿ⁺¹⁾⁄₂.

5. When n = 13, we get 'i' value of the median = ⁽ⁿ⁺¹⁾⁄₂  = ⁽¹³⁺¹⁾⁄₂ = ¹⁴⁄₂ = 7

6. From the fig.25.22(a), we can see that the 7^th value is indeed the median

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So when the number of values (n) in the list is an odd number, we can find the median by the following procedure:

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

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Case 2: When the list has an even number of numbers

1. Consider fig. 25.22(b) above. It has 12 (an even number) values sorted in ascending order. 

2. Each can be denoted as x_i. So we have x₁, x₂, x₃, . . . , upto x_n. where n = 12

3. 'i' denotes the position of each value in the sorted list

4. There are two values at the 'exact middle'. We will call them the 'middle pair'. For any sorted list having an even number of values, there will be a 'middle pair'.  

5. We want the value of 'i' for both the members of that pair. For calculating those value of 'i', we adopt the following procedure:

(a) There are equal number of values on either side of the middle pair. Let this equal numbers be 'x'

(b) So we get: 

x + 2 ( the middle pair) + x = n 

⇒ 2x + 2 = n. ⇒ 2x = n-2 ⇒ x = ^(n-2)⁄₂

(c) So there are ^(n-2)⁄₂ numbers on the left side of the middle pair

(d) That means, the 'i' value of the number just to the left of the pair = ^(n-2)⁄₂

(e) So the 'i' value of the first member of the pair = ^(n-2)⁄₂  + 1 = ^(n-2+2)⁄₂  = ⁿ⁄₂.

(f) So the 'i' value of the second member of the pair = ⁿ⁄₂  + 1 .

6. When n = 12, we get:

• 'i' of the first member of the middle pair = ⁿ⁄₂ = ¹²⁄₂ = 6

• 'i' of the second member of the middle pair = (ⁿ⁄₂  + 1) = 7

7. So the 6^th and 7^th values form the middle pair. From fig.25.22(b), we can see that, this is indeed true. 

8. Once we get the i values of both the members of the middle pair, we can take them out from the list

9. Then we calculate the mean of the two members. This mean is the median of the whole list

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So when the number of values (n) in the list is an even number, we can find the median by the following procedure:

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

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Now we will see some solved examples on the calculation of median

Solved example 25.22

The heights (in cm) of 9 students of a class are as follows:

Find the median of this data.

Solution:
There are 9 values. So n = 9. 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:

The numbers in yellow colour shows the sequence 'i' • i = ⁽ⁿ⁺¹⁾⁄₂ = ⁽⁹⁺¹⁾⁄₂ = ¹⁰⁄₂ = 5
• So the 5^th value 149 is the median

Solved example 25.23
The points scored by a Kabaddi team in a series of matches are as follows:  

Find the median of the points scored by the team.
Solution:
There are 16 values. So n = 16. 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  + 1)

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 = ¹⁶⁄₂ = 8. So the first member of the middle pair = 10
• (ⁿ⁄₂  + 1) = 9. So the second member of the middle pair = 14
• Median of the list = Mean of 10 and 14 =  ^(10+14)⁄₂ = ²⁴⁄₂ = 12

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We have completed the discussion on Median. In the next section we will see Mode

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Chapter 25 - Statistics

In the previous section we completed the discussion on Proportions. In this section we will see some topics in Statistics.

In our day to day life, we come across different types of information. For example:
• Temperatures recorded at major cities
• Profits obtained by a company in the last five years
• Performance of various parties in an election etc., 
We find such variety of information in news papers, televisions, online services etc., 
■ These information may be numerical. That is., in the form of numbers. For example:
• Profit of 2.5 crores in 2014
• Profit of 2.7 crores in 2015 etc., 
■ These information can also be in the form of graphs showing the profits in various years. 

Whether numerical or not, such information is collected for a specific purpose. They are called data. We have seen different types of data, and have performed various calculations on them. See chapter 1.

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Large amounts of data are seen in various fields. Such fields include whether forecasting, business management, science and engineering, sport tournaments etc., Just having a lot of data is not good enough. We must be able to extract useful information from them. Consider an example:

• A car manufacturing company makes different models of cars. There are big cars, medium cars and small cars.
• Many of those models are performing very well in the market
• A very few models are performing poorly
• Though the majority of the models are performing very well, the company is showing decreasing profits year after year. What could be the reason?
• The answer can be found out by carefully analysing the income and expense data. The poorly performing models may be using up large amounts of money for their making.

Statistics is the branch of mathematics which performs the following activities:
• Collection of data 
• Analysis of that data 
• Extraction of useful information from that data.

Primary data and Secondary data

Consider the following example:
■ You are given the task of recording the heights of 20 students in your class.
The task can be performed by two methods:
• Method 1 is by personally meeting each of the 20 students, measuring the heights, and then recording them.
• Method 2 is by taking the values from the register book kept at the physical training department.
■ Consider another example:
You are given the task of recording the number of absent students on each day for the current month
This task can also be performed by two methods:
• Method 1 is to take the number by yourself, at the time of calling attendance at the beginning of every day
• Method 2 is to take the numbers later at the end of the month, from the attendance register
■ The final data obtained by method 1 or method 2,  may or may not show the same values. But what is important is this:
• At the time of presenting the data, you must specify which method you used to collect the data.

■ In example 1, 
• If you used the method 1, 
    ♦ You were collecting the data yourself. 
    ♦ And you were collecting it for a particular purpose. The purpose is to ‘record the height of 20 students’
• If you used the method 2, 
    ♦ You were taking the values from a data collected by some one else. The purpose of that pre-existing data is not be the same as your present purpose. That pre-existing data was prepared for a general picture of the physical ability of students in the whole school.
■ In example 2, 
• If you used method 1, 
    ♦ You were collecting the data yourself. 
    ♦ And you were collecting it for a particular purpose. The purpose is to ‘record the number of absentees’
• If you used the method 2, you were taking the values from a data collected by some one else. The purpose of that pre-existing data may not be the same as your present purpose.
■ So the data collected by the two methods are different, and must be specified in the presentation.

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■ When the information is collected by the investigator herself/himself, with a definite objective in mind, the data obtained is called primary data.

■ When the information is collected from a source, which already had the information stored, the data obtained is called secondary data. 
When using secondary data, the following points should be noted:
• The pre-existing data was collected by some one else
• That pre-existing data was collected for a different purpose
• So the reliability of that pre-existing data should be verified.

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Presentation of Data

When the task of ‘collecting data’ is over, the next task is:
• To present the data in a meaningful and ‘easy to understand’ way. 
• Also, the presentation should show the main features of the data at a glance. 
Consider the following example:
1. The marks obtained by 12 students in a test are given below:

2. The data in this form is called raw data. We saw it earlier here.
3. From the raw data, we can see that:
Lowest score is 24, and the highest score is 90
4. Those lowest and highest scores can be found out easily if we arrange the raw data in ascending or descending order. The ascending order is shown below:

5. The difference of the highest and the lowest values in the data is called the range of the data. So, the range in this case is 90 – 24 = 66

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Presentation of data in ascending or descending order can be quite time consuming, particularly when the number of observations is large, as in the case of the next example:

1. The marks obtained (out of 100) by 35 students of Class IX of a school are given below:

2. We can see that some marks repeat. For example, 4 students have scored 58 marks. So the value 58 occurs 4 times. The number of times that a value occurs in the raw data is called frequency of that value. In our case, frequency of the value 58 is 4.
3. We find the frequencies of the different values, and make the frequency distribution table. For accuracy, we can use the method of tally marks as shown below. This also we have seen before. 

• From the above table, we can note that, there is no need to write all the 35 values. Because, repeating values are shown by 'frequency'. 
• The results in the above table can be presented in a more user friendly form, if we sort the scores in ascending order. This is shown below:

• The results in the above modified table can be still be made more user friendly, by using bar graph as shown below. We have seen bar graphs also earlier.

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In the next section we will see a case which will require a grouped frequency distribution table.

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