Measures of Central Tendency

Measures of Central Tendency
Go back to  'Data'

Mary asked Alex to count the total number of boys and girls in their grade.

Alex did this, and then Mary asked him to divide each number by the number of sections in their grade. 

Weirdly, Alex got an answer that ended up going into decimals! Does this imply that some sections have fractions of children?

Read on to understand what that fraction of a human actually means in the language of mathematics.

Lesson Plan

What Are the Measures of Central Tendency?

We come across new data every day.

We find them in newspapers, articles, in our bank statements, mobile and electricity bills.

Now the question arises if we can figure out some important features of the data by considering only certain representatives of the data.

This is possible by using measures of central tendency or averages.

Newspaper as an example of data gathered

A measure of central tendency describes a set of data by identifying the central position in the data set as a single value.

We can think of it as a tendency of data to cluster around a middle value.

In statistics, the three most common measures of central tendencies are mean, median, and mode.

Choosing the best measure of central tendency depends on the type of data we have.

Let’s begin by understanding the meaning of each of these terms.

Let’s begin by understanding the meaning of each of these terms.


What Is Mean?

This is the most common central tendency we know about and use. It is also known as average.

Mean is simply the sum of all the components in a group or collection, divided by the number of components.

It is denoted by \(\bar{x}\)

Example

To understand the definition, let us look at the weights of 8 boys in kilograms: 45, 39, 53, 45, 43, 48, 50, 45.

So, in the above example, there are 8 components.

\( \text{Therefore, the average of the group } \)
\(=\dfrac{45+ 39+ 53+ 45+ 43+48+ 50+45}{8}\)
\(=\dfrac{368}{8}\)
\(=46\)

Mean = \(\dfrac{\text{Sum of the terms}}{\text{Number of the terms}}\)

So, the average (or mean) weight of those 8 boys is 46 kgs. Let us see how it looks when we arrange the weights in ascending order: 39, 43, 45, 45, 45, 48, 50, 53.

We see that 4 boys have a weight less than the mean of the group, while 6 boys have weights higher than the mean of the group.

Play with the simulation below to calculate the mean of the number of trees.

Drag the sliders to set the number of trees in each pot and find their mean.

Let us see how to calculate the mean for different types of data.

Case 1:

Let there be \(n\) number of items in a list.

{\({x_1, x_2, x_3, … , x_n }\)}

The mean can be calculated using the formula given below.

 

OR

Case 2:

Let there be \(n\) number of items in a list.

{\({x_1, x_2, x_3, … , x_n }\)}

Let the frequency of each item be

{\({f_1, f_2, f_3, … , f_n }\)} respectively.

The mean can be calculated using the formula given below.

 

OR

Case 3:

When the items in a list are written in the form of a range, for example, 10-20, we need to first calculate the class mark.

Then, the mean can be calculated using the formula given below, where \(x_i\) will be the classmark for each item.


What Is the Median?

The value of the middle-most observation obtained after arranging the data in ascending order is called the median of the data.

Median Example

Case 1: Ungrouped Data

Step 1: Arrange the data in ascending or descending order.

Step 2: Let the total number of observations be \(n\).

To find the median, we need to consider if \(n\) is even or odd.

If \(n\) is odd, then use the formula:  

\(\text { Median = } (\dfrac {n+1}{2})^{th }\text {observation}\)

If \(n\) is even, then use the formula: 

\( \text { Median = } \dfrac {\dfrac{n}{2}^{th}\text {obs.}+ (\dfrac{n}{2}+1)^{th}\text {obs.}}{2}\)

Case 2: Grouped Data

When the data is continuous and in the form of a frequency distribution, the median is found as shown below:

Step 1: Find the median class.

Let \(n\) = total number of observations i.e. \(\sum f_i \) 

Note: Median Class is the class where \(\dfrac{n}{2}\) lies.

Step 2: Use the following formula to find the median.

\(\text { Median = } l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)

where,
\( l= \)lower limit of median class

\( c= \) cumulative frequency of the class preceding the median class

\(  f=\)frequency of the median class

\(  h=\)class size

Enter numbers separated by commas in the simulation below and hit calculate to determine the median of the set.


What is Mode?

In any collection of numbers, the number which occurs the most number of times is the mode.

Mode example

When we need to calculate the mode in the case of grouped frequency distribution, we will first identify the modal class, the class that has the highest frequency. Then, we will use the formula given below to calculate the mode.

Enter numbers separated by commas in the simulation below and hit calculate to determine the mode of the set.

 
important notes to remember
Important Notes
  1. The three measures of central tendency are mean, median, and mode.
  2. Mean is also known as the arithmetic mean and average.
  3. Data has to be arranged in ascending/descending order to find the middle value.
  4. The empirical relationship between mean, median, and mode is:
    (Mean – Mode) = 3 (Mean – Median)

Solved Examples

Example 1

 

 

The age of the members of a weekend baseball team has been listed below.

{42, 40, 50, 60, 35, 58, 32}

Find the median of the above set.

Solution

Step 1:

Arrange the data items in ascending order.

Original set:

{42, 40, 50, 60, 35, 58, 32}

Ordered Set:

{32, 35, 40, 42, 50, 58, 60}

Step 2:

Count the number of observations.

Number of observations \(n=7\)

If the number of observations is odd, then we will use the following formula:

 

\(\text{Median}=\dfrac{(n+1)^{th}}{2} \text{term}\)

Step 3:

Calculate the median using the formula.

\(\begin{align}\text{Median}&=\dfrac{(n+1)^{th}}{2} \text{term}\\&=\dfrac{(7+1)^{th}}{2}  \text{term}\\&=4^{th} \text{ term}\\&=42\end{align}\)

\(\therefore\) Median = 42
Example 2

 

 

Find the median marks for the following distribution:

Classes 0-10 10-20 20-30 30-40 40-50
Frequency 2 12 22 8 6

We need to calculate the cumulative frequencies to find the median. 

Solution

Calculation table: 

Classes Number of students Cumulative frequency
0-10 2 2
10-20 12 2 + 12 = 14
20-30 22 14 + 22 = 36
30-40 8 36 + 8 = 44
40-50 6 44 + 6 = 50

\( N=50 \)

\( \dfrac{N}{2} = \dfrac{50}{2}= 25\)

Median Class \( = 20-30 \) 

\( l = 20, f =22, c.f= 14, h =10 \)\(\)

Using Median formula: 

\begin{align}\text { Median  } &=l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\\&=20 + \dfrac{25-14}{22} \times 10 \\&= 20 + \dfrac{11}{22} \times 10 \\&=20+5= 25 \end{align}

\(\therefore\) Median = 25
Example 3

 

 

There are 30 students in Grade 8. The marks obtained by the students in mathematics are tabulated below. Calculate the mean marks.

Marks Obtained Number of students
100 2
95 7
88 10
76 6
69 5

Solution

The total number of students in Grade 8 = 30

\(\begin{align} &x_1=100 x_2=95 x_3=88 x_4=76 x_5=69 \\\\ &f_1=2 \;f_2=7 \;f_3=10 \;f_4=6 \;f_5=5\\  \end{align}\)

\[\begin{align} &x_1f_1= 100 \times 2= 200\\ &x_2f_2 = 95 \times 7 \;\, = 665\\ &x_3f_3 = 88 \times 10 = 880\\ &x_1f_1 = 76 \times 6 \;\,= 456\\ &x_1f_1 = 69 \times 5 \;\,= 345 \end{align}\]

\(  f_1x_1 + f_2x_2 + f_3x_3 + f_4x_4 + f_5x_5  \)
\(= 200 + 665 + 880 + 456 + 345 \)
\(= 2,546 \)
\( f_1 + f_2 + f_3 + f_4 + f_5  \)
\( = 2 + 7 + 10 + 6 + 5\)
\(= 30 \)

We will use the formula given below:

Mean formula

\[\begin{align}  \text{Mean marks}&= \frac{2546} {30} \\&= 84.87 \end{align}\]

\(\therefore\) Mean marks = 84.87
Example 4

 

 

There are 100 members in a basketball club. The different age groups of the members and the number of members in each age group are tabulated below. Calculate the mean age of the club members.

Age Group Number of Members
10-20 17
20-30 22
30-40 20
40-50 21
50-60 20

Solution

In this case, we first need to calculate the Class Mark for each age group.

We will use the formula given below and calculate the Class Mark for each age group.

Class Mark

Age Group Class Mark Number of Members
10-20 15 17
20-30 25 22
30-40 35 20
40-50 45 21
50-60 55 20

 Now,

\(\begin{align} &x_1=15 x_2=25 x_3=35 x_4=45 x_5=55 \\ &f_1=17 f_2=22 f_3=20 f_4=21 f_5=20   \end{align}\)

\[\begin{align}  &x_1f_1=15\times17=255\\ & x_2f_2= 25\times22= 550\\ & x_3f_3=35\times20=700\\ & x_1f_1=45\times21=945\\ & x_1f_1=55\times20=1100   \end{align}\]

\( f_1x_1 + f_2x_2 + f_3x_3 + f_4x_4 + f_5x_5\)
\( = 255 + 550 + 700 + 945 + 1100 \)
\( = 3550 \)
\( f_1 + f_2 + f_3 + f_4 + f_5 \)
\(= 17 + 22 + 20 + 21 + 20\) 
\(= 100 \)

We will use the formula given below:

Mean formula

\[\begin{align} \text{Mean Age } &= \frac {3550}{100} \\&=35.5 \end{align}\]

\(\therefore\) Mean age = 35.5 years
Example 5

 

 

Find the mode for the given data set.

{23, 30, 32, 34, 45, 45, 54, 54, 59, 68}

Solution

The numbers which have maximum frequency are 45 and 54

They both occur twice in the given data set.

\(\therefore\) Mode = {45, 54}
 
Challenge your math skills
Challenging Questions
  1. The table given below lists the height (in inch) of 30 students in Grade 9. Calculate the mode for the given dataset.
    Height (in inch) Number of students
    45-50 5
    50-55 7
    55-60 6
    60-65 9
    65-70 3

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about the measures of central tendency with the examples and interactive questions. Now you will be able to easily solve problems on measures of central tendency worksheet, measures of central tendency definition, best measures of central tendency, central tendency definition, average value, and range.

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Frequently Asked Questions(FAQs)

1. Why is the median resistant, but mean is not?

The mean is affected by extreme observations/values whereas the median is just the middle value of the ordered list.

The mean is sensitive to the outliers.

2. What is a central tendency?

Central tendency is a central value of a distribution. 

3. What is the average of the integers from 25 to 41?

The average of integers from 25 to 41 is 33             

4. How to find the median with even numbers?

Median of even number of numbers is calculated as:

Step 1: Arrange the data in ascending order

Step 2: Take the average of the two middle values.

This average is the median.

5. What are the measures of central tendency?

The measures of central tendency are:

Mean, median, and mode

  
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