Suppose we want to compare the age of students in two schools and determine which school has more aged students. If we compare on the basis of individual students, we cannot conclude anything. However, if for the given data, we get a representative value that signifies the characteristics of the data, the comparison becomes easy.Â

A certain value representative of the whole data and signifying its characteristics is called an average of the data. Three types of averages are useful for analyzing data. They are:Â

**Mean****Median****Mode**

In this article, we will study three types of averages for the analysis of the data.

## Mean

The mean (or average) of observations is the sum of the values of all the observations divided by the total number of observations. The mean of the data is given by **x = f**_{1}**x**_{1}** + f**_{2}**x**_{2}** + …. + f**_{n}**x**_{n}** /f1 + f2+… + fn. **The mean Formula is given by,

Mean = âˆ‘(f_{i}.x_{i})/âˆ‘f_{i}

**Methods for Calculating Mean**

**Methods for Calculating Mean**

**Method 1: Direct Method for Calculating Mean**

**Method 1: Direct Method for Calculating Mean**

** Step 1:** For each class, find the class mark x

_{i}, as

x=1/2(lower limit + upper limit)

** Step 2: **Calculate f

_{i}.x

_{i}for each i.

** Step 3:** Use the formula

**Mean = âˆ‘(f**

_{i}

**.x**

_{i}

**)/âˆ‘f**

_{i.}**Example: Find the mean of the following data.**

Class Interval |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
---|---|---|---|---|---|

Frequency |
12 |
16 |
6 |
7 |
9 |

**Solution:**

We may prepare the table given below:

Class Interval

Frequency Â f

_{i}Class Mark x

_{i}( f

_{i}.x_{i})0-10

12

5

60

10-20

16

15

240

20-30

6

25

150

30-40

7

35

245

40-50

9

45

405

Â

âˆ‘f

_{i}=50Â

âˆ‘f

_{i.}x_{i}=1100

Mean = âˆ‘(f_{i}.x_{i})/âˆ‘f_{i }= 1100/50 = 22

**Method 2: Assumed – Mean Method For Calculating Mean**

**Method 2: Assumed – Mean Method For Calculating Mean**

For calculating the mean in such cases we proceed as under.Â

** Step 1: **For each class interval, calculate the class mark x by using the formula:

**x**

_{i}

**= 1/2 (lower limit + upper limit).Â**** Step 2:** Choose a suitable value of mean and denote it by A. x in the middle as the assumed mean and denote it by A.

** Step 3:** Calculate the deviations d

_{i}= (x, -A) for each i.

** Step 4:** Calculate the product (f

_{i}x d

_{i}) for each i.

** Step 5:** Find n = âˆ‘f

_{i}

** Step 6:** Calculate the mean, x, by using the formula:

**X = A + âˆ‘f**

_{i}

**d**

_{i}

**/n.****Example: Using the assumed-mean method, find the mean of the following data:**

Class Interval |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
---|---|---|---|---|---|

Frequency |
7 |
8 |
12 |
13 |
10 |

**Solution:**

Let A=25 be the assumed mean. Then we have,

Class Interval Frequency

f

_{i}Mid value

x

_{i}Deviation

d

_{i}=(x_{i}-25)(f _{i}xd_{i})0-10 7 5 -20 -140 10-20 8 15 -10 -80 20-30 12 25=A 0 0 30-40 13 35 10 130 40-50 10 45 20 200 Â âˆ‘f _{i }= 50Â Â âˆ‘(f _{i}xd_{i}) = 100

Mean = X = A + âˆ‘f_{i}d_{i}/n = (25 + 110/50) = 27.2

**Check: ****Arithmetic Mean**

**Method 3: Step-Deviation method for Calculating Mean**

**Method 3: Step-Deviation method for Calculating Mean**

When the values of x, and f are large, the calculation of the mean by the above methods becomes tedious. In such cases, we use the step-deviation method, given below.Â

** Step 1: **For each class interval, calculate the class mark x, where

**X = 1/2 (lower limit + upper limit).Â**** Step 2: **Choose a suitable value of x, in the middle of the x, column as the assumed mean and denote it by A.

** Step 3: **Calculate h = [(upper limit) – (lower limit)], which is the same for all the classes.

** Step 4: **Calculate u

_{i}= (x

_{i}-A) /h for each class.

** Step 5: **Calculate fu for each class and hence find

**âˆ‘(f**

_{i}

**Ã— u**

_{i}

**).Â**** Step 6: **Calculate the mean by using the formula:

**x = A + {h Ã— âˆ‘(f**

_{i}

**Ã— u**

_{i}

**)/ âˆ‘f**

_{i}

**}****Example: Find the mean of the following frequency distribution:**

Class |
50-70 |
70-90 |
90-110 |
110-130 |
130-150 |
150-170 |
---|---|---|---|---|---|---|

Frequency |
18 |
12 |
13 |
27 |
8 |
22 |

**Solution:**

We prepare the given table below,

Class

Frequency

f

_{i}Mid Value

x

_{i}ui=(x

_{i}-A)/h=(x

_{i}-100)20(f

_{i}xu_{i})50-70

18

60

-2

-36

70-90

12

80

-1

-12

90-110

13

100=A

0

0

110-130

27

120

1

27

130-150

8

140

2

16

150-170

22

160

3

66

Â

âˆ‘f

_{i }= 100Â

Â

âˆ‘(f

_{i}xu_{i}) = 61A = 100, h = 20, âˆ‘f

_{i }= 100 and âˆ‘(f_{i }x u_{i}) = 61x = A + {h Ã— âˆ‘(f

_{i }x u_{i}) / âˆ‘f_{i}}=100 + {20 Ã— 61/100} = (100 + 12.2) =112.2

**Check: ****What is mean method formula?**

**Median**

**Median**

We first arrange the given data values of the observations in ascending order. Then, if n is odd, the median is the (n+1/2). And if n is even, then the median will be the average of the n/2th and the (n/2 +1)^{th} observation. The formula for Calculating Median:

Median, M_{e}= l + {h x (N/2 – cf )/f} Â Â ÂWhere,

Â

- l = lower limit of median class.
- h = width of median class.
- f = frequency of median class,Â
- c
_{f}= cumulative frequency of the class preceding the median class.- N = âˆ‘f
_{i}

**Example: Calculate the median for the following frequency distribution.**

Class Interval |
0-8 |
8-16 |
16-24 |
24-32 |
32-40 |
40-48 |
---|---|---|---|---|---|---|

Frequency |
8 |
10 |
16 |
24 |
15 |
7 |

**Solution:Â **

We may prepare cumulative frequency table as given below,

Class

Frequency

Cumulative Frequency

0-8

8

8

8-16

10

18

16-24

16

34

24-32

24

58

32-40

15

73

40-48

7

80

Â

N = âˆ‘f

_{i }= 80Â

Now, N = 80 = (N/2) = 40.Â

The cumulative frequency just greater than 40 is 58 and the corresponding class is 24-32.Â

Thus, the median class is 24-32.Â

Â l = 24, h = 8, f = 24, c

_{f}= c.f. of preceding class = 34, and (N/2) = 40.

Median, M_{e}= l+ h{(N/2-cf)/f}= 24 + 8 {(40 – 34)/ 24}

= 26Â

Hence,

median = 26.

**Check: ****Central Tendency**

**Mode**

**Mode**

It is that value of a variety that occurs most often. More precisely, the mode is the value of the variable at which the concentration of the data is maximum.Â

** Modal Class: **In a frequency distribution, the class having the maximum frequency is called the modal class. The formula for Calculating Mode: Â

M_{o}= x_{k}+ h{(f_{k}– f_{k-1})/(2f_{k}– f_{k-1 }– f_{k+1}Â)}Where,

- x
_{k}= lower limit of the modal class interval.- f
_{k}= frequency of the modal class.Â- f
_{k-1}= frequency of the class preceding the modal class.- f
_{k+1}= frequency of the class succeeding the modal class.- h = width of the class interval.

**Example 1: Calculate the mode for the following frequency distribution.**

Class |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |
60-70 |
70-80 |
---|---|---|---|---|---|---|---|---|

Frequency |
5 |
8 |
7 |
12 |
28 |
20 |
10 |
10 |

** Solution:**Â

Class 40-50 has the maximum frequency, so it is called the modal class.

x

_{k }= 40, h = 10, f_{k }= 28, f_{k-1 }= 12, f_{k+1 }= 20Mode,

M_{o}= x_{k}+ h{(f_{k}– f_{k-1})/(2f_{k}– f_{k-1 }– f_{k+1})}= 40 + 10{(28 – 12)/(2 Ã— 28 – 12 – 20)}

= 46.67

Hence,

mode = 46.67

**Important Result**

Relationship among mean, median and mode,

Mode = 3(Median) – 2(Mean)

**Example 2: Find the mean, mode, and median for the following data,**

Class |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
Total |
---|---|---|---|---|---|---|

Frequency |
8 |
16 |
36 |
34 |
6 |
100 |

**Solution:**

We have,

Class

Mid Value

x_{i}Frequency f

_{i}Cumulative Frequency

f

_{i . }x_{i}0-10

5

8

8

40

10-20

15

16

24

240

20-30

25

36

60

900

30-40

35

34

94

1190

40-50

45

6

100

270

Â

Â

âˆ‘f

_{i}=100Â

âˆ‘f

_{i}. x_{i}=2640

Mean = âˆ‘(f_{i}.x_{i})/âˆ‘f= 2640/100

= 26.4

Here, N = 100 â‡’ N / 2 = 50.Â

Cumulative frequency just greater than 50 is 60 and corresponding class is 20-30.Â

Thus, the median class is 20-30.Â

Hence, l = 20, h = 10, f = 36, c = Â c. f. of preceding class = 24 and N/2=50

Median, M_{e}= l + h{(N/2 – c_{f})/f}= 20+10{(50-24)/36}

Median = 27.2.Â

= (3 Ã— 27.2 – 2 Ã— 26.4) = 28.8.Mode = 3(median) â€“ 2(mean)

**Check: ****Mean, Median and Mode**

**Ogives**

**Ogives**

Let a grouped frequency distribution be given to us. On graph paper, we mark the upper-class limits along the x-axis and the corresponding cumulative frequencies along the y-axis. By joining these points successively by line segments, we get a polygon, called a cumulative frequency polygon. On joining these points successively by smooth curves, we get a curve, known as** ogive.**

**The step-by-step process for the construction of an Ogive graph:**

- First, we mark the upper limits of class intervals on the horizontal x-axis and their corresponding cumulative frequencies on the y-axis.
- Now plot all the corresponding points of the ordered pair given by (upper limit, cumulative frequency).
- Join all the points with a free hand.
- The curve we get is known as ogive.

**Types of Ogives**

**Types of Ogives**

**Less Than Ogive**

On graph paper, we mark the upper-class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.

- On joining these points successively by line segments, we get a polygon, called a cumulative frequency polygon.
- On joining these points successively by smooth curves, we get a curve, known as
**less than ogive.**

**More Than Ogive**

On graph paper, we mark the lower class limits along the x-axis and the corresponding cumulative frequencies along the y-axis.

- On joining these points successively by line segments, we get a polygon, called the cumulative frequency polygon.
- On joining these points successively by smooth curves, we get a curve, known as
**more than ogive.**

**Check: ****Relation Between Mean Median and Mode**

**Mean, Median, and Mode of Grouped Data – FAQs**

**Mean, Median, and Mode of Grouped Data – FAQs**

### What is Grouped Data?

Grouped data is data that has been organized into groups known as classes. It is often used in statistical analysis to simplify data and focus on key trends and patterns.

### How Do You Calculate the Mean of Grouped Data?

To calculate the mean of grouped data, multiply the midpoints of each class by their respective frequencies, sum these products, and then divide by the total number of data points.

### Can You Find the Median in Grouped Data?

Yes, the median can be found in grouped data. It is the value that divides the dataset into two equal parts. For grouped data, the median class is identified first, and then the median value is calculated using a formula that considers the cumulative frequency.

### How is the Mode Determined for Grouped Data?

The mode for grouped data is the value or class with the highest frequency. For a more precise value within a class, a modal class is identified, and the mode is calculated using a formula that accounts for the frequencies of the modal class and its adjacent classes.

### Is It Possible to Calculate All Three Measures for All Types of Data?

Yes and no. Continuous data can have a mean, median, and mode calculated. Ordinal data can have a median and mode, while nominal data typically only has a mode.

### Why Are These Measures Important?

Mean, median, and mode are measures of central tendency that provide insights into the distribution, concentration, and spread of data, helping in making informed decisions and comparisons.

### How Do You Handle Outliers in These Calculations?

Outliers can skew the mean significantly. While the mean is sensitive to outliers, the median and mode are more robust measures that are less affected by extreme values.

### Can These Measures Be Used for Comparative Analysis?

Yes, mean, median, and mode are often used to compare different datasets, such as comparing the average performance of students across different schools.

### What Are the Limitations of Using Mean, Median, and Mode?

Each measure has its limitations. The mean is affected by outliers, the median may not reflect the distribution’s shape, and the mode can be ambiguous in datasets with multiple peaks.

### How Do You Interpret These Measures in Real-World Data?

Interpreting these measures depends on the context. For example, the mean income of a region may give an idea of economic status, while the median can indicate the income level that divides the population into two equal groups, and the mode can show the most common income level.