Mean Deviation & Frequency Distribution

Continuous Frequency Data

Data represented in a tabular or graphical format denotes the frequency. The number of times an observation takes place within a particular class interval is known as frequency distribution. If the collection of data is large, for example, if we need to analyze the scores of 200 players, then such representation will be easily analyzed by using the concept of Grouping of Data according to the class intervals. In this article, we will discuss mean deviation for group data, continuous frequency data, mean deviation formula for group data, mean deviation formula for the ungrouped data, mean deviation formula for continuous series, etc.

Mean Deviation for grouped data Introduction

In the method of the frequency distribution of continuous type, class intervals or groups are arranged in a way that there is no gap between them and each class retains its respective frequency. The class intervals are selected in such a manner that they should be either mutually exhaustive and exclusive.

Mean deviation formula for group data

Here, you can see the mean deviation formula for group data

Mean deviation - ∑ f | X-X| / ∑ f  

Here, X Indicates the mean and is calculated as ∑ f x / ∑ f

X indicates different values of midpoints for class intervals

F indicates the different values of frequency

Midpoints are calculated as (lower limit + upper limit)/ 2

Let us understand the concept of mean deviation formula for group data with an example:

The below table represents the age group of employees working in some company.

 MEAN DEVIATION FROM MEAN FOR GROUP DATA

Age Group

(Class Interval)

Number of employees

(f)

Mid values

(x)

fx

Deviation from mean

dx – X-X

Absolute deviation

|dx||X- X|

F |dx|= |fX-  X|


0-10

5

5

25

-22

22

110

10-20

8

15

120

-12

12

96

20-30

15

25

375

-2

2

30

30-40

16

35

560

8

8

128

40-50

6

45

270

18

18

108

 

∑ f  =50

 

∑ f x = 1350

 

 

∑f |dx| = 472

 

X =∑ f x / ∑ f  = 1350/50

X= 27

Mean Deviation=∑ f | X-X| / ∑ f  

= 472/50

= 9.44

Hence, mean deviation is 9.44

Mean deviation for ungroup data.

Ungroup data is the type of distribution in which data is represented in a raw form. For example- the batsman scores for the last 5 matches are stated as 54, 76, 89 ,23 ,67. The mean deviation from the above-given data enables us to conclude his form and performance in the last 5 matches.

Mean Deviation formula for ungrouped data

Here, you can see mean deviation formula for ungrouped data

Mean deviation- [{∑ | X-a |} / n]

∑ [|X-a|] indicates the summation of the deviation for values from “a”

N indicates the number of observations

Frequency Distribution

Representation of data in a tabular or graphic format which states the frequency (number of times observation occurs within a particular interval) is known as frequency distribution. The importance of frequency distribution in statistics is immense. A well-structured frequency distribution creates the possibility of a detailed analysis of the structure of information. So the groups where population breaks down can easily be determined.

Frequency Distribution Table

It is a way of representing the data in a tabular format where each part of the data is assigned to its corresponding frequencies. The objective of the statistical representation of the data is to organize the data concisely so that the analysis of data becomes easy. For this reason, we organize the larger data in a tabular format which is called the frequency distribution table

Continuous frequency distribution

In the continuous frequency distribution, the data of the members are grouped into various class intervals and are related to their corresponding frequencies. In this continuous frequency data, two columns are given one for class intervals and one column for frequencies.

Let us understand the concept of continuous distribution through the continuous frequency data given below

Here, the scores of the crickets are given with their corresponding frequencies

Score / Class-interval

Frequency (f)

15-25

25

25-35

54

35-45

34

45-55

20

 

Mean Deviation formula for continuous series

Here, you can see mean deviation formula for continuous series

Mean Deviation=  ∑ f |X-Me| / N = ∑ f | D| / N

N Indicates the number of observation

F indicates the different values of frequency

X indicates different values of midpoints for class intervals

Me indicates median and it is calculated as 

∑f /N

Midpoints for the continuous series is calculated as (lower limit + upper limit)/ 2

Solved Examples

  1. Find the mean deviation for the following continuous data 

Marks Obtained

10-20

20-30

30-40

40-50

No. of students

5

15

25

35

Solution:

Marks obtained

Number of students

(f)

Midpoints (xi)

fi xi

|Xi –Me|

f|Xi –Me|

0-10

5

2

10

14.54

29.08

10-20

15

5

75

4.54

22.7

20-30

25

1

25

6.54

5.46

30-40

35

3

105

14.54

46.38

 

 

N=11

∑ f =215

∑ = 10

∑f |Xi –Me|

= 103.62

 

Median = ∑ f  / N = 215/11 = 19.54

Mean Deviation= ∑ f | D| / N = 103.62/11

= 9.42

2. Calculate the mean deviation about the median for the following ungroup data. 

X

5

7

8

9

10

11

13


Solution: 

X

|X-a|

5

3

7

1

8

0

9

1

10

2

11

3

13

5

 

∑= 15

 

As n is the odd, median is calculated as =Value of (n +1)/2th item = 8/2th = 4th item = 8

Therefore, value of a =8

Accordingly 

Mean deviation = 15/7 = 2.14

Fun Facts

  • The mean deviation is sometimes also called the mean absolute deviation because it is considered as the mean of the absolute deviation.

  • The statistic was established as a separate unit in 1979.

  • The department of Mathematics split statistics from physics in 1951.

  • Statistics play an important role in understanding the natural world and in technological innovation.

FAQ (Frequently Asked Questions)

1. Explain the term frequency distribution?

Frequency distribution

A frequency distribution is the representation of data in a tabular or graphical format to exhibit the number of observations within a given interval.

In statistics, the frequency distribution is termed as a table that illustrates the number of outcomes of a sample. Each entry existing in the table includes the count or frequency of occurrence of the values within a group.

Some common characteristics of the frequency distribution are:

  • Measures of the central tendency and location such as mean, median, and mode

  • The extent of the symmetry or asymmetry that is skewness.

  •  The flatness or peakedness is kurtosis.

  • Measures of the dispersion such as range, variance, and standard deviation.

2. Explain the term, mean deviation, and state its formulas.

The term mean deviation or absolute deviation is stated as the mean of the absolute deviation of the observations from the suitable average which can be the arithmetic mean, median, or mode. The mean deviation can be used to understand the dispersion of data from a measure of central tendency

Mean Deviation formulas

Mean deviation from mean = ∑ f | X-X

Mean deviation from median =∑ f | X-m|

Here,

Σ indicates the summation

X indicates the observation

X Indicates the mean

N indicates the number of observation

For frequency distribution, mean deviation is given by=  ∑ f | X-X| / ∑ f

Mean deviation about mode -∑f |X-Mode|/ ∑ f