Measures of Central Tendency

  • date 18th June, 2019 |
  • by Prwatech |
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Measures of Central Tendency Tutorial

 

Measures of central tendency tutorial: Sound knowledge of statistics helps an analyst to make effective business decision plans. In this tutorial, one can learn of all measures of central tendency in Data science to calculate mean, median and mode and under what conditions the measures are appropriate to be used in data science.

 

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Measures of Central Tendency Definition

 

Central Tendency is a descriptive summary of a dataset that represents the data into a single value.

 

measures of central tendency tutorial

 

We use Central Tendency when we need to denote any collection of data into a unique single numeric  value.

 

Measures of Central Tendency Example

 

Mr. A has scored the following marks:

 

Sr.No Subject Marks
1 English 72
2 Mathematics 65
3 Statistics 81
4 Hadoop 75
5 Python 83
  Total 376

 

Measures of Central Tendency Example

 

How to calculate Central Tendency?

 

It can be calculated over three types of series:

Individual Series: Series of Data consisting only of single values.

Example: 2,3,4,512,45,56,79

Discrete Series: Series of data having a value and a frequency of the given value.

    

Example:

 

X 5 10 15 20 25
F 6 8 12 6 3

 

Continuous Series: Series of data having a frequency in a continuous manner.

 

Example

 

Marks 0-10 10-20 20-30 30-40 40-50 50-60
Students 2 15 13 18 12 4

 

Types of Measures of Central Tendency

 

There are three types of Measures of Central Tendency.

 

Mean

Median

Mode

 

Mean

 

It is the most common method of finding Central Tendency. It is found by calculating the average value of the given dataset.

 

Individual Series

Types of Measures of Central Tendency _ Individual series mean

 

Discrete Series

Types of Measures of Central Tendency_Discrete series

 

Where –

      N = Frequency.

       f= frequency

       x=Item

 

Mean Example

 

Items 10 20 30 40 50
Frequency 2 5 6 9 4

 

Xi
Frequency
fi
fixi
10 2 20
20 5 100
30 6 180
40 9 360
50 4 200
Total 26 860

 

x̅=860/26

x̅=33.07

 

Continuous Series

Continuous series Mean Example

 

Items 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60
Frequency 2 5 6 9 4

measure of central tendency with example

 

Where –

N = Frequency.

f= frequency

x=item

 

Items Frequency Xi fiXi
10 – 20 2 15 30
20 – 30 5 25 125
30 – 40 6 35 210
40 – 50 9 45 405
50 – 60 4 55 220
Total 26 990

 

x̅= 990 / 26

x̅=38.07.

 

Median

 

The median is the value of the variable which divides the group into two equal parts, one part comprising all values greater and the other all values less than the median.

 

Individual series

 

series in which only values are given

Arrange data into ascending order

M=(N+1)/2

Where: N= No of Items

Individual series median example

 

50 40 60 80 12 5 44 41 29

 

Now arrange data into ascending order:

 

5 12 29 40 41 44 50 60 80

 

N = Total number of terms = 9

Now = (N+1)/2

= (9+1) / 2

= 10 / 2     =     5

Median = 5th term = 41.

When N is Even value:

 

50 40 60 80 12 5 44 41 29 89

 

5 12 29 40 41 44 50 60 80 89

 

N = Total number of terms = 9

Now = (N+1)/2

= (10+1) / 2

=  11 / 2     =     5.

Median = 42.5

 

Discrete Series

 

Arrange data into ascending order

Calculate Cumulative frequency

M=(N+1)/2

Where: N=total of frequency

 

Discrete Series median example

 

X 10 20 30 40 50 60 70
F 4 6 9 7 1 12 15

 

X F Cf Terms
10 4     =4 1-4
20 6 4+6=10 5-10
30 9 10+9=19 11-19
40 7 19+7=26 20-26
50 1 26+1=27 27-27
60 12 27+12=39 28-39
70 15 39+15=54 40-54
Total N=54 N=54

 

measure of central tendency example

 

Continuous Series

 

In this case, cumulative frequencies are taken and then the value from the class-interval in which (N/2)th term lies is taken. Using the formula.

measure of central tendency

Median can also be calculated from the formula given below:

M= L – Cf-N1/f × i:

Where L is the upper limit of the median class.

 

Important Working Hints

 

Take N1=N/2

Looking from the lowest to the upper side in the Ccolumn we get the value which is firstly numerically less than N1. It is the value of Cf.

We come one step down; make the terms in column ‘f and also the class interval to get f and L. L is taken from a lower limit of the class interval.

 

Class Interval 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Frequency 7 18 34 50 35 20 6

 

Solution:

 

Class Interval Frequency (f) Cumulative frequency (Cf)
0-10 7 7
10-20 18 25
20-30 34 59
30-40 50 109
40-50 35 144
50-60 20 164
60-70 6 170
N=170

 

 

measures of central tendency example_continous series

 

 

 

Mode

 

It is the measure of central tendency targets towards pointing out the value that occurs most frequently in a series. This value, when it represents the data is known as the mode of the series. Mode simply refers to the value that occurs the maximum number of times in a distribution.

 

Calculation of Mode

 

Individual Series

 

In the case of individual series, we just have to inspect the item that occurs most frequently in the distribution. Further, this item is the mode of the series.

Individual Series mode example

 

2 1 3 6 6 5 1 2 2 3

 

X Repetition
1 2
2 3
3 2
5 1
6 2

 

Here No 2 is repeated 3 times which is the highest.

Hence Mode = 2

 

Mode for Discrete Series

 

In discrete series, we have values of items with their corresponding frequencies. In essence, here the value of the item with the highest frequency will be the mode for the distribution.

 

Mode for Frequency Distribution

 

Lastly, for frequency distribution, the method for mode calculation is somewhat different. Here we have to find a modal class. The modal class is the one with the highest frequency value. The class just before the modal class is called the pre-modal class. Whereas, the class just after the modal class is known as the post-modal class. Lastly, the following formula is applied for calculation of mode:

 

 Mode = l + h [(f1-f0)/(2f1-f0-f2)]

 

l= The lower limit of the modal class

f, f, f = Frequency corresponding to the modal class, post-modal class, and pre-modal class

 

Mode Example

 

Class Interval 10-20 20-30 30-40 40-50 50-60
Frequency 3 10 15 10 2

 

Answer: As the frequency for class 30-40 is maximum, this class is the modal class. Classes 20-30 and 40-50 are pre-modal and post-modal classes respectively. The mode is:

 

Mode= 30 + 10×[(15-10)/(2×15-10-10)]= 30+ 5= 35

 

These are the basic measures of central tendency in data science that a data analyst should know. Here, we have covered the basics of measures of central tendency in data science, its definition, types of measures of central tendency with example and how to calculate mean, median and mode.

 

We hope you liked our article on Measures of central tendency tutorial. Interested in learning more? Then get advanced data science certification course in Bangalore from Prwatech, the world’s best data science training institute.

 

 

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