Measures of Central Tendency

Measures of central Tendency

Dept of AGB Veterinary College, SHIMOGA

Definition Measure of central tendency are typical values around which other values of distribution concentrate.

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Basic measures of Central tendency • • • • •

Arithmetic mean Median Mode Geometric mean Harmonic mean

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The Mean • Definition: The arithmetic average obtained by adding up all the scores and dividing by the total number of scores • Most commonly used measure of central tendency • Appropriate for variables that are intervalratio

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Calculating the mean

X 

X i  N

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Mean Example Scores on a exam 68 90 77 79 80 92

X

X  

68  90  77  79  80  92 X  6 486 X  6 X  81 STAT 512 VCS AGB 2018-19 RJ

N

i

Definition Mean – the average of a group of numbers.

2, 5, 2, 1, 5 Mean = 3 STAT 512 VCS AGB 2018-19 RJ

Mean is found by evening out the numbers

2, 5, 2, 1, 5

STAT 512 VCS AGB 2018-19 RJ

Mean is found by evening out the numbers

2, 5, 2, 1, 5 mean = 3

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How to Find the Mean of a Group of Numbers • Step 1 – Add all the numbers.

8, 10, 12, 18, 22, 26 8+10+12+18+22+26 = 96 STAT 512 VCS AGB 2018-19 RJ

How to Find the Mean of a Group of Numbers • Step 2 – Divide the sum by the number of addends.

8, 10, 12, 18, 22, 26 8+10+12+18+22+26 = 96 How many addends are there?

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How to Find the Mean of a Group of Numbers • Step 2 – Divide the sum by the number of addends. # of addends

16 6) 96 6 36 36

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sum

How to Find the Mean of a Group of Numbers The mean or average of these numbers is 16.

8, 10, 12, 18, 22, 26 STAT 512 VCS AGB 2018-19 RJ

What is the mean of these numbers?

7, 10, 16

11 STAT 512 VCS AGB 2018-19 RJ

What is the mean of these numbers?

2, 9, 14, 27

13 STAT 512 VCS AGB 2018-19 RJ

What is the mean of these numbers?

1, 2, 7, 11, 19

8 STAT 512 VCS AGB 2018-19 RJ

What is the mean of these numbers?

26, 33, 41, 52

38 STAT 512 VCS AGB 2018-19 RJ

Arithmetic Mean Denoted by symbol ‘X’ for sample mean and it is ‘µ’ for population mean.

It is obtained by adding together all the values and dividing this total by the number of observations in a set of data.

The MEAN is the arithmetic average, the average you are probably used to finding for a set of numbers - add up the numbers and divide by how many there are: (80 + 90 + 90 + 100 + 85 + 90) / 6 = 89 1/6.

STAT 512 VCS AGB 2018-19 RJ

Arithmetic mean X

or

=

∑x n μ= ∑x N

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Definition

Median is in the Middle STAT 512 VCS AGB 2018-19 RJ

Definition • Median – the middle number in a set of ordered numbers.

1, 3, 7, 10, 13 Median = 7 STAT 512 VCS AGB 2018-19 RJ

How to Find the Median in a Group of Numbers • Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 27 18, 19, 21, 24, 27 STAT 512 VCS AGB 2018-19 RJ

How to Find the Median in a Group of Numbers • Step 2 – Find the middle number.

21, 18, 24, 19, 27 18, 19, 21, 24, 27 STAT 512 VCS AGB 2018-19 RJ

How to Find the Median in a Group of Numbers • Step 2 – Find the middle number.

18, 19, 21, 24, 27 This is your median number.

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How to Find the Median in a Group of Numbers • Step 3 – If there are two middle numbers, find the mean of these two numbers.

18, 19, 21, 25, 27, 28 STAT 512 VCS AGB 2018-19 RJ

How to Find the Median in a Group of Numbers • Step 3 – If there are two middle numbers, find the mean of these two numbers.

21+ 25 = 46 median 23 2) 46 STAT 512 VCS AGB 2018-19 RJ

What is the median of these numbers?

16, 10, 7 7, 10, 16

10 STAT 512 VCS AGB 2018-19 RJ

What is the median of these numbers?

29, 8, 4, 11, 19 4, 8, 11, 19, 29

11 STAT 512 VCS AGB 2018-19 RJ

What is the median of these numbers?

31, 7, 2, 12, 14, 19 2, 7, 12, 14, 19, 31

13

12 + 14 = 26 2) 26 STAT 512 VCS AGB 2018-19 RJ

What is the median of these numbers?

53, 5, 81, 67, 25, 78 5, 25, 53, 67, 78, 81 60 53 + 67 = 120 2) 120 STAT 512 VCS AGB 2018-19 RJ

Definition

Mode is the most Popular STAT 512 VCS AGB 2018-19 RJ

Definition • Mode – the most popular or that which is in fashion.

Baseball caps are a la mode today.

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Definition

• Mode – the number that appears most frequently in a set of numbers.

1, 1, 3, 7, 10, 13 Mode = 1 STAT 512 VCS AGB 2018-19 RJ

How to Find the Mode in a Group of Numbers • Step 1 – Arrange the numbers in order from least to greatest.

21, 18, 24, 19, 18 18, 18, 19, 21, 24 STAT 512 VCS AGB 2018-19 RJ

How to Find the Mode in a Group of Numbers • Step 2 – Find the number that is repeated the most.

21, 18, 24, 19, 18 18, 18, 19, 21, 24 STAT 512 VCS AGB 2018-19 RJ

Which number is the mode?

29, 8, 4, 8, 19 4, 8, 8, 19, 29

8 STAT 512 VCS AGB 2018-19 RJ

Which number is the mode?

1, 2, 2, 9, 9, 4, 9, 10 1, 2, 2, 4, 9, 9, 9, 10

9 STAT 512 VCS AGB 2018-19 RJ

Which number is the mode?

22, 21, 27, 31, 21, 32 21, 21, 22, 27, 31, 32

21 STAT 512 VCS AGB 2018-19 RJ

Geometric mean • It is defined as Nth root of the product of the N number of observations ie G.M=N

x , x , x ........x 1

2

3

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N

Geometric mean

[

1 logx + logx + ........lo gx Log GM = 1 2 N N ∑logX Log GM = n GM= Antilog of ∑logX

n STAT 512 VCS AGB 2018-19 RJ

]

Geometric mean 29, 8, 4, 11, 19 G.M= 5

29 × 8 × 4 ×11×19

Log GM = ∑logX

n G.M= Antilog∑logx

n

Log G.M = log 29+log8+log4+log11+log19 5 STAT 512 VCS AGB 2018-19 RJ

Harmonic mean=

n ∑(1/X)

1

H.M=

1

1 +

x

1

1 + ...........

x

x

2

N n

Or

N HM =

1 1 1 + + ......... x x xn 1

2

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