stat 08,moctendency

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Jahangirnagar University © Islam, M.T. http://sites.google.com/site/kjatbd/

Measures of Central Tendency

Md. Tarikul Islam Jahangirnagar University, Bangladesh


The Course: Topics

No Topics

01 Basics of statistics and recap!

02 Collection of data

03 Presentation of data

04 Measures of central tendency

05 Measures of variation

06 Skewness, moments, and kurtosis

07 Correlation analysis

08 Regression analysis

09 Forecasting and time series analysis

10 Probability

11 Sampling

Conte

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Recap

q So far we have seen

ü  Recapping o  Research, types of research, and research methodology

Ø In core of all there is DATA

ü  Basics of statistics o  Data, types of data o  Place of data in statistics with the definition and

characteristics of statistics

ü Data collection and sampling o  What data to collect? From where? How? o  Sampling methods

ü Data presentation o  Classification, tabulation, and graphs


Topics Today

q  Topics Today

ü Central tendency calculation o  What is central tendency?

Ø Any idea?

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Central tendency

q Central tendency

ü Central tendency o  Mainly average calculation o  Average is neither the highest not the lowest value. Rather it’s

a value somewhere middle in the data series or data set o  That’s why this is called measures of central tendency


Why to know average?

q Why to know average?

ü  Reasons to know average o  One single value to represent a large number of population o  Enables comparison

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Features of a good average

q  Features of a good average

o  Should be easy to understand o  Should be simple to compute o  Should be based on all observations o  Should be rigidly defined o  Should be capable of further treatment o  Should have sampling stability o  Should not be affected by the presence of the extreme data


How to calculate average?

q How to calculate average?

ü Major ways o  Arithmetic mean o  Median o  Mode o  Geometric mean o  Harmonic mean

ü  In all cases there are two scenarios Ø One with ungrouped data Ø And the other with grouped data

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Arithmetic mean: Introduction

q  Introduction

o  Most popular and the easiest one of all averages »  Obtained by adding together all the observations

and then by dividing this total with the number of observations

o  Two ways to compute Ø Direct method Ø Short cut method

»  Also for two scenes - grouped and ungrouped data


AM: Ungrouped data-Direct Method

q  Example o  Given data

Ø 1487, 1493, 1502, 1446, 1475, 1492, 1572, 1516, 1468, 1489

o  Arithmetic Mean

q  Formulae o  Here

Ø X = particular observation Ø N = total number of observations

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AM: Ungrouped data-Short-Cut Method

q  Example

ü  Let’s assume A = 1460

o  So the Arithmetic Mean is


Ø A = Assumed or Arbitrary Mean, d = X - A, N = total number of observations

X d = X-1460

1487 +27

1493 +33

1502 +42

1446 -14

1475 +15

1492 +32

1572 +112

1516 +56

1468 +8

1489 +29

+340

!"#$! !!"#!" ! !"#"


AM: Grouped data-Direct Method

q  Example

ü  Arithmetic Mean


Ø f = frequency, X = class mid point, N = total frequency

Profits X f fx

200-400 300 500 150000

400-600 500 300 150000

600-800 700 280 196000

800-1000 900 120 108000

1000-1200 1100 100 110000

1200-1400 1300 80 104000

1400-1600 1500 20 30000

1400 848000

8480001400

!=!605.71

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AM: Ungrouped data-Short-Cut Method

q  Example

ü  Assumed mean = 900 ü  d = (X-900) ÷ (400-200) ü  Arithmetic Mean is


Ø A = assumed mean, f = frequency, N = total frequency, X

= class mid point, i = equal class interval, and

Profits X f d fd

200-400 300 500 -3 -1500

400-600 500 300 -2 -600

600-800 700 280 -1 -280

800-1000 900 120 0 0

1000-1200 1100 100 +1 +100

1200-1400 1300 80 +2 +160

1400-1600 1500 20 +3 +60

1400 -2060

900!+!!20601400

x200!=!605.71


Arithmetic mean: Pros and Cons

q Arithmetic mean

o  Strengths Ø Most widely used one in practice Ø Carries six out of seven featured of a good average

o  Weaknesses Ø Does not consider the presence of the extreme values in

the series Ø How?

»  The average of 10,15,20 is 15 »  The average of 1, 2, 42 is also 15

o  To avoid this one can put weights

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Arithmetic mean: Weighted

q Weighted Arithmetic Mean

o  We need to put weight if all the observations does not carry same importance in a particular population

o  Weight can be given in two ways Ø The actual weights Ø The imaginary or arbitrary weights

o  Formulae for the weighted arithmetic mean is o  Where

Ø X = variable Ø W = weights attached to the variable X

Xw!

!=!WX"W"


Arithmetic mean: Properties

q  The algebraic sum of the deviations of all observations from the mean is always zero q  Here mean is 30

X (X-‐X bar)

10 -‐20

20 -‐10

30 0

40 +10

50 +20

150 0

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q  The sum of the squared deviations of all observations from the mean is minimum q  Here mean is 30

X (X-‐X bar) (X-‐X bar)2

10 -‐20 400

20 -‐10 100

30 0 0

40 +10 100

50 +20 400

150 0 10000



q Properties

o  With the presence of mean and number of observations of two or more related groups we can compute the combined average for those groups with the formulae

X12!

!=!N1X1

!

!+!N2X2

!

N1!+!N

2

X123!

!=!N1X1

!

!+!N2X2

!

!+!N3X3

!

N1!+!N

2+!N

3

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Wrapping

q  Therefore ü Arithmetic Mean part is over ü Coming up

o  Median

Any Question?


Median

q Median

o  The value that is in the middle position of the all observations in an ordered sequence of values

o  It’s a positional average; half of the observations in a set of data are lower than it and half are greater than it

o  Calculation is also for Ø Ungrouped data and

»  When N = Odd number »  When N = Even number

Ø Grouped data

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Median: Ungrouped data, N = Odd

q  Example

ü  First arrange the data o  Either ascending or descending order o  Then use the formulae o  Median is

q  Formulae

o  Median would be the middle position, th position

o  If there are 7 data then 4th position would be median position

Sl. No Wages

1 1580

2 1600

3 1606

4 1640

5 1650

6 1660

7 1690

median!=!N+12

N+12!=!7+12!=!4th!position!=!1640


Median: Ungrouped data, N = Even

q  Example ü  First arrange the data

o  Either ascending or descending order o  Then use the formulae o  Median should be

o  4.5th position = (1640+1650) ÷ 2 = 1645

q  Formulae

o  Median would be the middle position, th position

o  Same formulae but the application is bit different

median!=!N+12

Sl. No Wages

1 1580

2 1600

3 1606

4 1640

5 1650

6 1660

7 1690

8 1700

N+12!=!8+12!=!4.5th!position

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Median: Grouped Data-1/2

q Grouped data

o  In the formulae

Ø L = Lower limit of the median class Ø P.C.F. = preceding cumulative frequency to the median

class Ø f = frequency of the median class Ø  i = class interval of the median class Ø N = total frequency

o  Therefore, first job is the determine the median class Ø That is N ÷ 2


Median: Grouped Data-2/2

Groups

Frequency-f

Cumulative f

0-10 4 4

10-20 12 16

20-30 24 40

30-40 36 76

40-50 20 96

50-60 16 112

60-70 8 120

70-80 5 125

q Here o  Median class = N ÷ 2 = 125 ÷ 2 = 62.5 o  This goes to the class 30-40

Ø So this is the median class

o  So the median is

30!+!62.5!4036

!x!10!=!36.25

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Median: Pros and Cons

q Pros and Cons

ü  Strengths o  Not influenced by extreme values and specially useful for the

open end distributions

ü Weaknesses o  This is positional average and therefore not determined by

each and every observations o  Also it can’t be used in algebraic treatment


Wrapping

q  Therefore ü Arithmetic Mean and median finished ü Coming up

o  Mode

Any Question?

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Mode

q Mode

o  The observation that comes most in a set of observations Ø It tells us about the single most occurred value Ø Graphically it is the value on the X-axis below the peak or

the highest value of the frequency curve

Ø In a set of data »  There can be more than one mode; we call it

bimodal or multimodal situation »  It means the data are not homogenous

Ø Calculation is for both »  Grouped and ungrouped data


Mode: Ungrouped Data

q Ungrouped data

o  For ungrouped data make tally and see which comes most times!

Size 5 6 7 8

Persons 12 15 25 30

Mode is 8 as it comes for the most times

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Mode: Grouped Data-1/2

q Grouped Data

ü Here o  L = Lower limit of the modal class o  f1 = frequency of the modal class o  f2 = frequency of the class preceding the modal class o  f2 = frequency of the class succeeding the modal class o  i = class interval of the modal class

ü  Therefore, first job is the determine the modal class o  That is the group having the highest frequency


Mode: Grouped Data-2/2

q Grouped data ü Here

o  Modal class is 64-66 o  So the mode is

Groups Frequency-f

Below 60 12

60-62 18

62-64 25

64-66 30

66-68 10

68-70 3

70-72 2

64+30!25

2x30!25!10x2!=!64.4

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Mode: Cautions in calculation

q Cautions in calculation

ü Cautions in calculation o  You can follow the formula only if the class interval is uniform

throughout

o  If there is one mode its called unimodal, having two is bimodal, and more than two is multimodal Ø In this case its not possible to calculate mode by using this

formula Ø May be the data set need to be changed; this is called ill

defined modes Ø You can adjust it by the formula mode = 3 median - 2

mean


Mean, mode, and median

q Mean, mode, and median

o  In any distribution three could be equal; its called symmetrical distribution

o  Conversely its called asymmetrical or skewed distribution o  In a moderately skewed distribution there is a relation in

between these three Ø Mode = 3 median - 2 mean

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Mode: In Graph

q  In graph


Mode: In Histogram

q  In Histogram

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Mode: Pros and Cons

q Pros and Cons

ü  Strength o  Its not affected by extreme values o  Can be used in qualitative judgments

ü Weaknesses o  Can be misleading as no rigs way to calculate and varied

ways give varied answers o  There can be more than one mode


Wrapping

q  Therefore ü Arithmetic Mean, median, and mode finished ü Coming up

o  Geometric mean

Any Question?

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Geometric mean: Defining

q  Defining the Geometric mean

o  GM is the measure that shows the average percentage change (increase/decrease) over the years Ø For example how much average increase in sales over

the last five years

o  This is defined as the Nth root of the product of N observations of a series

o  To make the calculations easier log is being used


Geometric mean: Formulae

q  GM is the Nth root, so

G.!M.!=! X1

N *X2*..... * X

N

log!G.!M.!=!logX

1+ logX

2+ ......logX

3

N!=!

logX!N

G.!M.!=!antilog!logX!N

"

#

$$

%

&

''

G.!M.!=!antilog!f!logX!N

"

#

$$

%

&

''

OR

OR

For ungrouped data

For grouped data

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GM-An example

q GM-An example

Ø The annual growth rate of factory output in 5 years are 5, 7.5, 2.5, 5, and 10 respectively. What is the compound rate of growth per annum for the period?


GM- and the solution!

q  GM- and the solution!

o  GM = AL (10.1259 ÷ 5) = AL (2.0252) = 105.9 o  The compound growth rate = 105.9-100 = 5.9%

Growth rate Output @ end of year Log X 5.0 105 2.0212

7.5 107.5 2.0314

2.5 102.5 2.0107

5.0 105 2.0212

10 110 2.0414

∑log X = 10.1259

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Harmonic mean

q Harmonic mean

o  This is based on the reciprocal of the numbers averaged

o  Formula for ungrouped data

o  Formula for grouped data

H.!M.!=!N

1X1

+1X2

+1X3

+ ...........+ 1XN

!

"##

$

%&&

!=!N

1X

!

"##

$

%&&'

H.!M.!=!N

f!!*!! 1X

!

"##

$

%&&'


HM: Example-Ungrouped data

ü  So harmonic mean is

X 1/X

10 0.10

20 0.05

25 0.40

40 0.025

50 0.02

∑1/X = 0.235

N1X

!!!=!!

50.235

!!=!!21.28

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HM: Example-Grouped data

ü  So the harmonic mean is Variable X F F*(1/X)

0-10 5 8 1.600

10-20 15 15 1.000

20-30 25 20 0.800

30-40 35 4 0.114

40-50 45 3 0.067

N = 50 f*1/x = 3.581

N

f!!*!! 1X

!!!=!!

503.581

!!=!!13.96


AM, GM, HM

q AM, GM, HM

o  If the data are not identical then in a distribution AM>GM>HM o  But it could be AM=GM=HM if the observations are identical

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When to use what?

q When to use what?

o  If the data are badly skewed then avoid mean o  If the data are gappy around the middle avoid median o  If the data are unequal in class interval avoid mode

o  Go through the book to know more about each measure; when not to use them!


General limitations on average

q General limitations on average

o  Simply one single value representing a group; might lead to wrong conclusion

o  It might give a value that is not even present in the data o  It might give absurd results like 4.5 persons o  It can’t give any idea about the formation of the series

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q Thank You!

q Any Question?!

stat 08,moctendency

Documents

sampling o

treatment o

defined o

good average o

types of data o place

o arithmetic mean q

average calculation

data set o thats