2 central tendency

8/4/2019 2 Central Tendency

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Properties – describing quantitative

data

Numerical values of an observation around which

most numerical values of other observations in the

data set show a tendency to cluster or group

Extent to which values are dispersed around the

central value called variation.

Extent of departure of numerical values from

symmetrical distribution around the central valuecalled skew ness



Requisites of a measure of central

tendency

It should be rigidly defined

It should be based on all the observations

Easy to understand and calculate Should have sampling stability

Should not be unduly affected by extreme

observation



MEASURES OF CENTRAL

TENDENCY

Averages of PositionThe Mode

The Median

Mathematical Averages

The Mean

The Symmetrical Distribution The Positively Skewed Distribution

The Negatively Skewed Distribution



Mode

A measure of central tendency Value that occurs most often

Not affected by extreme values

Used for either numerical or categorical data There may be no mode or several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode



Mode

Mode – measure of location recognized by the

location of the most frequently occurring

value of a set of data

Sales during 20 days period

53,56,57,58,58,60,61,63,63,64,64,65,65,67,68,71,71,71,71,74 (ascending order data)



Mode for frequency distribution

Sales Volume (Class

Interval)

No. of Days (Frequency)

53-56 257-60 4

61-64 5

65-68 4

69-72 4

72 and above 1

Frequency distribution of sales per day



Mode: The Category or Score with the

Largest frequency(or %)

The mode is always a category or score

The mode is not necessarily the category

with the majority(more than 50% of thecases)

The mode is the only measure of central

tendency for nominal variables Some distributions are bimodal



Mode for grouped data,

M0 = L + f m – f m-1 h

2 f m – f m-1 – f m+1



THE MEDIAN – measuring

qualitative characters

The median is a measure of centraltendency for variables which are at leastordinal.

The median represents the exact middleof a distribution.

It is the score that divides thedistribution into two equal parts



Finding the Median in sorted data

“How satisfied are you with your health insurance?

Responses of 7 Individuals

very dissatisfied

very satisfied

somewhat satisfied

very dissatisfied

somewhat dissatisfied

somewhat satisfied

very satisfied

Total(N) 7



To locate the median

Arrange the responses in order from lowest to highest

(or highest to lowest): Response

very dissatisfied

very dissatisfied

somewhat dissatisfied

somewhat satisfied ( The middle case =Median)

somewhat satisfied

very satisfiedvery satisfied

_________________________________________________



Summary :Locating the Median with

N=Odd

The median is the response associated with the

middle case.

You find the middle case by :(N + 1) 2

Since N= 7, the middle case is the (7 + 1)

2, or the 4th case

The response associated with the 4th case is

“somewhat satisfied”. Therefore the median is:

Somewhat satisfied.



To locate the median (N=Even)

Suicide rates of cities

7.44, 10.00, 12.26, 12.61, 13.38, 14.11, 14.30, 14.78

The median is located halfway between

the two middle cases. When the variable

is interval we can average the two middlecases.

Median = 12.61 + 13.38 = 12.99

2



Median

Robust measure of central tendency Not affected by extreme values

In an ordered array, the median is the

“middle” number If n or N is odd, the median is the middle number

If n or N is even, the median is the average of thetwo middle numbers

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5



Age of

Automobiles Frequency Cumalative Frequency

0-4 13 13

4-8 29 42

8-12 48 9012-16 22 112

16-20 8 120

120

Median

class

Median for grouped data

Med = L + (n /2) – cf h

f



Partition Values:

Quartiles, Deciles, and Percentiles

Quartiles – Divide an ordered data set into 4

equal parts - 2nd Quartile - Median

Deciles – Divide an ordered data set into 10

equal parts - 5th Decile - Median

Percentiles – Divide an ordered data set into

100 equal parts - 50th Percentile - Median



Quartile for a grouped data,

Qi = L + i(n /4) – cf h; i = 1,2,3

f

Decile for a grouped data,

Di = L + i(n /10) –

cf h; i = 1,2…9 f

Percentile for a grouped data,Pi = L + i(n /100) – cf h; i = 1,2…99

f



_____________________________

Mean. The arithmetic average obtained byadding up all the scores and dividing by the

total number of scores.___________________________________________________________



Objectives of an Average

Determine one single value that may be

used to describe the character sticks of

entire series. Facilitate comparison at a particular point of

time

Facilitate statistical inference Helps in decision making process



The Mean_________________________________________________________________Mean. The arithmetic average obtained by adding up all the scores anddividing by the total number of scores.

_________________________________________________________________

Y = raw scores of the variable y__Y = the mean of y

Y = the sum of all the y scores

N = the number of observations

N

Y Y

C i R i I di Ci i Fi di h M



CITY

Mumbai

Delhi

Kolkatta

Chennai

Banglore

Hyderabad

Baroda

Chandigarh

Meerut

Bhopal

Honolulu

Jaipur

Patna

Kanpur

Ajmer

Crime RATE per 1000

29.3

28.9

32.936.5

25

14.7

58.4

48.8

12.8

21.8

3.4

6.6

40.6

12.9

19.8

Total 392.4

16.2615

4.392

N

Y Y Mean

Crime Rate in Indian Cities:Finding the Mean



Sample statistic – a numerical value used as

a summary measure using data of thesample for estimation or hypothesis testing

Population parameter - a numerical value

used as a summary measure using data of

the population



Mean (Arithmetic Mean)

Mean (arithmetic mean) of data values

Sample mean

Population mean

1 1 2

n

i

i n

X X X X

X n n

1 1 2

N

i

i N

X X X X

N N

Sample Size

Population Size



Arithmetic mean of ungrouped raw

data

Direct method

Indirect method (short cut method)



Finding the mean in a frequency distribution

When data are arranged in a frequency distribution, we must

give each score its proper weight by multiplying it by its

frequency. We use the following formula to calculate the mean:

__

Y = Σ f Y N

where

__Y = the mean

f Y = a score multiplied by its frequency

Σ f Y = the sum of all the f Y’s N = the total number of cases in the distribution



Calculating the Mean from a

Frequency Distribution

# of Children(Y)

0

1

2

3

4

5

6

7

Total

Frequency(f)

12

25

733

333

183

26

15

12

1339

Frequency*Y(fY)

0

25

1466

999

732

130

90

84

3526

6.21339

3526

N

fY Y



Weighted Mean

Used when values are grouped by frequency or

relative importance

28

87

126

45

FrequencyDays to

Complete

28

87

126

45

FrequencyDays to

Complete

Example: Sample of26 Repair Projects

Weighted Mean Daysto Complete:

days 6.31 26

164

28124

8)(27)(86)(125)(4

w

xwX

i

iiW



Indirect method

The human resource manager at a city

hospital began a study of the overtime hours

of the registered nurses. Fifteen nurses were

selected at random and following overtime

hours were recorded during a month:

13 13 12 15 17 15 5 12 6 7 12 10 9 13 12

5 9 6 10 5 6 9 6 9 12



The following distribution gives the pattern of overtime work

done by 100 employees of a company. Calculate the average

overtime work done per employee

Overtime

No. of

Employees

Mid

Value d=(m-A)/5 fd

10-15 11 12.5 -2 -22

15-20 20 17.5 -1 -20

20-25 35 22.5 0 0

25-30 20 27.5 1 20

30-35 8 32.5 2 16

35-40 6 37.5 3 18

12

Arithmetic mean of grouped (classified) data

Direct & Step deviation method)



30

Geometric Mean Geometric Mean of a set of n numbers isdefined as the nth root of the product of

the n numbers and is used to averagepercents, indexes, and relatives.

The formula is: ( X i > 0)

More directly measures the change overmore than one period

Geometric Mean Arithmetic Mean

1 2G

n

n X X X X



Relationship between Mean,

Median and Mode

M0 = 3Median – 2Mean

OR

Mean – Mode = 3 (Mean – Median)



The Shape of Distributions

Distributions can be either symmetricalor skewed, depending on whether thereare more frequencies at one end of the

distribution than the other.

?



Symmetrical

Distributions

A distribution is symmetrical if thefrequencies at the right and left tails of the distribution are identical, so that if it

is divided into two halves, each will be themirror image of the other.

In a unimodal symmetrical distributionthe mean, median, and mode areidentical.



1.4. Shape of a Distribution

Describes how data is distributed

Symmetric or skewed

Mean = Median = ModeMean < Median < Mode Mode < Median < Mean

Right-SkewedLeft-Skewed Symmetric

(Longer tail extends to left) (Longer tail extends to right)



Choosing a Measure of Central Tendency

IF variable is Nominal..

– Mode

IF variable is Ordinal...

– Mode or Median(or both)

IF variable is Interval-Ratio and distribution is

Symmetrical…

– Mode, Median or Mean

IF variable is Interval-Ratio and distribution is

Skewed…

– Mode or Median



Calculate the mean, median and mode for the

following data pertaining to marks in statistics.

There are 80 students in class and the test is of 140 marks.

Marks more than No. of Students

0 80

20 76

40 50

60 28

80 18

100 9

120 3

2 central tendency

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