descriptive statistics - part 1

8/8/2019 Descriptive Statistics - Part 1

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12/18/2010 Unitedworld-PGPM 1

Statistics I

Prof. Madhu Iyengar

Descriptive Statistics


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Statistics & Descriptive Statistics

To behold is tolookbeyondthe fact; toobserve, to go beyond the

observation. Lookat the worldof people, andyouwillbe overwhelmedby

what yousee. But select from thatmass of humanity a well-chosen few,

andobserve them with insight, and they will tellyoumore than all the

multitudes together.

Descriptive orunivariate statistics are used todescribe a particular

sample or a particular individualwithin a sample.The data are limited to

describingonlyone variable, group, or individual. Any conclusions thatare made cannot be extended to anyone outside that particular sample.

12/18/2010 2Unitedworld-PGPM


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Lesson Plan


Measures of Central Tendency

Mean

Median

Mode

Weighted Mean

Combined Mean


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Need for Averaging

To find one value representing whole data

To enable comparison

To establish relationship

To derive inferences about population from sample

To aid decision making


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Central Tendency

A single value that represents the characteristics of

the entire available raw data is called the central

tendencyor the average .This also depicts the

behaviour of the data about the concentration of the

values in the central part of the distribution


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Arithmetic Mean

The most simple and frequently used average. Example:

No. of patients coming to the OPD per day if daily

data is given for a week.100 , 45, 79 , 47, 56, 87, 69

The Mean of the observation is :

110+ 45+ 79+ 47 + 56+ 87+69 /7

69 patients

x = Sum of all observations/number of observations=x/n orXX == Xi/Xi/ nn

ii!! XSXSnn


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Arithmetic Mean contd.

8

Y = 2 x + 3

Linear

Z = x2

Convex

W = X

Concave

For a frequency distribution

Mean = fx/f

For a Grouped Data (Class Interval Series)

Mean = fm/f

m = class mark/mid-point of the interval

12/18/2010 Unitedworld-PGPM


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Sample Mean f

58 45

64 42

77 8362 38

52 45

N = 253

fX

2610

2688

63912356

2340

7 fX = 16385

X = 7 fX/Nt

= 16385/253 = 64.76

Mean Discrete Series


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Exercise

Raw Data:10.3 4.9 8.9 11.7 6.3 7.7

nn

X =X = Xi/ nXi/ n

ii!! XSXSnn



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WEIGHTED MEAN

A Variant of Arithmetic Mean, weighted mean takes into

consideration the relative weight assigned to each of thevalues. Simple arithmetic mean assigns equal weightageto all observed values.

Used for calculating stock price movements, goodwill ofa company, market capitalisation etc.

WM = wx/w



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Weighted

Average -

Example

Product RoI Sales (Mn Rs) Weight RoIxW

A 10 400 0.20 2.00

B 30 200 0.10 3.00

C 5 900 0.45 2.25

D 20 500 0.25 5.00

Total 65 2000 1.00 12.25

Wt.Av.


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Positional Averages- Median

Median is the middle value of a series in any order

of magnitude

Median is the 50th percentile below which 50% of the

observations fall

Median divides the whole data set into equal halves


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Median contd..

For ungrouped data

Median = (n+1/2)th item

ForFrequency Distribution

Median = (n+1/2)th item falling under cumulative

frequency.

For grouped data

Median = l + (n/2 c) x i/f

l lower median class

i class interval

c- previous cumulative frequency

f- frequency of the median class


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Median

1. Measure of Central Tendency

2. Middle Value In Ordered Sequence

If Odd n, Middle Value of Sequence

If Even n, Average of 2 Middle Values

3.Position of Median in Sequence (n + 1) /2

4. Not Affected by Extreme Values


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Median Example

Odd-Sized Sample

Ordered: 21.5 22.6 22.8 23.7 24.1

PositioningPositioning PointPoint

Median = 22.8Median = 22.8

!! !! !!

n +1n +1

22

5 +15 +1

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Raw Data: 24.1 22.6 21.5 23.7 22.8

Position: 1 2 3 4 5


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Median Example

Even-Sized Sample

Raw Data: 10.3 4.9 8.9 11.7 6.3 7.7

Positioning Point

Median

!! !! !!

!! !!

n +1n +1

22

6 +16 +1

22

33 55

7.7 + 8.97.7 + 8.9

22

8.38.3

..

Ordered: 4.9 6.3 7.7 8.9 10.3 11.7Position: 1 2 3 4 5 6


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Median - frequency distribution

X freq Cumulative freq

10 3 3

11 2 5

12 4 9

14 1 1016 2 12

19 2 14

20 1 15

What is the median grade?

E.g.,A stats class received the following marks out of 20 on their

first exam.


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Marks obtained out of 125 Numbers of students

5 - 25 7

25 - 45 15

45 - 65 18

65 - 85 12

85 -105 6

105 -125 2

For a class consisting of 60 students, a test on English was conducted.

The marks obtained by the students are given below:

What is the median marks obtained by the students in the class?

Median Class Interval Series


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Mode

1.Measure of Central Tendency

2.Value That Occurs Most Often3.Not Affected by Extreme Values

4.May Be No Mode or Several Modes

5.May Be Used forNumerical & CategoricalData


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Mode - An example

The mode is the most frequently occurring number in a set

of data.

E.g., Find the mode of the following numbers

15, 20, 21, 23, 23, 23, 25, 27, 30

Also, if there are two modes, the data set is bimodal.

If there are more than two modes, the data set is said to be

multimodal.

In a class interval series, mode is calculated:

l + {(f-f1)/2f-f1-f2 }*i


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Find the mode for the class interval series given

Exercise


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Thinking Challenge

Youre a financial analyst.

You have collected the

following closing stock pricesof new stock issues: 17, 16,

21, 18, 13, 16, 12, 11.

Describe the stock prices

in terms ofcentral tendency.


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Central Tendency Solution

Mean

XX

XX

nn

XX XX XXii

ii

nn

!! !!

!!

!!

!!11 11 22 88

88

1717 1616 2121 1818 1313 1616 1212 1111

88

1515 55

00

..


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Median

Raw Data: 17 16 21 18 13 16 12 11

Ordered: 11 12 13 16 16 17 18 21

Position: 1 2 3 4 5 6 7 8

PositioninPositionin g Pointg Point

MedianMedian

!!

!!

!!

!!

!!

nn 11

22

88 11

2244 55

1616 1616

22

1616

..


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ModeMode

Raw Data: 17 16 21 18 13 16 12 11

Ordered: 11 12 13 16 16 17 18 21


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Summary of

Central Tendency Measures

Measure Equation Description

Mean Xi

/ n Balance Point

Median (n+1) Position2

Middle ValueWhen Ordered

Mode none Most Frequent


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Combined Mean


An analysis of monthly wages paid to the workers of two firms X and Y

belonging to the same industry gives the following results

Firm X Firm YNumber of workers 500 600

Average daily wage Rs.186.00 Rs.175.00

Find the combined mean.

X12 = (n1X1 + n2X2 )/(n1 + n2)

X12 = (500*186 +600*175) /(500+600) = 180


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A group of salesmen from the same industry consists of

some sales men who have 6 years of experience and

others who have 12 years of experience. Twenty-fivepercent of the salesmen in the group have 12 years of

experience and their average salary is Rs.10,000 per

month. The average salary for the entire group is

Rs.7,000.

What is the average salary of the salesmen who have 6years of experience?

Combined Mean- Exercise


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Questions


1. Mean of 100 observations is 40. But it was found

that, at the time of computation two items are wrongly

taken as 30 and 27 instead of 3 and 72. What is the

correct mean?2. What is the median value of the following data?

391, 384, 591, 407, 672, 522, 777, 753, 2488, 1490

3. The mean height of 25 male workers in a factory is

66 inches and the mean height of 35 female workers

in the same factory is 60 inches. Find the averageheight of workers in the factory .


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A Recap


Measures of Central Tendency

Mean

Median

Mode

Weighted Mean

Combined Mean


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Formulae Recap

Mean Raw Data

Discrete Series

Class Interval Series

Weighted Mean

Combined Mean

Median Raw Data

Discrete Series

Continuous SeriesMode Raw Data

Discrete Series

Continuous Series



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Q & A

Any Queries?

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