measurement scales - central tendency - dispersion

8/3/2019 Measurement Scales - Central Tendency - Dispersion

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Quantitative Methods 2010

Measurement Scales

Measures Of Central TendencyMeasures Of Dispersion

1Quant Methods 2010 - Kingston


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Overview

The 4 scales of measurement

Objectives of averaging

Requisites of a good average Types of averages

Mathematical and positional averages

Range, standard deviation and variance



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Scales Of Measurement

Quant Methods 2010 - Kingston 3


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Flat No. Refrigerator

101 LG102 Voltas

103 LG

104 LG

201 Samsung

202 Voltas

203 LG

204 Godrej

301 Godrej

302 LG

303 Voltas

304 Voltas

401 Samsung

402 LG403 LG

404 Godrej

501 Godrej

502 Voltas

503 LG

504 Voltas

Referigerator Models

In HILL CREST Society

Seat No. B.Sc. Physics

1450 Pass1451 Pass

1452 Second

1453 Pass

1454 Pass

1455 First

1456 Second

1457 Pass

1458 Pass

1459 First

1460 Second

1461 Pass

1462 Pass

1463 Second1464 Pass

1465 First

1466 Pass

1467 Pass

1468 Second

1469 Second

Passing Class Obtained

In B.Sc. Physics

Sample No Temp

1 4322 445

3 429

4 433

5 442

6 443

7 447

8 441

9 448

10 445

11 443

12 443

13 443

14 44615 447

16 441

17 439

18 440

19 447

20 442

Melting Temperature

Of Solder, C

Emp. No. mg/dL

2660 178.5

2661 152

2662 149.3

2663 164

2664 155

2665 138.5

2666 149.3

2667 146.5

2668 164.2

2669 178.8

2670 149.3

2671 138.2

2672 155.1

2673 146.5

2674 148.2

2675 167

2676 158.1

2677 143.2

2678 139.2

2679 143.3

Cholesterol Level

Of Managers


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Four Types Of Scales

Nominal (also called categorical) Measurement says which category observation falls in

Example: Refrigerators data (previous slide)

Central tendency: mode Neither mean nor median cannot be defined

Ordinal Measurement says what is rank of an observation compared to others.

Example: B.Sc results Central tendency: mode or median. Mean cannot be defined

Interval Measurement is position of observation on a scale with arbitrary zero.

Example: Melting temperature of solder

Central tendency: mode, median or arithmetic mean

Ratio Measurement is position of an observation on a scale with real zero.

Example: Cholesterol level of managers

Central tendency: mode, median or arithmetic mean


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Measures of Central Tendency



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Objectives Of Averaging

To get a single value that in some sense is

representative of the individual data-points.

To facilitate comparisons between data-sets.



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Requisites Of A Good Average

Should be

Well defined

Easy to compute

Capable of simple interpretation

Dependent on all observed values

Not influenced much by outliers

Similar from one random sample to next

Capable of mathematical manipulation



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Types Of Averages

9

Averages

Mathematical Positional

Arithmetic

Mean

Geometric

Mean

Harmonic

Mean

Median Mode

Quant Methods 2010 - Kingston


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Arithmetic Mean

Continuous, Ungrouped Data

If data is individual continuous observations

called xi , mean is written:

= (sum of all x) / total no of x

= / n

10

x

x

n

i

ix1

Company Name Profit, Rs Crore

Ashok Leyland 9.19

Classic Finance 4.27

Empire Finance 1.74

First Leasing 5.71

Llyods Finance 4.8

Total 25.71

Arith Mean (average) 5.142Quant Methods 2010 - Kingston


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Arithmetic Mean

Discrete, Ungrouped Data

If data is discrete values, called xi , with

frequencies (counts) fi :

= (sum of all frequency-value cross-products) /

(sum of all frequencies)

11

x

n

i

i

n

i

ii

f

xf

x

1

1

Profit, Rs. Crores No Of Companies Cross-Prod

16 15 240

20 12 240

24 8 192

25 7 175

31 8 248

Total 50 1095

Arith Mean: 21.9



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Arithmetic Mean

Grouped Data

If data is continuous values called xi , grouped

into classes, with mid-points mi and

frequencies fi :

= (sum of all freq-midpoints cross-prods) /

(sum of all frequencies)

12

n

i

i

n

i

ii

f

mf

x

1

1

Dividend, % No Of Companies Midpoint Cross-Prod

0 - 8 12 4 48

8 - 16 20 12 240

16 - 24 25 20 500

24 - 32 28 28 784

32 - 40 15 36 540

Total 100 2112

Average 21.12



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Weighted Average

If individual values are not to be treated as

equally important, but weights are assigned

(according to some rationale), we can

compute weighted average

13

n

i

i

n

i

ii

w

xw

x

1

1Company Name Profit, Rs Crore Weightage Cross-Prod

Ashok Leyland 9.19 5 45.95

Classic Finance 4.27 3 12.81

Empire Finance 1.74 5 8.7

First Leasing 5.71 4 22.84

Llyods Finance 4.8 1 4.8

Total 25.71 18 95.1

Arith Mean (average) 5.142 5.28



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Median

Middle value of a sorted series of data

Positional average

Value below which half the observations fall.

If there are odd number of data points, middle value isMedian.

If there are even number of data points, average of the two

central points is Median

14

Median

23 34 38 52 57 38

23 34 52 57 43



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Mode

If data is individual observations, most frequentlyoccurring value is Mode.

If data is grouped into frequency distribution,

Mode = , where

Lmo = lower limit of modal class

f1 and f2 = frequencies of preceding and succeeding

class respectively

f = frequency of modal class

W = class interval.

15

Wfff

ffLmo

21

1

2



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Examples For Mode

16

7

4

5

3

42

6

4

3

5

Mode 4

Sales, Crore No Of Co's

0 - 8 19

8 - 16 25

16 - 24 36

24 - 32 43

32 - 40 28

Lmo = 24, f2 = 28, f1 = 36, f = 43, W = 8

Mode = 26.55

0

10

20

30

40

50

0 - 8 8 - 16 16 - 24 24 - 32 32 - 40

Series1



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Measures Of Dispersion



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Range

Difference between highest and lowest data

point.

Convenient to calculate, but does not give

good idea of dispersion because

Many different dispersions can have same range.

Outliers give false idea of dispersion.



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Shortcomings Of Range

19

200 1000 200 1000 200 1000

200 1000

Each of these data-sets have very different dispersion, but same range.



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Mean Absolute Deviation (MAD)

Deviation of interest is deviation from mean.

Sum of deviation of individual data points from

Mean (with sign) will be 0.

MAD =

20

nxxi ||

x |x - x|

2 4

4 26 0

8 2

10 4

Mean 6 2.4Quant Methods 2010 - Kingston


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Standard Deviation & Variance

Concept of population and sample.

Population standard deviation denoted by

Sample standard deviation denoted by s.

Sometimes referred to as root-mean-square (RMS)deviation

Squares of standard deviations, 2 and s2 , are calledvariances (population and sample, respectively).

21

N

xi

i

2)(

1

)(2

n

xx

s

i

i

--- is population mean

--- x is sample mean



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Example Calculating s From Raw Data


PriceRs/Kg (x) (x - xbar) (x - xbar)^2

1 85.5 -1.36 1.8496

2 91.3 4.44 19.7136

3 81.9 -4.96 24.6016

4 84.7 -2.16 4.6656

5 83.6 -3.26 10.6276

6 90 3.14 9.8596

7 88.5 1.64 2.6896

8 92.2 5.34 28.5156

9 84.8 -2.06 4.243610 86.1 -0.76 0.5776

Ave, xbar 86.86

Sum 107.34

s 3.45


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Grouped Data

If data is grouped into classes with midpoints

mi , then

and


n

i

i

n

i

ii

f

mf

x

1

1

n

i

i

n

i

ii

f

xmf

s

1

1

2)(


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Example Calculating s from Grouped Data


Bin Frequency Mid-point, mCross-Prod, fm m - xbar (m - xbar)^2

80-81.9 1 81 81 -6 36

82-83.9 1 83 83 -4 16

84-85.9 3 85 255 -2 4

86-87.9 1 87 87 0 0

88-89.9 1 89 89 2 4

90-91.9 2 91 182 4 16

92-93.9 1 93 93 6 36

Sum f 10 870 112

Ave, xbar 87

s 3.35


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Some Properties Of Std Dev s

s is unaltered if every data point has constantadded to it. sx + c = sx


Price

Rs/Kg (x) (x - xbar) (x - xbar)^2

1 85.5 -1.36 1.8496

2 91.3 4.44 19.7136

3 81.9 -4.96 24.6016

4 84.7 -2.16 4.6656

5 83.6 -3.26 10.6276

6 90 3.14 9.85967 88.5 1.64 2.6896

8 92.2 5.34 28.5156

9 84.8 -2.06 4.2436

10 86.1 -0.76 0.5776

Ave, xbar 86.86

Sum 107.34

s 3.45

Price


1 90.5 -1.36 1.8496

2 96.3 4.44 19.7136

3 86.9 -4.96 24.6016

4 89.7 -2.16 4.6656

5 88.6 -3.26 10.6276

6 95 3.14 9.8596

7 93.5 1.64 2.6896

8 97.2 5.34 28.5156

9 89.8 -2.06 4.2436

10 91.1 -0.76 0.5776

Ave, xbar 91.86

Sum 107.34s 3.45


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Some Properties Of Std Dev s

s is unaltered if every data point has a constantmultiplied to it : scx = csx


Price

Rs/Kg (x) (x - xbar) x - xbar)^

1 171 -2.72 7.3984

2 182.6 8.88 78.8544

3 163.8 -9.92 98.4064

4 169.4 -4.32 18.6624

5 167.2 -6.52 42.5104

6 180 6.28 39.4384

7 177 3.28 10.7584

8 184.4 10.68 114.0624

9 169.6 -4.12 16.9744

10 172.2 -1.52 2.3104

Ave, xbar 173.72

Sum 429.38

s 6.91

Price


1 85.5 -1.36 1.8496

2 91.3 4.44 19.7136

3 81.9 -4.96 24.6016

4 84.7 -2.16 4.6656

5 83.6 -3.26 10.6276

6 90 3.14 9.8596

7 88.5 1.64 2.6896

8 92.2 5.34 28.5156

9 84.8 -2.06 4.2436

10 86.1 -0.76 0.5776

Ave, xbar 86.86

Sum 107.34

s 3.45


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Coefficient Of Variation (CV)

An s is good or bad (low or high)

depending on its relationship to the mean.

Example: s = 3 would be bad ifxbar = 5, but

good ifxbar = 500

CV = s/ xbar

Can be expressed in percent also ( Mult by 100)

CV < 20% considered good.



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Try This Yourself


Break-strength,

kg

1 55.8

2 49.3

3 52.64 57.4

5 58.8

6 47.5

7 52.1

8 61.7

9 53.9

10 56.6

1 Compute the standard deviation ofbreak-strength.

2 Make a new table, adding 5 to each

data point. Compute new standard

deviation. Verify that new standard

deviation is unaltered.

3 Make a new table, multiplying eachdata point by 2. Compute new

standard deviation. Verify that new

standard deviation is multiplied by 2.

4 Group data into classes of width 3,

and show frequency distribution.

Compute standard deviation ofgrouped data.


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Break-strength,

kg (x - xbar) (x - xbar)^2

1 55.8 1.23 1.51292 49.3 -5.27 27.7729

3 52.6 -1.97 3.8809

4 57.4 2.83 8.0089

5 58.8 4.23 17.8929

6 47.5 -7.07 49.9849

7 52.1 -2.47 6.1009

8 61.7 7.13 50.8369

9 53.9 -0.67 0.4489

10 56.6 2.03 4.1209

Ave, xbar 54.57

Sum 170.56

s 4.35

Working Of Q 1


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Adding 5 to each x

Break-

strength,


1 60.8 1.23 1.5129

2 54.3 -5.27 27.7729

3 57.6 -1.97 3.8809

4 62.4 2.83 8.0089

5 63.8 4.23 17.8929

6 52.5 -7.07 49.9849

7 57.1 -2.47 6.1009

8 66.7 7.13 50.8369

9 58.9 -0.67 0.4489

10 61.6 2.03 4.1209

Ave, xbar 59.57

Sum 170.56

s 4.35

Working Of Q 2


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Multiplying each x by 2

Break-

strength,


1 121.6 2.46 6.0516

2 108.6 -10.54 111.0916

3 115.2 -3.94 15.5236

4 124.8 5.66 32.0356

5 127.6 8.46 71.5716

6 105 -14.14 199.9396

7 114.2 -4.94 24.4036

8 133.4 14.26 203.3476

9 117.8 -1.34 1.7956

10 123.2 4.06 16.4836

Ave, xbar 119.14

Sum 682.24

s 8.71

Working Of Q 3


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Bin Frequency Mid-point, m Cross-Prod, fm m - xbar (m - xbar)^2

47.1- 50 2 48.5 97 -5.7 32.49

50.1-53 2 51.5 103 -2.7 7.29

53.1-56 2 54.5 109 0.3 0.09

56.1-59 3 57.5 172.5 3.3 10.89

59.1-62 1 60.5 60.5 6.3 39.69

62.1-65 0

Sum f 10 542 90.45

Ave, xbar 54.2

s 3.01

Working of Q 4


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Measurement Scales Central Tendency - Dispersion


measurement scales - central tendency - dispersion

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