Statistics-Measures of dispersions

MEASURES OF
DISPERSIONS
STATISTICS

MEASURES OF DISPERSIONS
• A quantity that measures the variability
among the data, or how the data one
dispersed about the average, known as
Measures of dispersion, scatter, or
variations.

2. Common Measures of
Dispersion
• The main measures of dispersion
1. Range
2. Mean deviation or the average deviation
3. The variance & the standard deviation

1. RANGE
• It is the difference between the largest and the smallest
observation in a set of data.
• Range = xm – xo
• Its relative measure known as coefficient of dispersion.
• Coefficient of dispersion =
• It is used in daily temperature recording stick prices rate
• It ignores all the information available in middle of data.
• It might give a misleading picture of the spread of data.
om
om
xx
xx
+
−

1. RANGE
• Example:
1. Find the range in the following data.
31,26,15,43,19,10,12,37
Range = xm – xo 33 = 43 – 10
2. Find the range in the following F.D. (Ungrouped)
5 = 8 – 3
Range 5 = 8 – 3
3. Find the range in the following data.
Range = 60 – 10 = 50
X 3 4 5 6 7 8
f 5 8 12 10 4 2
X 10 - 20 20 - 30 30- 40 40 – 50 50 - 60
f 5 8 12 10 4

MEAN (OR AVERAGE) DEVIATION
• It is defined as the “Arithmetic mean of the
absolute deviation measured either from
the mean or median.
• or for ungroup.
• or for grouped.
n
xx
DM
∑ −
=..
N
xxf∑ −
=
N
medianx∑ −
N
medianxf∑ −
=

• Example:
1. Calculate mean deviation from the FD (Ungrouped Data).
MD (x) = 33.6 / 20 = 1.68
X f f.x I x – 4.9 I f I x - 4.9 I
2 3 6 2.9 8.7
4 9 36 0.9 8.1
6 5 30 1.1 5.5
8 2 16 3.1 6.2
10 1 10 5.1 5.1
Total Σf =20 Σf.x =98 Σ f I x - 4.9 I = 33.6

• Exp: Calculate mean deviation from the FD (Grouped Data).
MD (x) = 33.6 / 20 = 1.68
M.D = 23.72 / 14 = 1.69
X f Class Mark
( x )
f.x I x – 6.57 I f I x – 6.57 I
2 – 4 2 3 6 3.57 7.14
4 - 6 3 5 15 1.57 4.71
6 – 8 6 7 42 0.43 2.58
8 – 10 2 9 18 2.43 4.86
10 – 12 1 11 11 4.43 4.43
Total Σf =14 Σ f.x =92 Σ f I x – 6.57 I =
23.72
=92/14=6.57ẋ

• It is an absolute measure.
• It is relative measure is coefficient of M.D.
• Coefficient of M.D. =
• It is based on all the observed values.
median
DM
or
mean
DM ....

• EXAMPLES

THE VARIANCE AND
STANDARD DEVIATION
• It is defined as “The mean of the squares
of deviations of all the observation from
their mean.” It’s square root is called
“standard deviation”.
• Usually it is denoted by (for population of
statistics) S2
(for sample)
• = for ungrouped
2
σ
2
σ
n
xx∑ − 2
)(

• = for grouped
• It is an absolute measure;
• It is relative measure is coefficient of
variation.
•
• Shortcut method
N
xxf∑ − 2
)(2
σ
100. ×=
µ
σ
VC 100
..
.. ×=
x
DS
VC
22
2








−=
∑∑
N
x
N
x
σ
22
2
.








−=
∑∑
N
fx
N
xf
σ
THE VARIANCE AND
STANDARD DEVIATION

• EXAMPLES
THE VARIANCE AND
STANDARD DEVIATION

VARIANCE AND STANDARD
DEVIATION• Example:
1. Calculate Variance and SD from the FD (Ungrouped Data).
Using Short cut method
var = (564 / 20) - (98 / 20) ^ 2 = 28.2 – 24.01 = 4.09
Sd = √ σ^2 = √ 4.09 = 2.02
X f f.x X^2 f.x^2
2 3 6 4 12
4 9 36 16 144
6 5 30 36 180
8 2 16 64 128
10 1 10 100 100
Total Σf =20 Σf.x = 98 Σ f.x^2=564
22
2
.








−=
∑∑
N
fx
N
xf
σ

VARIANCE AND STANDARD
DEVIATION
• Exp: Calculate Variance and Standard deviation from the FD (Grouped Data).
Using Short cut method:
var = (670 /14) - (92 / 14) ^ 2 = 47.85 – 43.18 = 4.67
Sd = √ σ^2 = √ 4.67 = 2.16
X f Class Mark
( x )
f.x x^2 f.x^2
2 – 4 2 3 6 9 18
4 - 6 3 5 15 25 75
6 – 8 6 7 42 49 294
8 – 10 2 9 18 81 162
10 – 12 1 11 11 121 121
Total Σf =14 Σ f.x =92 Σ f.x^2 =670
22
2
.








−=
∑∑
N
fx
N
xf
σ

16
Relative Measures ofRelative Measures of
DispersionDispersion
 Coefficient of Range
 Coefficient of Quartile Deviation
 Coefficient of Mean Deviation
 Coefficient of Variation (CV)
01:38 PM

17
Relative Measures of VariationRelative Measures of Variation
Largest Smallest
Largest Smallest
Coefficient of Range
X X
X X
−
=
+
3 1
3 1
Coefficient of Quartile Deviation
Q Q
Q Q
−
=
+
Coefficient of Mean Deviation
MD
Mean
=
01:38 PM

Coefficient of Variation (CV)Coefficient of Variation (CV)
Can be used to compare two or more
sets of data measured in different
units or same units but different
average size.
01:38 PM 18
100%
X
S
CV ⋅







=

19
Use of Coefficient of VariationUse of Coefficient of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Standard deviation = $5
but stock B is
less variable
relative to its
price
10%100%
$50
$5
100%
X
S
CVA =⋅=⋅







=
5%100%
$100
$5
100%
X
S
CVB =⋅=⋅







=
Both stocks
have the
same
standard
deviation
01:38 PM

20
Five Number SummaryFive Number Summary
The five number summary of a data set consists of the
minimum value, the first quartile, the second quartile, the
third quartile and the maximum value written in that order:
Min, Q1, Q2, Q3, Max.
From the three quartiles we can obtain a measure of central
tendency (the median, Q2) and measures of variation of the
two middle quarters of the distribution, Q2-Q1 for the
second quarter and Q3-Q2 for the third quarter.
01:38 PM

21
The weekly TV viewing times (in hours).
25 41 27 32 43 66 35 31 15 5
34 26 32 38 16 30 38 30 20 21
The array of the above data is given below:
5 15 16 20 21 25 26 27 30 30
31 32 32 34 35 37 38 41 43 66
01:38 PM

22
Hrs22.021}-0.25{2521obs.}5th-obs.0.25{6thobs.5th;Q1ofVALUE
obs.5.25thdatain theobs.th
4
1)1(20
;Q1ofLOCATION
=+=+
=
+
Hrs30.530}-0.50{3103obs.}10th-obs.0.50{11thobs.th10;Q2ofVALUE
obs.th50.10datain theobs.th
4
1)2(20
;2QofLOCATION
=+=+
=
+
Minimum value=5.0 Maximum value=66.0
Hrs36.535}-0.75{3735obs}15th-obs{16th75.0obs15th;3QofVALUE
obs.15.75thdatain theobs.th
4
1)3(20
;3QofLOCATION
=+=+
=
+
01:38 PM

23
Box and Whisker DiagramBox and Whisker Diagram
A box and whisker diagram or box-plot is a
graphical mean for displaying the five number
summary of a set of data. In a box-plot the first
quartile is placed at the lower hinge and the
third quartile is placed at the upper hinge. The
median is placed in between these two hinges.
The two lines emanating from the box are
called whiskers. The box and whisker diagram
was introduced by Professor Jhon W. Tukey.
01:38 PM

24
Construction of Box-PlotConstruction of Box-Plot
1. Start the box from Q1 and end at
Q3
2. Within the box draw a line to
represent Q2
3. Draw lower whisker to Min.
Value up to Q1
4. Draw upper Whisker from Q3
up to Max. Value
Q1
Q3
Q2
01:38 PM
Max
Value
Min
Value

25
Construction of Box-PlotConstruction of Box-Plot
1. Q1=22.0 Q3=36.5
2. Q2=30.5
3. Minimum Value=5.0
4. Maximum Value=66.0
70
60
50
40
30
20
10
0
01:38 PM

26
Interpretation of Box-PlotInterpretation of Box-Plot
70
60
50
40
30
20
10
0
Box-Whisker Plot is useful to identify
•Maximum and Minimum Values in the data
•Median of the data
•IQR=Q3-Q1,
Lengthy box indicates more variability in the data
•Shape of the data From Position of line within box
Line At the center of the box----Symmetrical
Line above center of the box----Negatively
skewed
Line below center of the box----Positively Skewed
•Detection of Outliers in the data
01:38 PM

27
OutliersOutliers
An outlier is the values that falls well outside the overall
pattern of the data. It might be
• the result of a measurement or recording error,
• a member from a different population,
• simply an unusual extreme value.
An extreme value needs not to be an outliers; it might,
instead, be an indication of skewness.
01:38 PM

28
Inner and Outer FencesInner and Outer Fences
If Q1=22.0 Q2=30.5 Q3=36.5
( )
( )


=+=
=−=
25.58IQR1.5QFenceInnerUpper
25.0IQR1.5QFenceInnerLower
:FencesInner
3
1
( )
( )


=+=
−=−=
0.80IQR3QFenceOuterUpper
5.21IQR3QFenceOuterLower
:FencesOuter
3
1
01:38 PM
Lower Inner Fence 22-1.5(36.5-22) = 0.25
Upper Inner Fence 36+1.5(36.5-22) = 58.25
Lower Outer Fence 22-3(36.5-22) = -21.5
Upper Outer Fence 36+3(36.5-22) = 80.0

29
Identification of the OutliersIdentification of the Outliers
1. The values that lie within inner
fences are normal values
2. The values that lie outside inner
fences but inside outer fences
are possible/suspected/mild
outliers
3. The values that lie outside outer
fences are sure outliers
80
70
60
50
40
30
20
10
0
Plot each suspected outliers with an asterisk
and each sure outliers with an hollow dot.
*
Only
66 is a
mild
outlier
01:38 PM

30
Box plots are
especially suitable for
comparing two or more
data sets. In such a
situation the box plots
are constructed on the
same scale.
Uses of Box and Whisker DiagramUses of Box and Whisker Diagram
Male
Female
01:38 PM

Standardized VariableStandardized Variable
A variable that has mean “0” and Variance “1” is
called standardized variable
Values of standardized variable are called
standard scores
Values of standard variable i.e standard scores
are unit-less
Construction
VariableofDeviationStandard
VariableofMeanVariable
Z
−
=
01:38 PM 31

X Z
3 25 -1.3624 1.85611.8561
6 4 -0.5450 0.29700.2970
11 9 0.81741 0.66820.6682
12 16 1.0899 1.18791.1879
32 54 0 4.009
5.13
4
54
8
4
32
2
==
===
∑
xS
n
X
X
2
)( XX −
67.3
8−
=
−
=
X
Sx
XX
Z
1
4
009.4
0
2
≅=
==
∑
zS
n
Z
Z
2
)( ZZ −
Variable Z has mean “0” and
variance “1” so Z is a standard
variable.
Standard Score at X=11 is
8174.0
67.3
811
=
−
=
−
=
Sx
XX
Z
01:38 PM
Standardized VariableStandardized Variable

33
The industry in which sales rep Mr. Atif works has mean
annual sales=$2,500
standard deviation=$500.
The industry in which sales rep Mr. Asad works has mean
annual sales=$4,800
standard deviation=$600.
Last year Mr. Atif’s sales were $4,000
and Mr. Asad’s sales were $6,000.
Performance evaluation by z-scoresPerformance evaluation by z-scores
Which of the representatives would you hire
if you have one sales position to fill?
01:38 PM

34
Performance evaluation by z-scoresPerformance evaluation by z-scores
3
500
500,2000,4
=
−
=
−
=
B
B
BB
B
Z
S
XX
Z
Sales rep. Atif
XB= $2,500
SΒ= $500
XB= $4,000
Sales rep. Asad
XP =$4,800
SP = $600
XP= $6,000
2
600
800,4000,6
=
−
=
−
=
P
P
PP
P
Z
S
XX
Z
Mr. Atif is the best choice
01:38 PM

35
valuesof68%aboutcontains1SX ±
The Empirical RuleThe Empirical Rule
X
68%
1SX ±
valuesof99.7%aboutcontains3SX ±
valuesof95%aboutcontains2SX ± 95%
X 2S±
X 3S±
99.7%
01:38 PM

Chebysev’s TheoremChebysev’s Theorem

37
A distribution in which the values equidistant from
the centre have equal frequencies is defined to be
symmetrical and any departure from symmetry is
called skewness.
1. Length of Right Tail = Length of Left
Tail
2. Mean = Median = Mode
3. Sk=0
a) Sk=(Mean-Mode)/SD
b) Sk=(Q3-2Q2+Q1)/(Q3-Q1)
01:38 PM
Measures of Skewness

38
A distribution is positively skewed, if the observations
tend to concentrate more at the lower end of the
possible values of the variable than the upper end. A
positively skewed frequency curve has a longer tail on
the right hand side
1. Length of Right Tail > Length of Left
Tail
2. Mean > Median > Mode
3. SK>0
b) Sk=(Q3-2Q2+Q1)/(Q3-Q1)
MeasuresMeasures ofof SkewnessSkewness
01:38 PM

39
A distribution is negatively skewed, if the
observations tend to concentrate more at the upper
end of the possible values of the variable than the
lower end. A negatively skewed frequency curve
has a longer tail on the left side.
1. Length of Right Tail < Length of Left
Tail
2. Mean < Median < Mode
3. SK< 0
b) Sk=(Q3-2Q2+Q1)/(Q3-Q1)
01:38 PM
Measures of
Skewness

01:38 PM 40
The Kurtosis is the degree of peakedness or flatness of a
unimodal (single humped) distribution,
• When the values of a variable are highly concentrated around
the mode, the peak of the curve becomes relatively high; the
curve is Leptokurtic.
• When the values of a variable have low concentration
around the mode, the peak of the curve becomes relatively
flat;curve is Platykurtic.
• A curve, which is neither very peaked nor very flat-toped, it
is taken as a basis for comparison, is called
Mesokurtic/Normal.
Measures of Kurtosis

4101:38 PM

42
1. If Coefficient of Kurtosis > 3 ----------------- Leptokurtic.
2. If Coefficient of Kurtosis = 3 ----------------- Mesokurtic.
3. If Coefficient of Kurtosis < 3 ----------------- is Platykurtic.
( )
( )
4
22
n X-X
Coefficient of Kurtosis=
X-X 
 
∑
∑
01:38 PM

Moments about origin
C.I f x f.x f. x^2 f.x^3 f.x^4
2.5 2.9 2 2.7 5.4 14.58 39.366 106.2882
3 3.4 7 3.2 22.4 71.68 229.376 734.0032
3.5 3.9 17 3.7 62.9 232.73 861.101 3186.074
4 4.4 25 4.2 105 441 1852.2 7779.24
4.5 4.9 20 4.7 94 441.8 2076.46 9759.362
5 5.4 12 5.2 62.4 324.48 1687.296 8773.939
5.5 5.9 9 5.7 51.3 292.41 1666.737 9500.401
6 6.4 8 6.2 49.6 307.52 1906.624 11821.07
Total (Σ) 100 453 2126.2 10319.16 51660.38
01:38 PM 43

Moments about origin
01:38 PM 44

Regression
X
Price
Y
(Quantity
Demanded) x*y x^2
15 440 6600 225
20 430 8600 400
25 450 11250 625
30 370 11100 900
40 340 13600 1600
50 370 18500 2500
Σx=180 Σy=2400 Σx.y=69650 Σx^2=6250
01:38 PM 45

Statistics-Measures of dispersions

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