Transcript 5.descriptive measures

Descriptive Measures
“While an individual is an insolvable
puzzle, in an aggregate he becomes a
mathematical certainty. You can, for
example, never foretell what any one
man do, but you can say with precision
what an average number will be upto”.
-Arthur Conan
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Measures of Central Tendency

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Mean
Median
Mode
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Requisites of a good measure
of central tendency
1.
2.
3.
4.
5.
It should be easy to calculate and
understand.
It should be rigidly defined.
It should be representative of the
data.
It should have sampling stability.
It should not be affected by extreme
values.
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Which measure is appropriate?
A person who does not know how to swim
has to cross a river from one bank to another
by walking into the water with a stick. He
tries to determine the length of stick on the
basis of average length of river.
Which measure of central location w.r.t
the depth of the river will save him from
getting drowned?
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Mean
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The FD below represents time in
seconds needed to serve a sample of
customers by a cashier at a discount
store.
Find the Mean.
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Time
20-30
30-40
40-50
50-60
60-70
70-80
Freq
6
16
21
29
25
22
Time
80-90
90-100
100-110
110-120
120-130
Freq
11
7
4
0
2
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Properties of Mean
1.
2.
3.
The sum of deviations of all individual
observations from mean is always
zero.
The sum of squares of deviations of
observations about the mean is
minimum.
Combine mean of set groups can be
obtained.
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X
X-Mean
Sqr (X-Mean) Sqr (X-10)
10
-20
400
0
20
-10
100
100
30
0
0
400
40
10
100
900
50
20
400
1600
Sum=1000
Sum=3000
Mean=30 Sum=0
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Limitations
1.
2.
3.
4.
Average may give a value that does
not exist in data
Average may give absurd results
Does not give idea about difference in
series
Affected by extreme values
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Weighted Arithmetic Mean
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Dave’s Giveaway Store advertises, “If
our prices are not equal or lower than
everyone else’s, you get it free.” One of
Dave’s customers came into the store
one day and threw on the counter bills
of sale for six items she bought from a
competitor for an average price less
than Dave’s.
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The items cost :
$1.29, $2.97, $3.49, $5.00, $7.50,$10.95.
Dave’s prices for the same six items are
$1.35, $2.89, $3.19, $4.98, $7.59, $11.50
Dave told, “My add refers to a weighted
average price for the same items. Our average
is low because our sales of these items have
been: 7, 9, 12, 8, 6, 3.
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Is Dave getting himself into our
out of trouble by talking about weighted
average?
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Median
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Ungrouped distribution
2
5
10
20
30
33
57
16
3
10
13
17
33
60
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Meridian Trucking company maintains
records on all its rolling equipment.
Here are weekly mileage records for its
trucks. Calculate the median miles a
truck travelled.
810
450 756 789 210 657 589 488 878 689
1450 560 469 890 987 559 788 943 447 775
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Median = 722.5 Miles
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Grouped distribution
Class
Freq
Class
Freq
18-22
120
38-42
184
22-26
125
42-46
162
26-30
280
46-50
86
30-34
260
50-54
75
34-38
155
54-58
53
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Median = 33.46
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Calculate median from the data pertaining
to profits (in Cr.) of 125 companies:
Profits
Less
Less
Less
Less
than
than
than
than
10
20
30
40
No. of
comp
4
16
40
76
Profits
Less
Less
Less
Less
than
than
than
than
50
60
70
80
No. of
comp
96
112
120
125
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Median Profit = 36.25 Cr.
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Advantages

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Easy to calculate and understand
Not effected by extreme observations
Can be calculated for an open class
Median can be found out for qualitative
descriptions
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Disadvantages

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Does not depend on all observations
Some accuracy is given up in choosing a
single value to represent the distribution.
Not capable of further algebraic
treatment
Not a good measure for estimation
purpose since it is more affected by
sampling fluctuations
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Properties

Sum of absolute deviations from
median is minimum
X
|X-med.|
|X-7|
4
6
4
2
3
1
8
0
1
10
2
3
12
4
Sum=12
5
Sum=13
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Quartiles
Deciles
Percentiles
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Mode
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Calculate Mode
Classes
Below 60
60-62
62-64
64-66
66-68
68-70
70-72
Freq.
12
18
25
30
10
3
3
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Measures of Variation
Also called Dispersion.
State the extent to which individual
values differ from mean.
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Significance

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To determine reliability of an average
To serve as a basis for the control of
variability
To compare two or more series with
regard to there variability
To facilitate the use of other statistical
measures
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

Absolute Variation
Relative Variation
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Methods of studying Variation
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Range
Inter-quartile Range or Quartile
deviation
Average deviation
Standard deviation
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