2. measures of dis[persion
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Measures of Measures of DispersionDispersion
Measures of Dispersion
• It is the measure of extent to which an individual items vary
• Synonym for variability• Often called “spread” or “scatter”• Indicator of consistency among a
data set• Indicates how close data are
clustered about a measure of central tendency
Compare the following Compare the following distributionsdistributions
o Distribution A
200 200 200 200 200
o Distribution B300 205 202 203 190
o Distribution C1 989 2 3 5
Arithmetic mean is same for all the series but distribution differ widely from one another.
Objectives of Measuring Objectives of Measuring VariationVariation
o To gauge the reliability of an average i.e. dispersion is small, means more reliable.
o To serve as a basis for the control of variability.
o To compare two or more series with regard to their variability.
The RangeThe Rangeo Difference between largest value and
smallest value in a set of datao Indicates how spread out the data areo Dependent on two extreme values.o Simple, easy and less time consuming.
o Calculation: Find largest and smallest number in
data set Range = Largest - Smallest
Uses of RangeUses of Range
o Quality Control
o Weather Forecasting
o Fluctuations in Share/Gold prices
The quartiles divide the data into four parts. There are a total of three quartiles which are usually denoted by Q1, Q2 and Q3.
Quartile Deviation/Inter-Quartile Range
The inter-quartile range is defined as the difference between the upper quartile and the lower quartile of a set of data.
Inter-quartile range = Q3 – Q1
Quartile Deviation/Semi Inter Quartile Range = Q3-Q1 / 2
ExampleExample
• Calculate Quartile Deviation-
Wages in Rs. per week
No. of wages earners
Less than 35 14
35-37 62
38-40 99
41-43 18
Over 43 7
Standard DeviationStandard Deviation• It is most important and widely used
measure of studying variation.
• It is a measure of how much ‘spread’ or ‘variability’ is present in sample.
• If numbers are less dispersed or very close to each other then standard deviation tends to zero and if the numbers are well dispersed then standard deviation tends to be very large.
For a set of ungrouped data x1, x2, …, xn,
n
xxf
n
xxxxxx
n
ii
n
1
21
222
21
)(
)()()( deviation Standard
data. of number total the is and mean the iswhere nx
For Ungrouped Data
•It is the average of the distances of the observed values from the mean value for a set of data
SD =Sum of squares of individual deviations from arithmetic mean
Number of items
Example:
ScoresDeviations From Mean
Squares of Deviations
01
03
05
06
11
12
15
19
34
37143
-13
-11
-09
-08
-03
-02
+01
+05
+20
+23
169
121
81
64
9
4
1
25
400
5291403
M = 143/10 = 14
No. of scores = 10
SD =1403
10= 11.8
n
ii
n
ii
n
nn
f
xxf
fff
xxfxxfxxf
1
1
21
21
2222
211
)(
)()()( deviation Standard
data. of number total the is and mean the isdata, of group th the offrequency the is where
nxifi
For Grouped Data
ExampleExampleCalculate standard deviation –
Marks No. of students
0-10 5
10-20 12
20-30 30
30-40 45
40-50 50
50-60 37
60-70 21
Uses of Standard Uses of Standard DeviationDeviation
o The standard deviation enables us to determine with a great deal of accuracy , where the values of frequency distribution are located in relation to the mean.
o It is also useful in describing how far individual items in a distribution depart from the mean of the distribution.
-1 +12.2 9.6
1411
25.812.4
68%
NORMAL DISTRIBUTION CURVE1 Standard Deviation
-2 +2018.2
1411
3713.8
95%
NORMAL DISTRIBUTION CURVE2 Standard Deviations
NORMAL DISTRIBUTION CURVE3 Standard Deviations
-3 +3016.8
1411
3715.2
99.7%
VarianceVarianceo = Variance= (standard deviation)2
o For comparing the variability of two or more distributions,
o Coefficient of variation = s.d. X 100o mean
2
. . 100CV Xx
More C.V., More variability
ExampleExample
Which organization is more uniform wages?
Organization A Organization B
Number of employees 100 200
Average wage per employee
5000 8000
Variance of wages per employee
6000 10000
MEASURE OF SHAPE- MEASURE OF SHAPE- SKEWNESSSKEWNESS
Distribution lacks symmetry
Data sparse at one end and piled up at the other .
- patients suffering from diabetes
Median is the best measure for skewed data , as it is not highly influenced by the frequency nor is pulled by extreme values.
SYMMETRIC
MODE = MEAN = MEDIAN
MEAN MODE MEDIAN
SKEWED LEFT(NEGATIVELY)
MEDIAN MEAN
MODE
SKEWED RIGHT(POSITIVELY)
Case Let for practiceCase Let for practice At one of the management institutes,
there are mainly six specialists available namely: Marketing, Finance, HR, Operations CRM and International Business.( See Table 1)
Calculate the average salaries for all the specializations, as also for the entire batch. Which specialization has the maximum variation in the salaries offered?
(Ans- Avg. salary for all specializations- 5.27, for mktg-5.24, finance-5.36, HR-4.88,Operations-5.4, CRM-5.5, IB-5.0. Max. Variation was in IB.
3-4 Lacs
4-5 Lacs
5-6 Lacs
6-7 Lacs
7-8 Lacs
Total
Mktg. 23 24 33 23 9 112
Finance 11 17 35 15 6 84
HR 1 7 4 1 0 13
Operations 1 2 5 1 1 10
CRM 1 2 5 2 1 11
IB 2 4 2 1 1 10
Total 39 56 84 43 18 240
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