measures of dispersion
TRANSCRIPT
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AGBS | Bangalore
Measures of Dispersion
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Summary Measures
Arithmetic Mean
Median
Mode
Describing Data Numerically
Variance
Standard Deviation
Coefficient of Variation
Range
Interquartile Range
Geometric Mean
Skewness
Central Tendency Variation ShapeQuartiles
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Quartiles
Quartiles split the ranked data into 4 segments with an equal number of values per segment
25% 25% 25% 25%
The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are larger)
Only 25% of the observations are greater than the third quartile
Q1 Q2 Q3
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Quartile Formulas
Find a quartile by determining the value in the appropriate position in the ranked data, where
First quartile position: Q1 = n/4
Second quartile position: Q2 = n/2 (the median position)
Third quartile position: Q3 = 3n/4
where n is the number of observed values
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(n = 9)
Q1 is in the 9/4 = 2.25 position of the ranked data
so Q1 = 12.25
Quartiles
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
Example: Find the first quartile
Q1 and Q3 are measures of noncentral location Q2 = median, a measure of central tendency
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Quartiles
Example:(continued)
Overtime Hours
10-15 15-20 20-25 25-30 30-35 35-40 Total
No. of employees
11 20 35 20 8 6 100
Calculate Median, First Quartile, 7th Decile & Interquartile Range
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Same center, different variation
Measures of Variation
Variation
Variance Standard Deviation
Coefficient of Variation
Range Interquartile Range
Measures of variation give information on the spread or variability of the data values.
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Range
Simplest measure of variation Difference between the largest and the smallest
values in a set of data:
Range = Xlargest – Xsmallest
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Range = 14 - 1 = 13
Example:
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Ignores the way in which data are distributed
Sensitive to outliers
7 8 9 10 11 12
Range = 12 - 7 = 5
7 8 9 10 11 12
Range = 12 - 7 = 5
Disadvantages of the Range
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 5 - 1 = 4
Range = 120 - 1 = 119
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Coefficient of Range
Coefficient of Range = [L-S]/[L+S]
Where,
L = Largest value in the data set
S = Smallest value in the data set
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Interquartile Range
Can eliminate some outlier problems by using the interquartile range
Eliminate some high- and low-valued observations and calculate the range from the remaining values
Interquartile range = 3rd quartile – 1st quartile = Q3 – Q1
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Interquartile Range
Median(Q2)
XmaximumX
minimum Q1 Q3
Example:
25% 25% 25% 25%
12 30 45 57 70
Interquartile range = 57 – 30 = 27
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Coefficient of QD
Coefficient of QD= [Q3-Q1]/[Q3+Q1]
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Average Deviation
Coefficient of AD = (Average Deviation)/(Mean or Median)
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Calculate the Coefficient of AD (mean)
Sales (in thousand Rs) No. of days
10-20 3
20-30 6
30-40 11
40-50 3
50-60 2
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Average (approximately) of squared deviations of values from the mean
Sample variance:
Variance
1-n
)X(XS
n
1i
2i
2∑
=
−=
Where = mean
n = sample size
Xi = ith value of the variable X
X
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Standard Deviation
Most commonly used measure of variation Shows variation about the mean Is the square root of the variance Has the same units as the original data Sample standard deviation:
1-n
)X(XS
n
1i
2i∑
=
−=
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Calculation Example:Sample Standard Deviation
Sample Data (Xi) : 10 12 14 15 17 18 18 24
n = 8 Mean = X = 16
4.3087
130
7
16)(2416)(1416)(1216)(10
1-n
)X(24)X(14)X(12)X(10S
2222
2222
==
−++−+−+−=
−++−+−+−=
A measure of the “average” scatter around the mean
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Population vs. Sample Standard Deviation
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Population vs. Sample Standard Deviation (Grouped Data)
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Calculate the Standard Deviation for the following sample
Number of pins knocked down in ten-pin bowling matches
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Solution
Number of pins knocked down in ten-pin bowling matches
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Calculate the Standard Deviation for the following sample
Heights of students
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Solution
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Calculate the Standard Deviation for the following sample
Number of typing errors
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Comparing Standard Deviations
Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21
11 12 13 14 15 16 17 18 19 20 21
Data B
Data A
Mean = 15.5 S = 0.926
11 12 13 14 15 16 17 18 19 20 21
Mean = 15.5 S = 4.567
Data C
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Measuring variation
Small standard deviation
Large standard deviation
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Advantages of Variance and Standard Deviation
Each value in the data set is used in the calculation
Values far from the mean are given extra weight (because deviations from the mean are squared)
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Coefficient of Variation
Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare two or more sets of
data measured in different units
100%X
SCV ⋅
=
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Comparing Coefficient of Variation
Stock A: Average price last year = $50 Standard deviation = $5
Stock B: Average price last year = $100 Standard deviation = $5
Both stocks have the same
standard deviation, but stock B is less
variable relative to its price
10%100%$50
$5100%
X
SCVA =⋅=⋅
=
5%100%$100
$5100%
X
SCVB =⋅=⋅
=
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Sample vs. Population CV
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Calculate!
Calculate the coefficient of variation for the following data set. The price, in cents, of a stock over five trading days
was 52, 58, 55, 57, 59.
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Solution
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Pooled Standard Deviation
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Pooled Standard Deviation
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Z Scores
A measure of distance from the mean (for example, a Z-score of 2.0 means that a value is 2.0 standard deviations from the mean)
The difference between a value and the mean, divided by the standard deviation
A Z score above 3.0 or below -3.0 is considered an outlier
S
XXZ
−=
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Z Scores
Example: If the mean is 14.0 and the standard deviation is 3.0,
what is the Z score for the value 18.5?
The value 18.5 is 1.5 standard deviations above the mean
(A negative Z-score would mean that a value is less than the mean)
1.53.0
14.018.5
S
XXZ =−=−=
(continued)
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Shape of a Distribution
Describes how data are distributed Measures of shape
Symmetric or skewed
Mean = Median Mean < Median Median < Mean
Right-SkewedLeft-Skewed Symmetric
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If the data distribution is approximately bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
The Empirical Rule
1σμ ±
μ
68%
1σμ ±
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contains about 95% of the values in the population or the sample
contains about 99.7% of the values in the population or the sample
The Empirical Rule
2σμ ±
3σμ ±
3σμ ±
99.7%95%
2σμ ±
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Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1)
Examples:
(1 - 1/12) x 100% = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) x 100% = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) x 100% = 89% ………. k=3 (μ ± 3σ)
Chebyshev Rule
withinAt least
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Calculate!
Calculate the mean & the standard dev. Calculate the percentage of
observations between the mean & 2σ±
Variable Frequency
44-46 3
46-48 24
48-50 27
50-52 21
52-54 5
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Find the correct Mean & Std. dev.
The Mean & Standard Deviation of a set of 100 observations were worked out as 40 & 5 respectively. It was later learnt that a mistake had been made in data entry – in place of 40 (the correct value), 50 was entered.
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Skewness
A fundamental task in many statistical analyses is to characterize the location and variability of a data set (Measures of central tendency vs. measures of dispersion)
Both measures tell us nothing about the shape of the distribution
It is possible to have frequency distributions which differ widely in their nature and composition and yet may have same central tendency and dispersion.
Therefore, a further characterization of the data includes skewness
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Positive & Negative Skew
Positive skewness There are more observations below the mean than
above it When the mean is greater than the median
Negative skewness There are a small number of low observations and a
large number of high ones When the median is greater than the mean
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Measures of Skew
Skew is a measure of symmetry in the distribution of scores
Positive Skew
Negative Skew
Normal (skew = 0)
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Measures of Skew
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The Rules
I. Rule One. If the mean is less than the median, the data are skewed to the left.
II. Rule Two. If the mean is greater than the median, the data are skewed to the right.
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Karl Pearson Coefficient of Skewness
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Karl Pearson Coefficient - Properties
Advantage – Uses the data completelyDisadvantage – Is sensitive to extreme values
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Calculate!
Calculate the coefficient of skewness & comment on its value
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Calculate!
Calculate the coefficient of skewness & comment on its value
Profits (lakhs) No. of Companies
100-120 17
120-140 53
140-160 199
160-180 194
180-200 327
200-220 208
220-240 2
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Bowley’s Coefficient of Skewness
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Bowley’s Coefficient - Properties
Advantage – Is not sensitive to extreme valuesDisadvantage – Does not use the data completely
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Calculate!
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Kelly’s Coefficient of Skewness
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Calculate!
Obtain the limits of daily wages of the central 50% of the workers
Calculate Bowley’s & Kelly’s Coefficients of Skewness
Wages No. of Workers
Below 200 10
200-250 25
250-300 145
300-350 220
350-400 70
Above 400 30
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Moments about the Mean