Transcript
Chap 3-1
SKEWNESS AND KURTOSIS
Curves representing the data points in the data set may be either symmetrical or skewed.
When the mean, mode and median do not have the same value in a distribution, then it is termed as skewed distribution
Chap 3-2
Shape of a Distribution
Describes how data is distributed Measures of shape
Symmetric or skewed
Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean
Right-SkewedLeft-Skewed Symmetric
Chap 3-3
Measures of Skewness
1. Karl Pearson’s measure 2. Bowley’s measure 3. Kelly’s measure 4. Moment’s measure
Chap 3-4
Karl Pearson’s formula :
Skewness MEAN-Mode Co-efficient of Skewness MEAN-MODE
Standard Deviation
Skewness when mode can not be determined
SK 3(Mean-Median) Standard Deviation
Chap 3-5
Given The data Calculate the Karl
Pearson’s coefficient of skewness
Sigma X 452 Sigma X2 24270 Mode 43.7 & N 10 Solution : Mean 452/10 45.2 SD (24270/10)- (45.2)2
19.59
SKp (45.2-43.7)/19.59
0.08 It shows there is positive skewness though it is marginal
Chap 3-6
X 10-20 20-30 30-40 40-50 50-60 60-70 70-80
f 18 30 40 55 38 20 16
Calculate the measure of skewness using the mean, median and standard deviation?
Sol. Midpoint 15 25 35 45 55 65 75 f 18 30 40 55 38 20 16 dx -3 –2 -1 0 1 2 3 fdx -54 –60 - 40 0 38 40 48 (-28) fdx2 162 120 40 0 38 80 144 (584)
cf 18 48 88 143 181 201 217
Chap 3-7
Mean 45- (28/217)10 43.71
Median 40+ (50-40)(109-88) 43.82 55 SD (584/217) - (-28/217)2 x 10 ->
16.4
Skewness 3(Mean-Median) -0.33 Coefficient of skewness Skp/SD -0.02 The result shows Distribution is negative
Skewed but it is negligible
Chap 3-8
BOWLEY’S MEASURE
Skewness Q3 +Q1 - 2Median Q3 - Q1
The value of this vary between +-1.
Chap 3-9
Kelly’s Measure
Coefficient of skewness p90 – 2 p50 + p10
P90-P10
Chap 3-10
MOMENTS
It is used to indicate peculiarities of a frequency distribution.
The utilities lies in the sense that they indicate different aspects of a given distribution.
We can measure the central tendency of a series, dispersion or variability, skewness and the peakedness of the curve.
Chap 3-11
First moment μ1 =∑ fi (xi – x)/N
Second moment μ2 = ∑ fi (xi – x)2/N
Third moment μ3 = ∑ fi (xi – x)3/N
Fourth moment μ4 = ∑ fi (xi – x)4/N
Chap 3-12
The first moment is zero.
The second indicates Variance.
The Third indicates skewness.
The fourth indicates Kurtosis.
Chap 3-13
KURTOSIS
Kurtosis is another measure of the shape of a frequency curve.
While Skewness signifies the extent of Asymmetry, Kurtosis measures the degree of peakedness of a frequency distribution.
Chap 3-14
Types of Curves
Leptokurtic Peaked Curve B2 > 3 Mesokurtic Normal Curve B2 3 Platykurtic Flat Curve B2 < 3
Chap 3-15
Coefficient of Kurtosis
K (Q3 – Q1)/2 P90 - P10
For Mesokurtic curve, MEAN is most Appropriate. For Leptokurtic curve, MEDIAN is most
Appropriate. For Platykurtic curve, Quartile is most
Appropriate.B2 μ4/ μ22
Chap 3-16
Exploratory Data Analysis
Box-and-whisker plot Graphical display of data using 5-number
summary
Median( )
4 6 8 10 12
XlargestXsmallest1Q 3Q
2Q
Chap 3-17
Distribution Shape and Box-and-Whisker Plot
Right-SkewedLeft-Skewed Symmetric
1Q 1Q 1Q2Q 2Q 2Q3Q 3Q3Q
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