Skewness-Kurtosisand

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Skewness and Kurtosis Skewness

Measure of the degree of asymmetry of a frequency distribution Skewed to left Symmetric or unskewed Skewed to right

Kurtosis Measure of flatness or peakedness of a frequency distribution

Platykurtic (relatively flat) Mesokurtic (normal) Leptokurtic (relatively peaked)

Skewed to left

Negative Skewness

Symmetric

Symmetric

Positive SkewnessSkewed to right

Symmetric Bimodal Distribution

700600500400300200100

40

30

20

10

0

X

F

r

e

q

u

e

n

c

y

10

15

35

20

35

15

10

Mean = Median

700600500400300200100

40

30

20

10

0

X

F

r

e

q

u

e

n

c

y

10

15

35

20

35

15

10

Mean = Median

Symmetric distribution with two Modes

Kurtosis

Platykurtic - flat distribution

Kurtosis

Mesokurtic - not too flat and not too peaked

Kurtosis

Platykurtic - flat distribution

Kurtosis

Mesokurtic - not too flat and not too peaked

Kurtosis

Leptokurtic - peaked distribution

Relations between the Mean and Standard Deviation

Chebyshevs TheoremApplies to any distribution, regardless of shapePlaces lower limits on the percentages of observations

within a given number of standard deviations from the mean

Empirical RuleApplies only to roughly mound-shaped and symmetric

distributionsSpecifies approximate percentages of observations within a

given number of standard deviations from the mean

1 12

1 1434 75%

1 13

1 1989 89%

1 14

1 1161516 94%

2

2

2

= = =

= = =

= = =

Chebyshevs Theorem At least of the elements of any distribution

lie within k standard deviations of the mean

At least

Lie within

Standarddeviationsof the mean

2

3

4

21

1

k

For roughly mound-shaped and symmetricdistributions, approximately:

68% 1 standard deviation of the mean

95% Lie within

2 standard deviations of the mean

All 3 standard deviations of the mean

Empirical Rule