SKEWNESS and kurtosis

J001, J009, J015, J019MBA (TECH), EXTC

2

SKEWNESS

•Distributions (aggregations of observations) can be spread evenly around both sides of the central tendency, like so:

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Such distributions are considered symmetrical with no skew.

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Mean = Median =

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SKEWNESS

•As scores are weighted and distribute unevenly around the median, the mean is “pulled” toward the extreme outlier and it diverges away from the median.

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75

Mean = Median =

2120

SKEWNESS

• When the outlying scores are on the higher end of the scale the distribution becomes positively skewed.

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2120

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SKEWNESS

• When the outlying scores are on the lower end of the scale the distribution becomes negatively skewed.

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2120

_

Example: skewness

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Example: skewness

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Problem: College Men’s Heights

Height (inches)

Class Mark, x Frequency, f

59.5–62.5 61 5

62.5–65.5 64 18

65.5–68.5 67 42

68.5–71.5 70 27

71.5–74.5 73 8

xf (x−x̅) (x−x̅)²f (x−x̅)³f

305 -6.45 208.01 -1341.7

1152 -3.45 214.25 -739.15

2814 -0.45 8.51 -3.83

1890 2.55 175.57 447.7

584 5.55 246.42 1367.63

∑ 6745 n/a 852.75 −269.33x̅, m2, m3 67.45 n/a 8.5275 −2.6933

Finally, the skewness isg1 = m3 / m2

3/2 = −2.6933 / 8.52753/2 = −0.1082

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kurtosis

• Distributions of data and probability distributions are not all the same shape. Some are asymmetric and skewed to the left or to the right. Many times, there are two values that dominate the distribution of values.

Kurtosis is the measure of the peak of a distribution, and indicates how high the distribution is around the mean.

Types of kurtosis

MesokurticA distribution identical to the normal distribution

Leptokurtic

A distribution that is more peaked than normal

Platykurtic

A distribution that is less peaked than normal

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Formulae: kurtosis

•Moment coefficient of kurtosis One measure of kurtosis uses the fourth moment about fourth power of standard deviation in dimensionless form:

Problem: test scores

67 24.125 338742.19 677484.37562 19.125 133784.49 267568.98557 14.125 39806.485 119419.45452 9.125 6933.1643 6933.1643147 4.125 289.53149 1737.1889642 -0.875 0.5861816 6.4479980537 -5.875 1191.3284 9530.6269532 -10.88 13986.758 41960.274227 -15.88 63511.875 127023.7522 -20.88 189891.68 379783.36

16311447.6

Xi

65-69 260-64 255-59 350-54 145-49 640-44 1135-39 830-34 325-29 220-24 2

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C.I . fi

Where,SD = 7.22

(leptokurtic)

CASE STUDY II

THANK YOU