Calculating Normality of Distribution

Calculations for normality of distribution include Skewness, Kurtosis and Chi-squared goodness-of-fit.

The tables for skewness and kurtosis are taken from Quality Control and Industrial Statistics, pp. 956-957. See "References" at the end of this section for more information.

When the values of skewness and kurtosis are tested for normality, the Moments Hypothesis tests are used

Skewness

If the distribution is normal, there is a strong probability (95% or 99%, depending on how you have configured the program) that the skewness will not exceed the listed value.

Samples

Critical value

25

1.073

30

0.985

35

0.932

40

0.869

45

0.825

50

0.787

60

0.723

70

0.673

80

0.631

90

0.596

100

0.567

125

0.508

150

0.464

175

0.430

200

0.403

250

0.360

300

0.329

400

0.285

500

0.255

750

0.208

1000

0.180

Skewness describes how non-symmetrical the data are on either side of a vertical axis through the mean. Calculations for skewness are as follows:

image\IMG00104.gif

Kurtosis

If the distribution is normal, there is a 99% probability that the kurtosis will not be below the low value or above the high value.

Samples

Low value

High value

25

-1.28

2.30

30

-1.21

2.21

40

-1.11

2.04

50

-1.05

1.88

75

-0.92

1.59

100

-0.82

1.39

125

-0.76

1.24

150

-0.71

1.13

200

-0.63

0.98

250

-0.58

0.87

300

-0.54

0.79

500

-0.43

0.60

1000

-0.32

0.41

2000

-0.23

0.28

5000

-0.15

0.17

Kurtosis describes the peakedness of the data. When compared to a normal curve, is the curve for this data higher or flatter? Calculations for kurtosis are as follows:

image\IMG00105.gif

Chi-squared goodness of fit

The Chi-squared goodness of fit is used to test the standard deviation for a distribution.

image\IMG00106.gif

The validity of this statistic is affected by the scaling factors chosen in constructing the histogram.

Assumptions when calculating for normality of distribution:

image\IMG00107.gif

References:

Beyer, William H. Standard Mathematical Tables, 25th Edition. CRC Press, 1981. p. 507.

Duncan, Acheson J., Quality Control and Industrial Statistics, Fourth Edition. Homewood, IL: R.D. Irwin, 1974.

Geary, R.C. and E.S. Pearson. Biometrika. Biometrika Office, University College, 1938.