Testing for Normality of Distribution (the Kolmogorov-Smirnov test)

Testing for Normality of Distribution (Kolmogorov-Smirnov test) using the Online Normal Distribution Calculator

Many statistical tests (t-test, f-test, one-way ANalysis Of VAriance ANOVA) assume that data used are drawn from a normal population. Although the chi-squared test can be used to test this assumption it should be used only if there are 50 or more data points, so it is of limited value in analytical work, when we often have only a small set of data. There Kolmogorov-Smirnov test (a nonparametric test).

The Kolmogorov-Smirnov test – included in SPSS - has been used in a previous post entitled "Statistical Treatment of Analytical Data - One-Sample t-test in Chemical Analysis" for testing that data is normally distributed.

The principle of the method involves comparing the sample cumulative distribution function with the cumulative distribution function of the hypothesized distribution. If the experimental data depart substantially from the expected distribution, the two functions will be widely separated. If, however, the data are closely in accord with the expected distribution, the two functions will never be very far apart. The test statistic is given by the maximum difference between the two functions (Dx)exp and is compared in the usual way with a set of tabulated values (D)crit.

When the Kolmogorov–Smirnov method is used to test whether a distribution is normal, the original data are transformed into the standard normal variable, z.

This is done by using the equation:

z = (x – μ) / s

where μ is the mean and s the standard deviation of the data.

The data are next transformed by using the above equation and then the Kolmogorov–Smirnov method is applied. This test is illustrated in the example given in “Statistical Treatment of Analytical Data - One-Sample t-test in Chemical Analysis” using SPSS. The Kolmogorov–Smirnov test in SPSS shows that the data tested are normally distributed at the 95% confidence level (Figure I.1)

The same data are tested below using an online Normal Distribution Calculator (Kolmogorov-Smirnov test).

The data are inserted or copied in the yellow-labeled cells and the confidence level is selected from the drop-down list (in this case 95%). The median, 15% trimmed mean, mean, standard deviation, # of data, Dexp, Dcrit and the Result is calculated (Fig. I.2).

The result is consistent with the SPSS test showing that the tested data are normally distributed. A first indication of normally distributed data is given by the fact that mean≈ median ≈ 15% trimmed mean.

The Online Normal Distribution Calculator is given below:

---

Relevant Posts - Relevant Videos

Statistical Treatment of Analytical Data - One-Sample t-test in Chemical Analysis

Comparing several group means by ANOVA using SPSS

Testing for Normality of Distribution

---

References

1. D.B. Hibbert, J.J. Gooding, "Data Analysis for Chemistry", Oxford Univ. Press, 2005
2. J.C. Miller and J.N Miller, “Statistics for Analytical Chemistry”, Ellis Horwood Prentice Hall, 2008
3. Steven S. Zumdahl, “Chemical Principles” 6th Edition, Houghton Mifflin Company, 2009
4. D. Harvey, “Modern Analytical Chemistry”, McGraw-Hill Companies Inc., 2000
5. R.D. Brown, “Introduction to Chemical Analysis”, McGraw-Hill Companies Inc, 1982
6. S.L.R. Ellison, V.J. Barwick, T.J.D. Farrant, “Practical Statistics for the Analytical Scientist”, 2nd Edition, Royal Society of Chemistry, 2009
7. A. Field, “Discovering Statistics using SPSS” , Sage Publications Ltd., 2005

---

Key Terms

statistical tests, normal population, chi-squared test, data points, plotting a histogram, QQ plot, Kolmogorov-Smirnov test