How to Test for Normality (Bell Curve Distribution)


Many measurements in real life have a distribution pattern that resembles a bell curve, formally known as the "normal distribution" or "gaussian" curve. For example, IQ is distributed normally. Or, if you flip a coin a hundred times, the expected number of heads follows a normal distribution. The logarithm of human height and foot length is normal as well (often called log-normal).

In many statistical applications (such as quality control and error analysis) distributions are assumed to be normal. However, this is something that always needs to be verified, or else the analysis will be incorrect.

The steps below describe an easy method for non-statisticians to check if a distribution is more or less normal.

  • First, use a large enough random sample size for the normality test. To accurately verify whether or not a distribution is normal, you should have at least 50 data points.

  • Next, compute the average (mean), median, range, and standard deviation of the sample. Call these numbers A, M, R, and D.

  • Check if the average and the median are relatively close, considering the range of the sample. Closeness is relative, but a good standard to use is that difference between the average and median is at most 1% of the range.

    One of the hallmarks of normal distributions is that they are symmetric, that is, the mean and the median are equal. If your random sample comes from a population that is normally distributed, then the average and the median should be close.

  • Next, use the standard deviation to check the 68-95-99.7 rule. In a normal distribution, 68% of the data points lie within 1 standard deviation of the mean, 95% lie within 2 s.d., and 99.7% lie within 3 s.d.

  • If the results of Steps 3 and 4 are positive, then there is a good chance that the distribution is normal.

  • Statisticians and data analysts use more powerful mathematical tests for normality, such as the Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk tests, named after their inventors.

    You can purchase add-ins that run with Excel to perform these more rigorous tests. Analyse-It is popular program that works seamlessly with Excel to run tests for normality, as well as other statistical calculations.

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