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When can we say that populations are
Introduction:
Several tests of normality exist, using which you can verify whether a particular data follows the normal distribution.
Usually, before conducting a formal test, we prefer to take the help of graphical methods, to see if the data may be assumed to follow the normal distribution, at least approximately. A few such graphical methods are:
- Histogram of the data , superimposed with a normal probability curve,
- Normal probability plot with confidence interval,
- Normal quantile-quantile (QQ) plot.
- Boxplot, etc.
Explanation:
If the graphical display appears to show at least an approximate normal distribution, then a formal test can be used to verify the normality. A few such tests are as follows:
- Pearson’s Chi-squared test for goodness of fit,
- Shapiro-Wilk test,
- Kolmogorov-Smirnov test, etc.
The Pearson’s Chi-squared test is discussed here.
Pearson’s Chi-squared test for goodness of fit:
Suppose the data set can be divided into n categories or classes, with observed frequency in the ith class as Oi and expected frequency in the ith class as Ei (i = 1, 2, …, n). Further, assume that the data is obtained from a simple random sampling, the total sample size is large, each cell count (for each category) is at least 5 and the observations are independent.
Then, the degrees of freedom, df = (number of categories) – (number of parameters in the model) – 1. For n categories in the data set and 2 parameters (mean and variance) of the normal distribution, df = n – 3.
The test statistic for the test is given as, χ2 = Σ [(Oi – Ei)2/ Ei], where the summation is done over all i = 1, 2, …, n.
The observed frequencies will be known from the data set. The expected frequencies for a normal distribution can be obtained by multiplying the total sample size, say, N, by the normal probability for the corresponding class (obtained from a standard normal table or any software such as, EXCEL, MINITAB, etc.).
The corresponding p-value for the test can be used to check whether the data follows normal distribution or not.
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