  How do I run a the test of normality?  Assumptions of normality

Question

How do I run a the test of normality?  Assumptions of normality

Step 1

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.
Step 2

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.

Step 3

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 [(OiEi...

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