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Data Analysis [2-15]

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Normality Test

You are often required to check and ensure the data follows a normal distribution prior to many statistical analysis.

​This can be done by using:

• (1) Numerical Method or
• (2) Graphical Method

Example

Data Ticket; Input Team $ Game MPrice; Datalines; Raptors 1 194 Raptors 2 258 Raptors 3 82 Raptors 4 113 Raptors 5 106 Raptors 6 187 Raptors 7 135 Raptors 8 50 Raptors 9 161 Raptors 10 70 Raptors 11 120 Raptors 12 154 Raptors 13 55 Raptors 14 201 Raptors 15 260 Raptors 16 51 Raptors 17 186 Raptors 18 129 Raptors 19 133 Raptors 20 13 Raptors 21 175 Raptors 22 98 Raptors 23 126 Raptors 24 84 Raptors 25 153 Raptors 26 71 Raptors 27 114 Raptors 28 89 Raptors 29 67 Raptors 30 81 Raptors 31 104 Raptors 32 114 Raptors 33 300 Raptors 34 191 Raptors 35 138 Raptors 36 81 Raptors 37 122 Raptors 38 98 Raptors 39 68 Raptors 40 168 Raptors 41 158 Raptors 42 89 Raptors 43 65 Raptors 44 241 Raptors 45 222 Raptors 46 88 Raptors 47 85 Raptors 48 1 Raptors 49 243 Raptors 50 85 Raptors 51 150 Raptors 52 141 Raptors 53 126 Raptors 54 152 Raptors 55 67 Raptors 56 121 Raptors 57 144 Raptors 58 58 Raptors 59 171 Raptors 60 174 Raptors 61 130 Raptors 62 200 Raptors 63 11 Raptors 64 95 Raptors 65 111 Raptors 66 41 Raptors 67 186 Raptors 68 61 Raptors 69 119 Raptors 70 50 Raptors 71 73 Raptors 72 6 Raptors 73 69 Raptors 74 167 Raptors 75 264 Raptors 76 163 Raptors 77 109 Raptors 78 226 Raptors 79 246 Raptors 80 181 Raptors 81 72 Raptors 82 111 ; Run;

Note: there are 82 observations in this data set. Not all of them are shown in the image above.


The TICKET data set contains 3 variables:

• Team: Raptors (GO RAPS GO)
• Game: The nth game of the season
• MPrice: Median Ticket Price

An analyst is interested in checking whether the median selling price (i.e. MPrice) follows a normal distribution.


1. Numerical Method

The numerical method is based on 4 normality test results.

Example

​Proc Univariate Data=Ticket normal;
Var Mprice;
Run;

​Adding the NORMAL option to Proc Univariate creates an additional table with the list of normality tests in the output:

Note: this is the 4th table in the output.
​

The numerical method looks at the 4 normality testings:

• ​​Shapiro-Wilk
• Kolmogorov-Smirnov
• Cramer-von Mises
• Anderson-Darling

Since all of the p-values are greater than 0.05, the median ticket price is assumed to be normally distributed.

Quick Resources to learn about:

• p-value
• normality test

2. Graphical Method

The graphical method looks at the stem-and-leaf plot, box plot, and normality probability plots as well as the histogram.

Example
​
Proc Univariate Data=Ticket plots;
Var Mprice;    
Histogram;
Run;

The PLOTS option generates 3 plots:

1. Stem-and-leaf plot (or a horizontal bar chart)
2. Box plot
3. Normal probability plot

The HISTOGRAM statement plots the histogram as well:

All of the graphs show a distribution that is fairly close to a normal distribution.

Quick Resources to learn about:

• normal probability plot

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Exercise

Locate the CARS data set from the SASHelp library.

Analyze whether the MSRP follows a normal distribution using both the numerical and graphical methods.

What would be your conclusion and why?

Next

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Need some help? 

Get Hint

HINT:
What are the p-values from the normality tests? How do the plots look like?

Get Solution

SOLUTION:
Proc Univariate Data=SASHelp.cars normal plots;
Var MSRP;
histogram;
Run;

The data does not seem to be normally distributed.

The p-values are all less than 0.05, which rejects the hypothesis that the data is normally distributed.

Both of the bar chart and box plot show asymmetric distribution with the majority of the MSRP falling in the range of $15,000 to $45,000.

The normal probability plot bends up to the left of the normal line. This shows a long tail to the right.

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