# The Normality Test

Simple and Done in Excel

The normality test is used to determine whether a data set resembles the normal distribution. If the data set can be modeled by the normal distribution, then statistical tests involving the normal distribution and t distribution such as **Z test**,

**t tests**,

**F tests**, and

**Chi-Square tests**can performed on the data set.

There are a number of well-known normality tests such as

**Kolmogorov Smirnov Test**,

**Shapiro Wilk Test**, and the

**Anderson Darling Test**. In this article we will describe two normality tests that can be performed with Excel, but are much simpler than the above tests.

# The Normality Test

The Most Basics Ones

**The Histogram - The Simplest Normality Test **

Probably the easiest normality test is to plot the data in an Excel histogram and then compare the histogram to a normal curve. This method works much better with larger data sets. It is extremely simple to perform in Excel. Here is an example of how a Histogram is used in Excel as the most basic Normailty test:We are going to evaluate the following data for Normality using a Histogram:

After the input data is arranged as above, we need to determine how we want the data to be grouped when it is broken down into a Histogram. Excel calls the groups "bins." We need to determine the upper and lower range of each bin. When the data is inserted into Excel, we need only to provide the lower boundary of each bin. Here is how is I have arbitrarily set up lower boundaries for each bin:

Now we are ready to create a Histogram with Excel. Access the Excel Histogram in Excel 2003 from:

**Tools / Data Analysis / Histogram**. A dialogue box will appear. The following dialogue box is shown completed. Highlight the input data and bin range data by selecting yellow-colored data cells as is shown above. Your dialogue box will look like this one when you are ready to create the Histogram:

Hitting the OK button will give a completed Histogram that will look like this:

Compare this to a Normal curve with the same mean and standard deviation as follows:

In this case, the data does appear to have been drawn from a Normally-distributed population

**The Normal Probability Plot - **

A Simple, Quick Normality Test for Excel

Another normality test that is very easy to implement in Excel is called the Normal Probability Plot. There are 2 ways to create the Normal Probability Plot. They both create the same output. I use the 1st method because it is accompanied with an explanation of why the method works. I personally have difficulty with applying a method that I don't understand. Here are both methods, starting with my preferred choice:A Simple, Quick Normality Test for Excel

**Creating the Normal Probability Plot - Method 1**

One characteristic that defines the Normal distribution is that Normally-distributed data will have the same amount of area of Normal curve between each point. For example, if there were 7 sampled points total that were perfectly Normally-distributed, The area under the Normal curve between each point would contain 1/7 of the total area under the Normal curve.

The area under the Normal curve between 2 points can be shown graphically as follows:

**Calculating the CDF**

We can obtain the normal curve area between two sample points (on the X-axis) by using the

**Cumulative Distribution Function (CDF).**The CDF at any point on the x-axis is the total area under the curve to the left of that point. We can obtain the percentage of area in normal curve for each regionby subtracting the CDF at the x-Value of region's lower boundary from the CDF at the x-Value of the region's upper boundary.

The normal distribution that we are trying to fit data has as its two and only parameters the sample's mean and standard deviation.

The CDF of this normal distribution at any point on the x-Axis can be determined by the following Excel formula:

CDF = NORMDIST ( x Value, Sample Mean, Sample Standard Deviation, TRUE )

Once again, this formula calculate the CDF at that x Value, which is the area under the normal curve to the left of the x Value. That normal curve has as its parameters the sample's mean and standard deviation.

**Graphical Interpretation of the CDF**

**CDF (65% of Curve Area From Upper Boundary of Region)**

**Click on Image To See Larger Version****MINUS**

**CDF (25% of Curve Area From Lower Boundary of Region)**

**Click On Image To See Larger Version****EQUALS**

**25% of Curve's Total Area Is Inside Region**

**Click On Image To See Larger Version**Given the above, here are the Steps to creating a Normal Probability Plot to evaluate the Normality of sampled data.

Here is a set of 7 sampled points that we are going test for Normality using the Normal Probability Plot:

From these samples, we need to calculate sample size (count - number of samples), sample mean, and sample stadard deviation. Here are those calculations:

Given the above sample size, mean, and standard deviation, if the sample were perfectly Normally-distributed, the sample would have been as follows:

If there are 7 sampled data points that were perfectly Normally distributed, there would be 1/7 of the total Normal curve area between each sampled point.

The Z Score at each sampled point are found with the following Excel formula:

NORMSINV (CDF at each Sample Point)

The Expected Sample Values are found by the following Excel formula:

NORMINV (CDF at Sample Point, Sample Mean, Sample Stan. Dev.)

A graph of Expected Sample Values vs. Z Score will be a straight line, as follows:

We now observe the actual data samples compared to the Expected Data Samples for Normally-distributed data having the same mean and standard deviation:

We now wish to see how close the Actual Sample Values graph to the staright line of the Expected Sample Values, as follows:

We can see that the Actual Sample Data (in purple) maps closely to the Expected Sample Values (in dark blue) so we conclude that the data appears to be derived from a Normally-distributed population.

One caution: A larger sample size (at least 50) should be used to obtain valid results. The small sample size (7) was used here for simplicity.

**************************************************************

**Creating the Normal Probability Plot - Method 2**

The data set is ranked in order and then plotted on a graph. Each point in the data set represents a y value of a plotted point. The x values of the points are Normal Order Statistic Medians. The closer than the graph is to a straight line, the more closely the data set resembles the normal distribution. Correlation analysis can also be performed the data set (called the Order Responses) and the Normal Order Statistic Medians. The closer the correlation coefficient is to 1, the more the data set resembles the normal distribution.

**An Example**

An example is the best way to illustrate the Normal Probability Plot. Evaluate the following data set of 6 points for normality:

{66, 76, 17, 23, 44, 41}

The rank of each data point is:

5, 6, 1, 2, 4, 3

The data in ranked order is:

{17, 23, 41, 44, 66, 76}

Now we have to calculate the

**Normal Order Statistic Medians**. We know that we have 6 points so n = 6. The

**Normal Order Statistic Medians**are given by the following formula:

**N(i)**= G(U(i))

**U(i) are the Uniform Order Statistic Medians**defined by this formula:

m(i) = 1 - m(n) for i = 1

m(i) = (i - 0.3175)/(n + 0.365) for i = 2, 3, ..., n-1

m(i) = 0.5(1/n) for i = n

**G is called the Percent Point of the Normal Distribution**. It is the inverse of the cumulative distribution function. In Excel, it would be the NORMSINV(x) function. It tells you the probability the x has a value of m(i) or less. Variable x is normally distributed on a standard normal curve (µ = 0 and σ = 1).

Given the above information, here is how the

**Normal Order Statistic Medians**are calculated:

n = 6

Now calculate

**U(i) – the Uniform Order Statistic Medians**.

**U(i)**are the

**Uniform Order Statistic Medians**defined by this formula:

m(i) = 1 - m(n) for i = 1

m(i) = (i - 0.3175)/(n + 0.365) for i = 2, 3, ..., n-1

m(i) = 0.5(1/n) for i = n

i = 1

m(1) = 1 – m(n) = 1 – m(6) = 1 – 0.0833 = 0.9167

i = 2

m(2) = (i - 0.3175)/(n + 0.365) = (2 – 0.3175) / (6 + 0.365) = 0.2639

i = 3

m(3) = (i - 0.3175)/(n + 0.365) = (3 – 0.3175) / (6 + 0.365) = 0.4208

i = 4

m(4) = (i - 0.3175)/(n + 0.365) = (4 – 0.3175) / (6 + 0.365) = 0.5776

i = 5

m(5) = (i - 0.3175)/(n + 0.365) = (5 – 0.3175) / (6 + 0.365) = 0.7345

i = 6

m(6) = m(i) = 0.5(1/n) for i = n = m(i) = 0.5(1/6) = 0.0833

So,

U(1) = 0.9167

U(2) = 0.2639

U(3) = 0.4208

U(4) = 0.5776

U(5) = 0.7345

U(6) = 0.0833

The

**Normal Order Statistic Medians**are given by the following formula:N(i) = G(U(i)) --> G(U(i)) is the inverse of the cumulative distribution function. It tells the x value that corresponds to the probability U(i) that a random sample taken from a standardized normally distributed population will have a value of x or less.

This is found in Excel by the following formula:

N(i) = G(U(i)) = NORMSINV(U(i))

So, the

**Normal Order Statistic Medians**are given by:G(U(i)) = NORMSINV(U(i))

N(1) = NORMSINV(U(1)) = NORMSINV(0.9167) = 1.383

N(2) = NORMSINV(U(2)) = NORMSINV(0.2639) = -0.631

N(3) = NORMSINV(U(3)) = NORMSINV(0.4208) = - 0.200

N(4) = NORMSINV(U(4)) = NORMSINV(0.5776) = 0.196

N(5) = NORMSINV(U(5)) = NORMSINV(0.7345) = 0.626

N(6) = NORMSINV(U(6)) = NORMSINV(0.8908) = -1.383

The above are the X values of the data points whose Y values are the ranked point in the data set. The ranked data set is:

{17, 23, 41, 44 66, 76}

So, the following points can be plotted:

(1.383, 17) (-0.631, 23) (-0.200, 41) (0.196, 44) (0.626, 66) (-1.383, 76)

The final graph will resemble a chart such as this:

The closer that the plotted resembles a straight line, the closer the data set resembles the normal distribution. You can also run correlation analysis between the data set of Ordered Responses and the Normal Order Statistic Medians. The closer the correlation coefficient is to 1, the more closely the data set resembles the normal distribution.

There are other well-known Normality tests such as the

**Kolmogorov-Smirnov Goodness-of-Fit Test**, the

**Anderson-Darling Goodness-of-Fit Test**,

**The Shapiro-Wilk Test**, and the

**Chi-Square Goodness-of-Fit Test.**I will very shortly publish an article or two in this blog which will detail how to do these tests in Excel.

If you are going to perform any statistical analysis that uses the normal distribution or t distribution such as Z test, t tests, F tests, and chi-square tests, you should first test your data set for normality. The Normal Probability Plot described in this article is probably the easiest and quickest way to do it in Excel.

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- Overview of the 1-Sample t-Test in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- Overview of 2-Independent-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2-Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- Overview of z-Based Confidence Intervals of a Population Mean in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Simple Linear Regression Example in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Multiple Linear Regression Example in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression Performed in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer-Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA Example in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared (?2) in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi (?) – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared (?2) in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers

- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables
- SEO Functions in Excel
- Time Series Analysis in Excel

Where are you getting 1/14, 3/14, 5/14 ... for CDF at each sample point in the first Normal Plat example?

ReplyDeleteThe values of 1/14 etc. are the probability intervals. You get them as follows: start with 1/(2*n), so 1/(2*7), then add 2/2n to the previous value. You'll get 1/14, 3/14, 5/14, etc.

ReplyDeleteAs the author said, there's a 1/7th distance between each probability value, or 1/n in general.

quite helpful to young statisticians and can do it alone

ReplyDeleteQuite helpful.

ReplyDeleteHow does m(6) = 0.5(1/6) = 0.8909? By my math, 0.5*(1/6) = 0.0833

ReplyDeleteJust fixed it. Thanks for catching that.

ReplyDeleteExcel is also very useful as SPSS.

ReplyDeleteThis is extremely useful

ReplyDeleteThis was of very timely help. Thanks much

ReplyDeleteI found the first method of creating the normal probability plot helpful. But when I use the data from the second method for the first, the corresponding correlations do not agree. It looks like it is because of an error in the expression for m(n). You use 0.5(1/n) when it should be 0.5 to the 1/n power. When I make this change the correlations are close.

ReplyDeleteIn your first method you can quantify what you see by using the R² function --> =rsq(actual values;z-scores at each sample point if sample is normal distributed). The more closer to 1 the more the actual values are normal distributed.

ReplyDeleteA formula for Uniform Order Statistic Medians is wrong. m(i) = 0.5 ^ (1/n) for i = n. That is m(6) = m(i) = 0.5 ^ (1/n) for i = n = m(i) = 0.5 ^ (1/6) = 0.8909. It results in a smoother line for your example. Source: http://www.itl.nist.gov/div898/handbook/eda/section3/normprpl.htm

ReplyDelete