The **normal probability plot** is a graphical technique for normality testing: assessing whether or not a data set is approximately normally distributed. In other words, a normal probability plot is a graphical technique to identify substantive departures from normality. The normal probability plot is one type of quantile-quantile (Q-Q) plot.

A Normal Probability Plot compares the values in a data set (on the vertical axis) with their associated quantile values derived from a standardized normal distribution (on the horizontal axis). In other words, it plot graph Z-scores against the data.

**Why You Use a Normal Probability Plot**?

The normal probability plot is formed by plotting the sorted data with an approximation to the means or medians of the corresponding order statistics. It is used to determine if a small set of data come from a normal distribution. In other words, if we get a straight line from the plot, we can say the process is normally distributed. Thus, it is a good option to determine the Process capability.

## When Do You Use a Normal Probability Plot

The data are drawn against a theoretical normal distribution in a manner that the points should form almost a straight line. Departures from this straight line indicate departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures.

## How to Construct a Normal Probability Plot

1. Arrange the values in ascending order. In other words, arrange the n number of values from minimum to maximum.

2. Arrange a rank order number(i) from 1 to n. Here n is the total number of samples

3. Calculate the cumulative probability for each rank order from1 to n values

f(i) = (i-0.375)/(n+0.25)

4. For each value of cumulative probability, determine the z-value from the standard normal distribution.

5. Create a scatter plot with the sorted data versus corresponding z-values

6. Finally, analyze the graph. If all the points are roughly on the straight line, then determine it follows the normal distribution.

## Example of Using the Normal Probability Plot in a Six Sigma Project

**Example:** The data in the table below is a random sample of 16 individuals wait time in the coffee shop. Is there evidence to support the belief that the variable of waiting time follows a normal distribution?

**Step 1:** Arrange data in ascending order

**Step 2:** Then, assign a rank order number(i) from 1 to n. For instance, the total number of samples n equals to 16.

**Step 3: **Calculate the cumulative probability

f(i) = (i-0.375)/(n+0.25)

for i=1, f(1) =(1-0.375)/(16+0.25) = 0.0385

for i=2, f(2) = (2-0.375)/(16+0.25) = 0.1000, similarly calculate for other values.

**Step 4:** Further, determine the z value for each cumulative probability

**Step5:** Then, create a scatter plot with the sorted data versus corresponding z-values

**Step 6:** Analyze the graph: To explain, from the above graph it is almost clear that Normal probability plot is close enough to linear. Hence we conclude that wait time follows a normal distribution.

## Comments (4)

need the data set to understand the concept

Data set added!

Can you please explain how to determine the Z value. Can you share any reference to determine the Z value as shown in the table on step 4.

Hi Manimaran,

Thank you for the comment. We have a graphic under step 4 that shows you how to look up the z value.

Do you see on the chart how the cumulative probability is 0.0384?

If you look for that value in the z chart (on our diagram that’s where the 2 red rectangles instersect), you’ll see a value of 0.0385. That’s close enough for our purposes.

You then trace out to the 2 axis to get the actual value. Again, following each of the red rectangles in the picture you get -1.77.

You repeat this process for every value in the chart.

Does that make sense?

Best, Ted.