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Is it uniform?

Jan 8, 2019 7:11 AM

José G. Ramírez @ZenEos, Ph.D., W.L. Gore and Associates, Inc.

**NOTE: This article originally appeared in JMPer Cable, Issue 27, Winter 2011. The author recently co-authored with Brenda Ramirez a SAS Press Book, Douglas Montgomery's Introduction to Statistical Quality Control: A JMP® Companion.**

If you have taken a basic course in probability and statistics you were probably introduced to the Uniform distribution. Don’t blame yourself if it isn’t foremost in your memory; the Uniform distribution doesn’t have much appeal because it describes a phenomenon with constant probability. It seems hard to find applications in engineering and science for which the probability of occurrence is constant for all the values in a given interval. Pseudo-random numbers between 0 and 1 come to mind since any number in this interval should have the same probability of occurring. The Uniform distribution has sometimes been used as a model for the distribution of traffic along a straight road.

In general, probability distributions are mathematical models, or equations, used to describe and quantify the degree of uncertainty that we observe in our data. Under a continuous uniform probability model, the likelihood of observing a value *x *in the interval [θ,θ + σ] is constant and equal to, 1/σ or 1 divided by length of the interval. The mathematical equation describing a continuous uniform (rectangular) probability density function is

One of the readers of our blog, StatInsights, asked how to fit a Uniform distribution in JMP. Uniform is not one of the choices in the Continuous Fit contextual menu of the Distribution platform. However, Beta is one of the choices.

Uniform (θ,θ + σ) is Beta (1, 1,θ, σ).

Fortunately, there is a convenient relationship between a Beta distribution, which is one of the JMP choices, and the Uniform distribution. The general form of the Beta probability density function is

where *B*(α,β) is the Beta function. This pdf generalizes the standard 2-parameter Beta distribution, *Beta(α,β)*, from the interval [0,1] to an arbitrary bounded interval [θ,θ + σ]. When the shape parameter values are *α*= 1 and β = 1, then the beta function *B*(1, 1) = 1 and the pdf has constant probability equal to 1/σ. In other words, if X is distributed as uniform in the interval (θ,θ + σ), then X is distributed as Beta(1, 1) with threshold θ and scale σ.

Let’s explore this relationship. The histogram in *Figure 1 *shows 100 simulated observations from a Uniform distribution in the interval (0,1). These observations were generated using the Random Uniform generator in the Formula Editor.

As expected, this distribution has a somewhat rectangular shape. To fit a Uniform distribution to this data, select Continuous Fit > Beta from the contextual menu (red triangle) on the histogram Uniform(0, 1) title bar.

*Figure 2 *shows the histogram with the superimposed Beta fit, which has a rectangular shape. The fitted Beta Parameters Estimates show α=1.07 and β=1.03 (close to 1 — but how close?). The 95% confidence intervals for α and β both include 1, indicating that there is not enough evidence to say that they are different from 1. In practical terms, that means α and β are assumed to be 1 so the Uniform distribution does a good job at describing this data. The threshold parameter, θ, is 0.009 and scale parameter, σ, is 0.977 supporting a Uniform(0, 1). (Note that θ and σ are not maximum likelihood estimates. JMP sets θ to the minimum data value, and σ to the range (=maximum-minimum) of the data). Finally the Diagnostic Plot shows the points hovering closely around the line, and within the 95% confidence bands.

For a Uniform(θ, θ + σ), the mean is equal to θ + σ/2 and the standard deviation is equal to σ/√12 . The Moments table in *Figure 3 *shows the mean to be close to 0.5 and the standard deviation to be close to 1/√12 = 0.2886. Both the 95% confidence intervals for the mean and standard deviation contain 0.5 and 0.2886, respectively.

**What to Look for When Fitting the Beta(1, 1, θ, σ))** To use the beta distribution fit to see if the Uniform distribution is a good approximation for data, follow these steps.

- Fit a Beta distribution in the Distribution Select Continuous Fit > Beta from the red triangle menu on the Histogram title bar.
- Look at the Parameter Estimates report and verify that the estimates for the shape parameters α and β are close to
- Verify that the 95% confidence intervals for α and β include 1.
- Select Diagnostic Plot from the red triangle menu on the Fitted Beta title bar. The points on the diagnostic plot should fall close to the straigh
- The Uniform(θ, θ + σ) parameters are given by the threshold θ and scale σ.
- Select Confidence Intervals > .95 from the red triangle menu on the Histogram to see 95% confidence intervals for the mean and standard deviation (see
*Figure 3*). - Check that θ + σ/2 (mean) and σ/√12 ( Dev.) are within the intervals.

**Example: The Brisbane Baby Boom Data **> December 18, 1997, was a record-breaking day for Brisbane in Queensland, Australia. Forty-four babies were born in a 24-hour period at the Mater Mothers' Hospital.

*Figure 4 *shows the histogram of the number of minutes since midnight for each birth occurring that day. It is reasonable to believe that in a 24-hour period a birth can occur at any minute, and the histogram seems to support that. Will a Uniform distribution fit the data well?

*Figure 5 *shows the results for the Beta fit. The Parameter Estimates, α=1.357 and β=1.154, are in the vicinity of one, and their 95% confidence intervals contain 1. The Diagnostic Plot looks reasonably straight. The Threshold = 5 and the Scale =1430, suggests that a Uniform(5, 1435) distribution describes the data. In other words, the probability of being born on December 18,1997, in any given minute in a 24-hour interval, at the Mater Mothers' Hospital in Brisbane, Australia is 1/1430 = 0.0007 or 0.07%.

*Figure 6 *shows the moments and 95% confidence intervals for the mean and standard deviation. For a Uniform(5, 1435) the mean should be close to 5 + 1430/2 = 720, and standard deviation close to 1450/√12 = 412.8054. The 95% confidence interval for the mean does contain 720, and the one for the standard deviation contains 412.8054.

Next time you are wondering if your data can be described by a Uniform distribution, think ‘beta’ and use the Distribution platform with Continuous Fit **> **Beta fit.

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