Probability Distribution Plot Minitab

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Lyric Maro

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Aug 4, 2024, 4:13:29 PM8/4/24

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Whenwe take pictures with a digital camera or smartphone, what the device really does is capture information in the form of binary code. At the most basic level, our precious photos are really just a bunch of 1s and 0s, but if we were to look at them that way, they'd be pretty unexciting.

We encounter a similar situation when we try to use statistical distributions and parameters to describe data. There's important information there, but it can seem like a bunch of meaningless numbers without an illustration that makes them easier to interpret.

For instance, if you have data that follows a gamma distribution with a scale of 8 and a shape of 7, what does that really mean? If the distribution shifts to a shape of 10, is that good or bad? And even if you understand it, how easy would it be explain to people who are more interested in outcomes than statistics?

A building materials manufacturer develops a new process to increase the strength of its I-beams. The old process fit a gamma distribution with a scale of 8 and a shape of 7, whereas the new process has a shape of 10.

But if we go in Minitab to Graph > Probability Distribution Plot, select the "View Probability" option, and enter the information about these distributions, the impact of the change will be revealed.

The probability distribution plots make it easy to see that the shape change increases the number of acceptable beams from 91.4% to 99.5%, an 8.1% improvement. What's more, the right tail appears to be much thicker in the second graph, which indicates the new process creates many more unusually strong units. Hmmm...maybe the new process could ultimately lead to a premium line of products.

In the pilot study, the mean improvement is small, and so is the standard deviation. When the company's board looks at the numbers, they don't see the benefits of approving the program, given its cost.

By overlaying the before and after distributions, the specialist makes it very easy to see that price differences using the new system are clustered much closer to zero, and most are in the 0.5% acceptable range. Now the board can see the impact of adopting the new system.

An electronics manufacturer counts the number of printed circuit boards that are completed per hour. The sample data is best described by a Poisson distribution with a mean of 3.2. However, the company's test lab prefers to use an analysis that requires a normal distribution and wants to know if it is appropriate.

The manufacturer can easily compare the known distribution with a normal distribution using the probability distribution plot. If the normal distribution does not approximate the Poisson distribution, then the lab's test results will be invalid.

Just like your camera when it assembles 1s and 0s into pictures, probability distribution plots let you see the deeper meaning of the numbers that describe your distributions. You can use these graphs to highlight the impact of changing distributions and parameter values, to show where target values fall in a distribution, and to view the proportions that are associated with shaded areas. These simple plots also clearly and easily communicate these advanced concepts to a non-statistical audience that might be confused by hard-to-understand concepts and numbers.

Scientists who use the Hubble Space Telescope to explore the galaxy receive a stream of digitized images in the form binary code. In this state, the information is essentially worthless- these 1s and 0s must first be converted into pictures before the scientists can learn anything from them.

The same is true of statistical distributions and parameters that are used to describe sample data. They offer important information, but the numbers can be meaningless without an illustration to help you interpret them. For instance, what does it mean if your data follow a gamma distribution with a scale of 8 and a shape of 7? If the distribution shifts to a shape of 10, is that good or bad? And how would you explain all of this to an audience that is more interested in outcomes than in statistics?

A building materials manufacturer develops a new process to increase the strength of its I-beams. The output shows that the old process fit a gamma distribution with a scale of 8 and a shape of 7, whereas the new process has a shape of 10. The manufacturer does not know what this change in the shape parameter means.

The fabrication department of a farm equipment manufacturer counts the number of tractor chassis that are completed per hour. A Poisson distribution with a mean of 3.2 best describes the sample data. However, the test lab prefers to use an analysis that requires a normal distribution and wants to know if it is appropriate. If the normal distribution does not approximate the Poisson distribution, then the test results are invalid.

The scores in the region of interest (115-135) represent 14.9% of the population. This somewhat small percentage suggests that the analyst may have to expend extra effort to find a sufficient number of qualified subjects.

Probability distribution plots provide valuable insight because they reveal the deeper meaning of your distributions. Use these graphs to highlight the effect of changing distributions and parameter values, to show where target values fall in a distribution, and to view the proportions that are associated with shaded areas. These simple plots also clearly and easily communicate these advanced concepts to a non-statistical audience.

Minitab can be used to find the proportion of a normal distribution in a given range. The default in Minitab is to construct a standard normal distribution (i.e., z distribution), but the mean and standard deviation of the distribution can be edited. The following pages walk through how to construct normal distributions to find the proportion greater than a given value, the proportion less than a given value, or the proportion between two given values.

Later in this lesson, we'll see that these procedures may be used to find the p value for a given test statistic. For a right-tailed test, the p value is the area greater than the test statistic. For a left-tailed test the p value is the area less than the test statistic. For a two-tailed test, the p value is the total area in the left and right tails that is more extreme than the test statistic.

Scenario: Vehicle speeds at a highway location have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What is the probability that a randomly selected vehicle will be going 73 mph or slower?

The following two examples use Minitab to find the area under a normal distribution that is greater than a given value. The first example uses the standard normal distribution (i.e., z distribution), which has a mean of 0 and standard deviation of 1; this is the default when first constructing a probability distribution plot in Minitab. The second example models a normal distribution with a mean of 65 and standard deviation of 5.

Question: Vehicle speeds at a highway location have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What is the probability that a randomly selected vehicle will be going more than 73 mph?

In the following examples we will use Minitab to find the area under a normal distribution between two values. The first example uses the z distribution and the second example uses a normal distribution with a mean of 65 and standard deviation of 5.

Question: Vehicle speeds at a highway location have a normal distribution with a mean of 65 mph and a standard deviation of 5 mph. What proportion of vehicles are deviating from the mean by 10 mph or more? In other words, what proportion are going less than 55 mph or more than 75 mph?

Essentially same question as was asked here, but I want to do it in Python. I have used scipy stats to get a probplot, but I want to recreate the confidence interval curves and I'm not sure how to proceed. Can anyone point me in a direction??

I have an answer for the first part of the task but I am not sure how minitab calculates the confidence interval. None of the definitions I found yields something similar. Here is the code for the base plot and the fit:

There are different shapes, models and classifications of probability distributions including the ones discussed in the probability distributions article. It is always a good practice to know the distribution of your data before proceeding with your analysis. Once you find the appropriate model, you can then perform your statistical analysis in the right manner. Minitab can be used to find the appropriate probability distribution of your data.

You may use the Individual Distribution Identification in Minitab to confirm that a particular distribution best fits your current data. It allows to easily compare how well your data fit various different distributions.

A given distribution is a good fit if the data points approximately follow a straight line and the p-value is greater than 0.05. In our case, the data does not appear to follow a normal distribution as the points are not close to a straight line. You may transform your non-normal data using the Box-Cox or Johnson transformation methods so that it follows a normal distribution. You can then use the transformed data with any analysis that assumes the data follow a normal distribution.

You may also use the Probability Distribution Plots in Minitab to clearly communicate probability distribution information in a way that can be easily understood by non-experts. These plots can be used for example to highlight the effect of changing the distribution parameters or to show where target values fall in a distribution. Select Graph > Probability Distribution Plot, and then choose one of the following options: