How Do We Graphically Check Whether A Data Set Is A Drawn From A Normal Distribution?

You're probably familiar with data that follow the normal distribution. The normal distribution is that nice, familiar bong-shaped bend. Unfortunately, not all data are normally distributed or as intuitive to empathize. Yous can picture the symmetric normal distribution, but what almost the Weibull or Gamma distributions? This uncertainty might leave you feeling unsettled. In this mail, I show you how to identify the probability distribution of your information.

You might think of nonnormal data every bit abnormal. However, in some areas, you should really expect nonnormal distributions. For example, income information are typically right skewed. If a process has a natural limit, data tend to skew abroad from the limit. For example, purity can't exist greater than 100%, which might cause the data to cluster near the upper limit and skew left towards lower values. On the other hand, drill holes can't exist smaller than the drill bit. The sizes of the drill holes might be right-skewed away from the minimum possible size.

Data that follow any probability distribution can exist valuable. Withal, many people don't feel equally comfy with nonnormal information. Let's shed light on how to identify the distribution of your information!

We'll learn how to identify the probability distribution using body fat percentage data from middle school girls that I collected during an experiment. Yous can download the CSV information file: body_fat.

Related post: Agreement Probability Distributions and The Normal Distribution

Graph the Raw Data

Let's plot the raw data to meet what it looks like.

The histogram gives us a practiced overview of the data. At a glance, we can see that these data clearly are non unremarkably distributed. They are right skewed. The peak is around 27%, and the distribution extends further into the college values than to the lower values. Acquire more well-nigh skewed distributions. Histograms can besides place bimodal distributions.

These data are not normal, merely which probability distribution do they follow? Fortunately, statistical software tin help us!

Related posts: Using Histograms to Understand Your Data, Dot Plots: Using, Examples, and Interpreting, and Assessing Normality: Histograms vs. Normal Probability Plots

Using Distribution Tests to Identify the Probability Distribution that Your Data Follow

Distribution tests are hypothesis tests that decide whether your sample information were drawn from a population that follows a hypothesized probability distribution. Like any statistical hypothesis examination, distribution tests have a null hypothesis and an culling hypothesis.

H₀: The sample data follow the hypothesized distribution.
H_ane: The sample data do not follow the hypothesized distribution.

For distribution tests, small p-values bespeak that you tin reject the null hypothesis and conclude that your data were non drawn from a population with the specified distribution. However, we desire to identify the probability distribution that our information follow rather than the distributions they don't follow! Consequently, distribution tests are a rare case where y'all look for loftier p-values to place candidate distributions.

Before we exam our data to identify the distribution, here are some measures y'all need to know:

Anderson-Darling statistic (Advertizement): There are different distribution tests. The exam I'll use for our data is the Anderson-Darling exam. The Anderson-Darling statistic is the exam statistic. It's like the t-value for t-tests or the F-value for F-tests. Typically, you don't translate this statistic directly, just the software uses it to summate the p-value for the examination.

P-value: Distribution tests that have high p-values are suitable candidates for your data's distribution. Unfortunately, it is not possible to calculate p-values for some distributions with three parameters.

LRT P: If y'all are considering a three-parameter distribution, assess the LRT P to determine whether the third parameter significantly improves the fit compared to the associated two-parameter distribution. An LRT P value that is less than your significance level indicates a significant comeback over the 2-parameter distribution. If you see a college value, consider staying with the ii-parameter distribution.

Annotation that this case covers continuous information. For categorical and discrete variables, you lot should use the chi-foursquare goodness of fit exam.

Goodness of Fit Exam Results for the Distribution Tests

I'm using Minitab, which can test 14 probability distributions and 2 transformations all at once. Let'southward accept a expect at the output beneath. Nosotros're looking for higher p-values in the Goodness-of-Fit Examination table below.

Every bit we expected, the Normal distribution does not fit the data. The p-value is less than 0.005, which indicates that we can reject the nix hypothesis that these information follow the normal distribution.

The Box-Cox transformation and the Johnson transformation both have high p-values. If nosotros demand to transform our data to follow the normal distribution, the high p-values indicate that we can apply these transformations successfully. Nonetheless, nosotros'll disregard the transformations considering nosotros desire to identify our probability distribution rather than transform it.

The highest p-value is for the three-parameter Weibull distribution (>0.500). For the 3-parameter Weibull, the LRT P is pregnant (0.000), which ways that the third parameter significantly improves the fit.

The lognormal distribution has the next highest p-value of 0.345.

Let'due south consider the 3-parameter Weibull distribution and lognormal distribution to be our top two candidates.

Related post: Understanding the Weibull Distribution

Using Probability Plots to Place the Distribution of Your Data

Probability plots might be the best style to determine whether your data follow a detail distribution. If your data follow the straight line on the graph, the distribution fits your information. This process is simple to do visually. Informally, this process is called the "fat pencil" test. If all the information points line up within the surface area of a fat pencil laid over the center straight line, you can conclude that your information follow the distribution.

Probability plots are too known every bit quantile-quantile plots, or Q-Q plots. These plots are similar to Empirical CDF plots except that they transform the axes and so the fitted distribution follows a straight line.

Q-Q plots are specially useful in cases where the distribution tests are too powerful. Distribution tests are similar other hypothesis tests. As the sample size increases, the statistical power of the test as well increases. With very large sample sizes, the exam can take and so much power that trivial departures from the distribution produce statistically significant results. In these cases, your p-value volition be less than the significance level even when your data follow the distribution.

The solution is to appraise Q-Q plots to identify the distribution of your data. If the data points autumn forth the direct line, you can conclude the data follow that distribution even if the p-value is statistically significant.

The probability plots below include the normal distribution, our top 2 candidates, and the gamma distribution.

The data points for the normal distribution don't follow the center line. Even so, the data points do follow the line very closely for both the lognormal and the three-parameter Weibull distributions. The gamma distribution doesn't follow the middle line quite also as the other ii, and its p-value is lower. Again, it appears like the option comes down to our top two candidates from before. How practice we choose?

An Additional Consideration for Three-Parameter Distributions

Three-parameter distributions have a threshold parameter. The threshold parameter is also known as the location parameter. This parameter shifts the unabridged distribution left and correct along the x-axis. The threshold/location parameter defines the smallest possible value in the distribution. Yous should use a iii-parameter distribution only if the location truly is the lowest possible value. In other words, use field of study-area knowledge to help you cull.

The threshold parameter for our information is xvi.06038 (shown in the table below). This cutoff point defines the smallest value in the Weibull distribution. However, in the full population of middle schoolhouse girls, information technology is unlikely that there is a strict cutoff at this value. Instead, lower values are possible even though they are less probable. Consequently, I'll option the lognormal distribution.

Related post: Understanding the Lognormal Distribution

Parameter Values for Our Distribution

We've identified our distribution as the lognormal distribution. Now, we demand to find the parameter values for it. Population parameters are the values that define the shape and location of the distribution. We just need to await at the distribution parameters table below!

Our torso fat per centum data for middle school girls follow a lognormal distribution with a location of 3.32317 and a calibration of 0.24188.

Below, I created a probability distribution plot of our two elevation candidates using the parameter estimates. You can run across how the three-parameter Weibull distribution stops abruptly at the threshold/location value. However, the lognormal distribution continues to lower values.

Identifying the probability distribution that your data follow can exist critical for analyses that are very sensitive to the distribution, such as capability analysis. In a future blog post, I'll evidence you what else you can do by simply knowing the distribution of your data. This post is all continuous data and continuous probability distributions. If you have discrete data, read my postal service nearly Goodness-of-Fit Tests for Discrete Distributions.

Finally, I'll close this mail service with a graph that compares the raw data to the fitted distribution that we identified.

Annotation: I wrote a different version of this postal service that appeared elsewhere. I've completely rewritten and updated it for my blog site.