In this lab, you’ll investigate the probability distribution that is most central to statistics: the normal distribution. If you are confident that your data are nearly normal, that opens the door to many powerful statistical methods. Here we’ll use the graphical tools of JASP to assess the normality of our data and also learn how to generate random numbers from a normal distribution.

Getting Started

Load module

In this lab, we will use the Distributions module in JASP. Click the plus button at the top right part of the screen, and add the Distributions module to JASP. When you do this, you will see a new icon in the top bar.

The data

This week you’ll be working with fast food data. This data set contains data on 515 menu items from some of the most popular fast food restaurants worldwide, and you can load it from here. Let’s take a quick peek at the first few rows of the data.

You’ll see that for every observation there are 17 measurements, many of which are nutritional facts.

You’ll be focusing on just three columns to get started: restaurant, calories, calories from fat.

Let’s first focus on just products from McDonalds and Dairy Queen.

  1. Make a plot (or plots) to visualize the distributions of the amount of calories from fat of the options from each of these two restaurants. How do their centers, shapes, and spreads compare? Hint: There are many ways to do this. One way would be to apply a filter to only include McDonals’ foods, and create your plots and find descriptive statistics. Then you could apply a filter to only include Dairy Queen. Alternatively, you could apply a filter to include only McDonald’s and Dairy Queen, then split by restaurant to do this in one step. To create a filter for a categorical variable, you can use the filter menu like we have done before, or click the name of the variable to click levels of the variable to include them or not include them (using the check marks and X’s in the left column).

The normal distribution

In your description of the distributions, did you use words like bell-shaped or normal? It’s tempting to say so when faced with a unimodal symmetric distribution.

To see how accurate that description is, you can plot a normal distribution curve on top of a histogram to see how closely the data follow a normal distribution. This normal curve should have the same mean and standard deviation as the data.

You’ll be focusing on calories from fat from Dairy Queen products, so let’s first filter out everything except for those products.

Open the Distributions menu and select Normal in the continuous section. We want to work with our actual data, so click Generate and Display Data to show those options. Move cal_fat into the Get variable from data set box so that we can inspect it further.

Next we want to compare this to the normal distribution, so open the Assess Fit section of this same page. Check the box Histogram vs. theoreitcal pdf to overlay a density histogram with a normal distribution with the same mean and standard deviation.

The difference between a frequency histogram and a density histogram is that while in a frequency histogram the heights of the bars add up to the total number of observations, in a density histogram the areas of the bars add up to 1. The area of each bar can be calculated as simply the height times the width of the bar. Using a density histogram allows us to properly overlay a normal distribution curve over the histogram since the curve is a normal probability density function that also has area under the curve of 1. Frequency and density histograms both display the same exact shape; they only differ in their y-axis. You can verify this by creating a frequency histogram of the same variable in the Descriptives menu.

  1. Based on the this plot, does it appear that the data follow a nearly normal distribution?

Evaluating the normal distribution

Eyeballing the shape of the histogram is one way to determine if the data appear to be nearly normally distributed, but it can be frustrating to decide just how close the histogram is to the curve. An alternative approach involves constructing a normal probability plot, also called a normal Q-Q plot for “quantile-quantile”.

Again within the Assess Fit section, check the Q-Q plot box to have JASP generate the Q-Q plot for this data.

The x-axis values correspond to the quantiles of a theoretically normal curve with mean 0 and standard deviation 1 (i.e., the standard normal distribution). The y-axis values correspond to the quantiles of the original unstandardized sample data. However, even if we were to standardize the sample data values, the Q-Q plot would look identical. A data set that is nearly normal will result in a probability plot where the points closely follow a diagonal line. Any deviations from normality leads to deviations of these points from that line.

The plot for Dairy Queen’s calories from fat shows points that tend to follow the line but with some errant points towards the upper tail. You’re left with the same problem that we encountered with the histogram above: how close is close enough?

A useful way to address this question is to rephrase it as: what do probability plots look like for data that I know came from a normal distribution?

We can answer this by simulating data from a normal distribution. Make a note of the mean and standard deviation from our cal_fat variable - we’re going to need them. Go back to the data and remove the filter - we don’t want the filter to affect the data we’re going to generate. Back in the Normal distributions menu, we can now click the Show Distribution menu. We’re going to specify the mean and standard deviation for our distribution, so in the Parameters pulldown menu, select \(\mu, \sigma\), and in the boxes below, type the mean and standard deviation of our variable in.

Now open the Generate and Display Data section. We’re going to simulate data by drawing points from a normal distribution with the same mean and standard deviation as our data. Since we have 42 observations in our dataset, enter 42 into the Number of samples box, and give the variable the name sim_norm, then click Draw samples.

We now have sim_norm as a variable that we can work with, so we can inspect it. You can take a look at the shape of our simulated data set, sim_norm, as well as its normal probability plot.

  1. Make a normal probability plot (Q-Q plot) of sim_norm. Do all of the points fall on the line? How does this plot compare to the probability plot for the real data?

Even better than comparing the original plot to a single plot generated from a normal distribution is to compare it to more plots. Click the Draw samples button again to generate another set of data using the same parameters.

  1. Does the normal probability plot for the calories from fat look similar to the plots created for the simulated data? That is, do the plots provide evidence that the calories from fat are nearly normal?

  2. Using the same technique, determine whether or not the calories from McDonald’s menu appear to come from a normal distribution.

Normal probabilities

Okay, so now you have a slew of tools to judge whether or not a variable is normally distributed. Why should you care?

It turns out that statisticians know a lot about the normal distribution. Once you decide that a random variable is approximately normal, you can answer all sorts of questions about that variable related to probability. Take, for example, the question of, “What is the probability that a randomly chosen Dairy Queen product has more than 600 calories from fat?”

If we assume that the calories from fat from Dairy Queen’s menu are normally distributed (a very close approximation is also okay), we can find this probability by calculating a Z score and consulting a Z table (also called a normal probability table). In JASP, this is done in the Show Distributions section of the Normal menu. Within this menu, make sure parameters is set to \(\mu, \sigma\), and mean and standard deviation are correct, then check the Probability box under Highlight. Select “from 0 to \(\infty\)”, then chance the 0 to 600.

Assuming a normal distribution has allowed us to calculate a theoretical probability. If we want to calculate the probability empirically, we need to determine how many observations fall above 600. We could do this by creating a new variable which is 0 or 1 depending on whether cal_fat is greater than 600, then take the mean of this column.

Although the probabilities are not exactly the same, they are reasonably close. The closer that your distribution is to being normal, the more accurate the theoretical probabilities will be.

  1. Write out two probability questions that you would like to answer about any of the restaurants in this dataset. Calculate those probabilities using both the theoretical normal distribution as well as the empirical distribution (four probabilities in all). Which one had a closer agreement between the two methods?

More Practice

  1. Now let’s consider some of the other variables in the dataset. Out of all the different restaurants, which ones’ distribution is the closest to normal for sodium?

  2. Note that some of the normal probability plots for sodium distributions seem to have a stepwise pattern. why do you think this might be the case?

  3. As you can see, normal probability plots can be used both to assess normality and visualize skewness. Make a normal probability plot for the total carbohydrates from a restaurant of your choice. Based on this normal probability plot, is this variable left skewed, symmetric, or right skewed?
    Use a histogram to confirm your findings.

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