What are Normal Probability Plots?
Normal distribution is a major player in statistics and the foundation of many tests and models. Since so much in the real world tends to follow a normal distribution, it’s an essential concept for data analysis.
A normal distribution probability plot is a way to see if a dataset approximates normal distribution. It’s a quick and easy way to check if you can use a statistical procedure that relies on the data being normally distributed.
There are statistical tests to check if your data are normally distributed. But visual inspection is important and can give you insights that a test might miss. For example, a normal distribution probability plot can expose skewness or outliers. It’s always a good idea to use a plot in conjunction with a statistical test.
What is a Normal Probability Plot?
A normal probability plot is a way to visualize whether your data follows a normal distribution. It compares your data to a perfect normal distribution.
Here’s how it works: Imagine you rank your data from lowest to highest and calculate the percentile for each value. Then, for each percentile, you find the corresponding Z-score – how many standard deviations away from the mean that percentile would be in a perfect normal distribution. Finally, you plot each data point against its Z-score.
If your data is normally distributed, all the points on the plot will fall pretty close to a straight line. If the points stray far from the line, your data might not be normal.
The plot essentially compares the “quantiles” (the value below which a certain percentage of your data falls) of your data to the quantiles of a standard normal distribution.
Interpreting normal probability plots
A normal probability plot is a visual tool that helps you assess whether a dataset follows a normal distribution. It plots your sample data against a theoretical normal distribution in such a way that the points should form an approximate straight line. If the data are normally distributed, the points will fall close to this line. Deviations from this line suggest departures from normality.
Identifying Normality
When you look at a normal probability plot, data that fall closely to a straight line suggest that the data are normally distributed. The line doesn’t have to be perfectly straight, but it should be relatively linear. A correlation coefficient close to 1 indicates strong linearity, which suggests normality. Lower correlation coefficients suggest deviations from normality.
Detecting Non-Normality with Normal Probability Plots
Normal probability plots can also help you spot issues like skewness, outliers, and kurtosis.
- Skewness: If your data are skewed, the points on the normal probability plot will form an S-shaped curve instead of a straight line.
- Outliers: An outlier will appear as a point that falls far away from the general pattern of the other points. They often show up at the extreme ends of the plot.
- Kurtosis: Kurtosis refers to the shape of the tails of the distribution. If the data have heavy tails (leptokurtic), the ends of the plot will curve away from the straight line. If the data have light tails (platykurtic), the ends of the plot will curve toward the straight line.
Statistical Tests for Normality
While a normal probability plot gives you a visual sense of whether your data are normally distributed, statistical tests can provide more concrete evidence. Here are a few commonly used tests.
Anderson-Darling Test
The Anderson-Darling test checks whether a sample of data fits a specific distribution. This test puts more weight on the tails of the distribution than some other tests.
The test results are expressed as a p-value. If the p-value is small (typically less than 0.05), that indicates the data is not normally distributed. A larger test statistic indicates a greater departure from normality.
Shapiro-Wilk Test
The Shapiro-Wilk test is another statistical test used to assess normality, and it’s considered one of the more powerful tests, especially when you’re working with smaller sample sizes.
As with the Anderson-Darling test, a small p-value indicates that your data is not normally distributed. You can use the Shapiro-Wilk test on samples up to a size of 2000.
Skewness and Kurtosis
Skewness and kurtosis are two measures that can help describe the shape of your distribution.
- Skewness measures the asymmetry of the distribution.
- Kurtosis measures the “tailedness” of the distribution.
Positive skewness means the tail is longer on the right side, while negative skewness means the tail is longer on the left. High kurtosis means heavier tails and a sharper peak, while low kurtosis means lighter tails and a flatter peak.
What if my data isn’t normal?
Sometimes, no matter how hard you try, your data just won’t play nice and follow a normal distribution. Don’t despair! You have options.
One common trick is to use data transformations. These mathematical functions reshape your data to make it look more normal. Think of it like stretching or squeezing a rubber band to fit a specific shape. Common transformations include:
- Logarithmic transformation: Great for data with a “long tail” on the right side (positive skewness).
- Square root transformation: Useful when you’re dealing with counts (like the number of events in a period).
- Box-Cox transformation: A more advanced technique that can handle a wider range of non-normal shapes.
But what if transformations don’t work, or you’re uncomfortable changing your data? That’s where nonparametric tests come in. These statistical tests don’t rely on the assumption that your data is normally distributed. They’re like the rebels of the statistical world, playing by their own rules. Examples include the Mann-Whitney U test and the Kruskal-Wallis test.
Creating Normal Probability Plots
Normal probability plots can be created using spreadsheet software like Excel or statistical programming languages like R or Python. Here’s a quick rundown of how to create these plots in each program.
In Excel
To create a normal probability plot in Excel, you’ll first need to sort your data from smallest to largest. Next, calculate the empirical cumulative distribution function (ECDF) for each data point. This involves determining the proportion of data points that are less than or equal to each value. Then, calculate the corresponding z-scores for each ECDF value using the inverse of the standard normal cumulative distribution function. Finally, plot your original data points against the calculated z-scores.
In R
R makes creating normal probability plots relatively straightforward. The `qqnorm()` function generates the plot, and the `qqline()` function adds a reference line representing a perfect normal distribution. By visually comparing your data points to this line, you can quickly assess normality.
In Python
Python offers a convenient way to generate normal probability plots using the `probplot()` function from the `scipy.stats` module. You’ll also need `matplotlib` for plotting. Simply pass your data to the `probplot()` function, and it will generate the plot, allowing you to visually assess how well your data conforms to a normal distribution.
Applications of Normal Probability Plots
Normal probability plots aren’t just theoretical tools; they have a variety of real-world applications. Here are a few examples:
Manufacturing and Quality Control
In manufacturing, these plots are used to keep an eye on the consistency of production. By plotting product dimensions or other key quality characteristics, manufacturers can quickly see if the data is following a normal distribution. If the data starts to stray from that straight line, it could signal that something’s off with the manufacturing process or that defects are creeping in.
This is where methodologies like Six Sigma and Statistical Process Control (SPC) come in. Normal probability plots are a key part of these strategies, helping to improve processes, keep variation in check, and spot potential problems early on.
Scientific Experiments
Many statistical tests that scientists use rely on the assumption that the data is normally distributed. Before running those tests, researchers can use normal probability plots to make sure that assumption holds true. If the plot shows a significant departure from normality, it might be necessary to use different statistical methods or transform the data.
Financial Modeling
In finance, it’s often assumed that financial data like stock returns follow a normal distribution. Normal probability plots offer a quick way to check this assumption. If the plot shows that the data isn’t normally distributed, it could mean that more sophisticated models are needed to accurately represent the behavior of the financial data.
Case Studies
Want to see how normal probability plots play out in the real world? Here are a couple of examples.
Semiconductor Manufacturing
A semiconductor fabrication plant was carefully monitoring the dimensions of its chips, using normal probability plots to keep an eye on the manufacturing process. Any variations from what was considered “normal” immediately triggered a closer look. In one instance, the normal probability plot showed a clear deviation, which led the engineers to discover a malfunction in one of their key pieces of equipment. Quick action, thanks to the plot!
Clinical Trials
Researchers running a clinical trial were analyzing patient response data when their normal probability plot showed some unexpected results. The data wasn’t following a normal distribution! This non-normal effect prompted the researchers to dig deeper, looking at different patient subgroups and questioning the effectiveness of the treatment for various populations. The normal probability plot was the key to unlocking more nuanced findings.
Best practices and limitations of normal probability plots
Normal probability plots can be a quick and easy way to assess whether your data is normally distributed. But like any statistical tool, there are some best practices to keep in mind, along with a few limitations.
Best practices
- Use a sufficiently large sample size. A sample of at least 25 or 30 data points is best for reliable results.
- Check assumptions. Make sure you’re working with continuous data that was obtained through random sampling.
- Supplement visual inspection with numerical tests. Don’t rely on the plot alone.
Limitations of normal probability plots
- Interpretation can be subjective. What looks “straight” to one person might look curved to another.
- Subtle deviations can be hard to detect. This is especially true when you’re working with smaller sample sizes.
- They’re not a substitute for formal statistical tests. A normal probability plot is a useful tool, but it shouldn’t be the only tool you use to assess normality.
Summary
Normal probability plots are powerful visual aids for determining whether your data are normally distributed. They quickly show you any departures from normality, such as skewness, kurtosis, or outliers that might affect your statistical analysis.
For the most robust analysis, combine visual inspection of normal probability plots with statistical tests and data transformations. This approach helps you get a complete picture of your data’s distribution and make informed decisions about which statistical methods are most appropriate.
Whether you’re in manufacturing, science, finance, or another field, normal probability plots are a versatile tool for understanding and interpreting your data.