Use a variances test to determine whether the variability of two groups differs. In this post, we’ll work through a two-sample variances test that Excel provides. Even if Excel isn’t your primary statistical software, this post provides an excellent introduction to variance tests. Excel refers to this analysis as F-Test Two-Sample for Variances. [Read more…] about How to Test Variances in Excel
Use two-way ANOVA to assess differences between the group means that are based on two categorical factors. In this post, we’ll work through two-way ANOVA using Excel. Even if Excel isn’t your main statistical package, this post is an excellent introduction to two-way ANOVA. Excel refers to this analysis as two factor ANOVA. [Read more…] about How to do Two-Way ANOVA in Excel
Outliers are unusual values in your dataset, and they can distort statistical analyses and violate their assumptions. Unfortunately, all analysts will confront outliers and be forced to make decisions about what to do with them. Given the problems they can cause, you might think that it’s best to remove them from your data. But, that’s not always the case. Removing outliers is legitimate only for specific reasons. [Read more…] about Guidelines for Removing and Handling Outliers in Data
Outliers are data points that are far from other data points. In other words, they’re unusual values in a dataset. Outliers are problematic for many statistical analyses because they can cause tests to either miss significant findings or distort real results.
Unfortunately, there are no strict statistical rules for definitively identifying outliers. Finding outliers depends on subject-area knowledge and an understanding of the data collection process. While there is no solid mathematical definition, there are guidelines and statistical tests you can use to find outlier candidates.
In this post, I’ll explain what outliers are and why they are problematic, and present various methods for finding them. Additionally, I close this post by comparing the different techniques for identifying outliers and share my preferred approach.
Outliers and Their Impact
Outliers are a simple concept—they are values that are notably different from other data points, and they can cause problems in statistical procedures.
To demonstrate how much a single outlier can affect the results, let’s examine the properties of an example dataset. It contains 15 height measurements of human males. One of those values is an outlier. The table below shows the mean height and standard deviation with and without the outlier.
Throughout this post, I’ll be using this example CSV dataset: Outliers.
|With Outlier||Without Outlier||Difference|
|2.4m (7’ 10.5”)||1.8m (5’ 10.8”)||0.6m (~2 feet)|
|2.3m (7’ 6”)||0.14m (5.5 inches)||2.16m (~7 feet)|
From the table, it’s easy to see how a single outlier can distort reality. A single value changes the mean height by 0.6m (2 feet) and the standard deviation by a whooping 2.16m (7 feet)! Hypothesis tests that use the mean with the outlier are off the mark. And, the much larger standard deviation will severely reduce statistical power!
Before performing statistical analyses, you should identify potential outliers. That’s the subject of this post. In the next post, we’ll move on to figuring out what to do with them.
There are a variety of ways to find outliers. All these methods employ different approaches for finding values that are unusual compared to the rest of the dataset. I’ll start with visual assessments and then move onto more analytical assessments.
Let’s find that outlier! I’ve got five methods for you to try.
Sorting Your Datasheet to Find Outliers
Sorting your datasheet is a simple but effective way to highlight unusual values. Simply sort your data sheet for each variable and then look for unusually high or low values.
For example, I’ve sorted the example dataset in ascending order, as shown below. The highest value is clearly different than the others. While this approach doesn’t quantify the outlier’s degree of unusualness, I like it because, at a glance, you’ll find the unusually high or low values.
Graphing Your Data to Identify Outliers
Boxplots, histograms, and scatterplots can highlight outliers.
Boxplots display asterisks or other symbols on the graph to indicate explicitly when datasets contain outliers. These graphs use the interquartile method with fences to find outliers, which I explain later. The boxplot below displays our example dataset. It’s clear that the outlier is quite different than the typical data value.
You can also use boxplots to find outliers when you have groups in your data. The boxplot below shows a different dataset that has an outlier in the Method 2 group. Click here to learn more about boxplots.
Histograms also emphasize the existence of outliers. Look for isolated bars, as shown below. Our outlier is the bar far to the right. The graph crams the legitimate data points on the far left.
Click here to learn more about histograms.
Most of the outliers I discuss in this post are univariate outliers. We look at a data distribution for a single variable and find values that fall outside the distribution. However, you can use a scatterplot to detect outliers in a multivariate setting.
In the graph below, we’re looking at two variables, Input and Output. The scatterplot with regression line shows how most of the points follow the fitted line for the model. However, the circled point does not fit the model well.
Interestingly, the Input value (~14) for this observation isn’t unusual at all because the other Input values range from 10 through 20 on the X-axis. Also, notice how the Output value (~50) is similarly within the range of values on the Y-axis (10 – 60). Neither the Input nor the Output values themselves are unusual in this dataset. Instead, it’s an outlier because it doesn’t fit the model.
This type of outlier can be a problem in regression analysis. Given the multifaceted nature of multivariate regression, there are numerous types of outliers in that realm. In my ebook about regression analysis, I detail various methods and tests for identifying outliers in a multivariate context.
For the rest of this post, we’ll focus on univariate outliers.
Using Z-scores to Detect Outliers
Z-scores can quantify the unusualness of an observation when your data follow the normal distribution. Z-scores are the number of standard deviations above and below the mean that each value falls. For example, a Z-score of 2 indicates that an observation is two standard deviations above the average while a Z-score of -2 signifies it is two standard deviations below the mean. A Z-score of zero represents a value that equals the mean.
To calculate the Z-score for an observation, take the raw measurement, subtract the mean, and divide by the standard deviation. Mathematically, the formula for that process is the following:
The further away an observation’s Z-score is from zero, the more unusual it is. A standard cut-off value for finding outliers are Z-scores of +/-3 or further from zero. The probability distribution below displays the distribution of Z-scores in a standard normal distribution. Z-scores beyond +/- 3 are so extreme you can barely see the shading under the curve.
In a population that follows the normal distribution, Z-score values more extreme than +/- 3 have a probability of 0.0027 (2 * 0.00135), which is about 1 in 370 observations. However, if your data don’t follow the normal distribution, this approach might not be accurate.
Z-scores and Our Example Dataset
In our example dataset below, I display the values in the example dataset along with the Z-scores. This approach identifies the same observation as being an outlier.
Note that Z-scores can be misleading with small datasets because the maximum Z-score is limited to (n−1) / √ n.* Indeed, our Z-score of ~3.6 is right near the maximum value for a sample size of 15. Sample sizes of 10 or fewer observations cannot have Z-scores that exceed a cutoff value of +/-3.
Also, note that the presence of the outlier throws off the Z-scores because it inflates the mean and standard deviation as we saw earlier. Notice how all the Z-scores are negative except the outlier’s value. If we calculated Z-scores without the outlier, they’d be different! Be aware that if your dataset contains outliers, Z-values are biased such that they appear to be less extreme (i.e., closer to zero).
Using the Interquartile Range to Create Outlier Fences
You can use the interquartile range (IQR), several quartile values, and an adjustment factor to calculate boundaries for what constitutes minor and major outliers. Minor and major denote the unusualness of the outlier relative to the overall distribution of values. Major outliers are more extreme. Analysts also refer to these categorizations as mild and extreme outliers.
The IQR is the middle 50% of the dataset. It’s the range of values between the third quartile and the first quartile (Q3 – Q1). We can take the IQR, Q1, and Q3 values to calculate the following outlier fences for our dataset: lower outer, lower inner, upper inner, and upper outer. These fences determine whether data points are outliers and whether they are mild or extreme.
Values that fall inside the two inner fences are not outliers. Let’s see how this method works using our example dataset.
Related post: Percentiles: Interpretations and Calculations
Calculating the Outlier Fences Using the Interquartile Range
Using statistical software, I can determine the interquartile range along with the Q1 and Q3 values for our example dataset. We’ll need these values to calculate the “fences” for identifying minor and major outliers. The output below indicates that our Q1 value is 1.714 and the Q3 value is 1.936. Our IQR is 1.936 – 1.714 = 0.222.
To calculate the outlier fences, do the following:
- Take your IQR and multiply it by 1.5 and 3. We’ll use these values to obtain the inner and outer fences. For our example, the IQR equals 0.222. Consequently, 0.222 * 1.5 = 0.333 and 0.222 * 3 = 0.666. We’ll use 0.333 and 0.666 in the following steps.
- Calculate the inner and outer lower fences. Take the Q1 value and subtract the two values from step 1. The two results are the lower inner and outer outlier fences. For our example, Q1 is 1.714. So, the lower inner fence = 1.714 – 0.333 = 1.381 and the lower outer fence = 1.714 – 0.666 = 1.048.
- Calculate the inner and outer upper fences. Take the Q3 value and add the two values from step 1. The two results are the upper inner and upper outlier fences. For our example, Q3 is 1.936. So, the upper inner fence = 1.936 + 0.333 = 2.269 and the upper outer fence = 1.936 + 0.666 = 2.602.
Using the Outlier Fences with Our Example Dataset
For our example dataset, the values for these fences are the following: 1.048, 1.381, 2.269, 2.602. Almost all of our data should fall between the inner fences, which are 1.381 and 2.269. At this point, we look at our data values and determine whether any qualify as being major or minor outliers. 14 out of the 15 data points fall inside the inner fences—they are not outliers. The 15th data point falls outside the upper outer fence—it’s a major or extreme outlier.
The IQR method is helpful because it uses percentiles, which do not depend on a specific distribution. Additionally, percentiles are relatively robust to the presence of outliers compared to the other quantitative methods.
Boxplots use the IQR method for calculating the inner fences. Typically, I’ll use boxplots rather than calculating the fences myself when I want to use this approach. Of the quantitative approaches in this post, this is my preferred method.
Finding Outliers with Hypothesis Tests
You can use hypothesis tests to find outliers. Many outlier tests exist, but I’ll focus on one to illustrate how they work. In this post, I demonstrate Grubbs’ test, which tests the following hypotheses:
- Null: All values in the sample were drawn from a single population that follows the same normal distribution.
- Alternative: One value in the sample was not drawn from the same normally distributed population as the other values.
Let’s perform this hypothesis test using our sample dataset. Grubbs’ test assumes your data are drawn from a normally distributed population, and it can detect only one outlier. If you suspect you have additional outliers, use a different test.
Grubbs’ outlier test produced a p-value of 0.000. Because it is less than our significance level, we can conclude that our dataset contains an outlier. The output indicates it is the high value we found before.
If you use Grubbs’ test and find an outlier, don’t remove that outlier and perform the analysis again. That process can cause you to remove values that are not outliers.
Challenges of Using Outlier Hypothesis Tests: Masking and Swamping
When performing an outlier test, you either need to choose a procedure based on the number of outliers or specify the number of outliers for a test. Grubbs’ test checks for only one outlier. However, other procedures, such as the Tietjen-Moore Test, require you to specify the number of outliers. That’s hard to do correctly! After all, you’re performing the test to find outliers! Masking and swamping are two problems that can occur when you specify the incorrect number of outliers in a dataset.
Masking occurs when you specify too few outliers. The additional outliers that exist can affect the test so that it detects no outliers. For example, if you specify one outlier, and there are two, the test can miss both outliers.
Conversely, swamping occurs when you specify too many outliers. In this case, the test identifies too many data points as being outliers. For example, if you specify two outliers, and there is only one, the test might determine that there are two outliers.
Because of these problems, I’m not a big fan of outlier tests. More on this in the next section!
My Philosophy about Finding Outliers
As you saw, there are many ways to identify outliers. My philosophy is that when analyzing data, you must go into the analysis with in-depth knowledge about all the variables. Part of this knowledge is knowing what values are typical, unusual, and impossible.
I find that when you have this in-depth knowledge, it’s best to use the more straightforward, visual methods. At a glance, data points that are potential outliers will pop out under your knowledgeable gaze. Consequently, I’ll often use boxplots, histograms, and good old-fashioned data sorting! These simple tools provide enough information for me to find unusual data points for further investigation.
Typically, I don’t use Z-scores and hypothesis tests to find outliers because of their various complications. Using outlier tests can be challenging because they usually assume your data follow the normal distribution, and then there’s masking and swamping. Additionally, the existence of outliers makes Z-scores less extreme. It’s ironic, but these methods for identifying outliers are actually sensitive to the presence of outliers! Fortunately, as long as researchers use a simple method to display unusual values, a knowledgeable analyst is likely to know which values need further investigation.
In my view, the more formal statistical tests and calculations are overkill because they can’t definitively identify outliers. Ultimately, analysts must investigate unusual values and use their expertise to determine whether they are legitimate data points. Statistical procedures don’t know the subject matter or the data collection process and can’t make the final determination. You should not include or exclude an observation based entirely on the results of a hypothesis test or statistical measure.
At this stage of the analysis, we’re only identifying potential outliers for further investigation. It’s just the first step in handling them. If we err, we want to err on the side of investigating too many values rather than too few.
In my next post, I’ll explain what you’re looking for when investigating outliers and how that helps you determine whether to remove them from your dataset. Not all outliers are bad and some should not be deleted. In fact, outliers can be very informative about the subject-area and data collection process. It’s important to understand how outliers occur and whether they might happen again as a normal part of the process or study area.
Use one-way ANOVA to determine whether the means of at least three groups are different. Excel refers to this test as Single Factor ANOVA. This post is an excellent introduction to performing and interpreting one-way ANOVA even if Excel isn’t your primary statistical software package. [Read more…] about How to do One-Way ANOVA in Excel
Excel can perform various statistical analyses, including t-tests. It is an excellent option because nearly everyone can access Excel. This post is a great introduction to performing and interpreting t-tests even if Excel isn’t your primary statistical software package.
In this post, I provide step-by-step instructions for using Excel to perform t-tests. Importantly, I also show you how to select the correct form of t-test, choose the right options, and interpret the results. I also include links to additional resources I’ve written, which present clear explanations of relevant t-test concepts that you won’t find in Excel’s documentation. And, I use an example dataset for us to work through and interpret together! [Read more…] about How to do t-Tests in Excel
I’m thrilled to release my new ebook! Introduction to Statistics: An Intuitive Guide for Analyzing Data and Unlocking Discoveries. [Read more…] about New eBook Release! Introduction to Statistics: An Intuitive Guide
Statistical power is the ability of a hypothesis test to detect an effect that exists in the population. Clearly, a high powered study is a good thing just for being able to identify these effects. Low power reduces your chances of discovering real findings. However, many analysts don’t realize that low power also inflates the effect size.
In this post, I show how this unexpected relationship between power and exaggerated effect sizes exists. I’ll also tie it to other issues, such as the bias of effects published in journals and other matters about statistical power. I think this post will be eye-opening and thought provoking! As always, I’ll use many graphs rather than equations. [Read more…] about Low Power Tests Exaggerate Effect Sizes
The Monty Hall Problem is where Monty presents you with three doors, one of which contains a prize. He asks you to pick one door, which remains closed. Monty opens one of the other doors that does not have the prize. This process leaves two unopened doors—your original choice and one other. He allows you to switch from your initial choice to the other unopened door. Do you accept the offer?
If you accept his offer to switch doors, you’re twice as likely to win—66% versus 33%—than if you stay with your original choice.
The solution to the Monty Hall Problem is tricky and counter-intuitive. It did trip up many experts back in the 1980s. However, the correct answer to the Monty Hall Problem is now well established using a variety of methods. It has been proven mathematically, with computer simulations, and empirical experiments, including on television by both the Mythbusters (CONFIRMED!) and James Mays’ Man Lab. You won’t find any statisticians who disagree with the solution.
In this post, I’ll explore aspects of this problem that have arisen in discussions with some stubborn resisters to the notion that you can increase your chances of winning by switching!
The Monty Hall problem provides a fun way to explore issues that relate to hypothesis testing. I’ve got a lot of fun lined up for this post, including the following!
- Using a computer simulation to play the game 10,000 times.
- Assessing sampling distributions to compare the 66% percent hypothesis to another contender.
- Performing a power and sample size analysis to determine the number of times you need to play the Monty Hall game to get an answer.
- Conducting an experiment by playing the game repeatedly myself, record the results, and use a proportions hypothesis test to draw conclusions! [Read more…] about Revisiting the Monty Hall Problem with Hypothesis Testing
Causation indicates that an event affects an outcome. Do fatty diets cause heart problems? If you study for a test, does it cause you to get a higher score?
In statistics, causation is a bit tricky. As you’ve no doubt heard, correlation doesn’t necessarily imply causation. An association or correlation between variables simply indicates that the values vary together. It does not necessarily suggest that changes in one variable cause changes in the other variable. Proving causality can be difficult.
If correlation does not prove causation, what statistical test do you use to assess causality? That’s a trick question because no statistical analysis can make that determination. In this post, learn about why you want to determine causation and how to do that. [Read more…] about Causation versus Correlation in Statistics
Observational studies use samples to draw conclusions about a population when the researchers do not control the treatment, or independent variable, that relates to the primary research question.
In my previous post, I show how random assignment reduces systematic differences between experimental groups at the beginning of the study, which increases your confidence that the treatments caused any differences between groups you observe at the end of the study.
Unfortunately, using random assignment is not always possible. For these cases, you can conduct an observational study. In this post, learn about observational studies, why these studies must account for confounding variables, and how to do so. I’ll close this post by reviewing a published observational study about vitamin supplement usage. [Read more…] about Observational Studies Explained
Random assignment uses chance to assign subjects to the control and treatment groups in an experiment. This process helps ensure that the groups are equivalent at the beginning of the study, which makes it safer to assume the treatments caused any differences between groups that the experimenters observe at the end of the study. [Read more…] about Random Assignment in Experiments
The scientific method is a proven procedure for expanding knowledge through experimentation and analysis. It is a process that uses careful planning, rigorous methodology, and thorough assessment. Statistical analysis plays an essential role in this process.
In an experiment that includes statistical analysis, the analysis is at the end of a long series of events. To obtain valid results, it’s crucial that you carefully plan and conduct a scientific study for all steps up to and including the analysis. In this blog post, I map out five steps for scientific studies that include statistical analyses. [Read more…] about 5 Steps for Conducting Scientific Studies with Statistical Analyses
Percentiles indicate the percentage of scores that fall below a particular value. They tell you where a score stands relative to other scores. For example, a person with an IQ of 120 is at the 91st percentile, which indicates that their IQ is higher than 91 percent of other scores.
Percentiles are a great tool to use when you need to know the relative standing of a value. Where does a value fall within a distribution of values? While the concept behind percentiles is straight forward, there are different mathematical methods for calculating them. In this post, learn about percentiles, special percentiles and their surprisingly flexible uses, and the various procedures for calculating them. [Read more…] about Percentiles: Interpretations and Calculations
I’m thrilled to announce the release of my first ebook! Regression Analysis: An Intuitive Guide for Using and Interpreting Linear Models.
If you like the clear writing style I use on this website, you’ll love this book! The end of the post displays the entire table of contents! You can also download a Free Sample that includes the complete Table of Contents and the first two chapters. Go to My Store to download the ebook sample. [Read more…] about New eBook Release! Regression Analysis: An Intuitive Guide
To determine whether the difference between two means is statistically significant, analysts often compare the confidence intervals for those groups. If those intervals overlap, they conclude that the difference between groups is not statistically significant. If there is no overlap, the difference is significant.
While this visual method of assessing the overlap is easy to perform, regrettably it comes at the cost of reducing your ability to detect differences. Fortunately, there is a simple solution to this problem that allows you to perform a simple visual assessment and yet not diminish the power of your analysis.
In this post, I’ll start by showing you the problem in action and explain why it happens. Then, we’ll proceed to an easy alternative method that avoids this problem. [Read more…] about Using Confidence Intervals to Compare Means
Can high p-values be helpful? What do high p-values mean?
Typically, when you perform a hypothesis test, you want to obtain low p-values that are statistically significant. Low p-values are sexy. They represent exciting findings and can help you get articles published.
However, you might be surprised to learn that higher p-values, the ones that are not statistically significant, are also valuable. In this post, I’ll show you the potential value of a p-value that is greater than 0.05, or whatever significance level you’re using. [Read more…] about Can High P-values Be Meaningful?
Histograms are graphs that display the distribution of your continuous data. They are fantastic exploratory tools because they reveal properties about your sample data in ways that summary statistics cannot. For instance, while the mean and standard deviation can numerically summarize your data, histograms bring your sample data to life.
In this blog post, I’ll show you how histograms reveal the shape of the distribution, its central tendency, and the spread of values in your sample data. You’ll also learn how to identify outliers, how histograms relate to probability distribution functions, and why you might need to use hypothesis tests with them.
[Read more…] about Using Histograms to Understand Your Data
Graphing your data before performing statistical analysis is a crucial step. Graphs bring your data to life in a way that statistical measures do not because they display the relationships and patterns. In this blog post, you’ll learn about using boxplots and individual value plots to compare distributions of continuous measurements between groups. You’ll also learn why you need to pair these plots with hypothesis tests when you want to make inferences about a population. [Read more…] about Boxplots vs. Individual Value Plots: Graphing Continuous Data by Groups
Post hoc tests are an integral part of ANOVA. When you use ANOVA to test the equality of at least three group means, statistically significant results indicate that not all of the group means are equal. However, ANOVA results do not identify which particular differences between pairs of means are significant. Use post hoc tests to explore differences between multiple group means while controlling the experiment-wise error rate.
In this post, I’ll show you what post hoc analyses are, the critical benefits they provide, and help you choose the correct one for your study. Additionally, I’ll show why failure to control the experiment-wise error rate will cause you to have severe doubts about your results. [Read more…] about Using Post Hoc Tests with ANOVA