# Outliers in Statistics: How to Find and Deal with Them in Your Data

One thing many people forget when dealing with data: outliers.

Even in a controlled online A/B test, your data set may be skewed by extremities. How do you deal with them? Do you trim them out, or is there another way?

How do you even detect the presence of outliers—and how extreme they are?

If you’re optimizing your site for revenue, you should care about outliers. This post dives into the nature of outliers, how to detect them, and popular methods for dealing with them.

## What is an outlier in statistics?

An outlier in statistics is an observation that lies an abnormal distance from other values in a random sample from a population.

There is, of course, a degree of ambiguity. Qualifying a data point as an anomaly leaves it up to the analyst or model to determine what is abnormal—and what to do with such data points.

• Mild outliers lie beyond an “inner fence” on either side.
• Extreme outliers are beyond an “outer fence.”

Why do outliers occur and how to find them in a dataset? According to Tom Bodenberg, chief economist and data consultant at Unity Marketing, “It can be the result of measurement or recording errors, or the unintended and truthful outcome resulting from the set’s definition.”

Outliers in data may contain valuable information. Or be meaningless aberrations caused by measurement and recording errors. In any case, they can cause problems with repeatable A/B test results, so it’s important to question and analyze outliers.

Why are they occurring? Where—and what—might the meaning be? The answer could differ from business to business, but it’s important to have the conversation rather than ignore the data, regardless of the significance.

The real question, though, is, “How do outliers affect your testing efforts?”

## How data sets with outliers affect A/B testing

Though outlier statistics show up in many analyses, for conversion optimization you should mostly be concerned about tests in which you’re optimizing for revenue metrics, like Average Order Value or Revenue Per Visitor.

Taylor Wilson, Senior Optimization Analyst at Brooks Bell, explains a few scenarios in which that could happen:

In this particular situation, resellers were the culprit—customers who buy in bulk with the intention of reselling items later. Far from your typical customer, they place unusually large orders, paying little attention to the experience they’re in.

It’s not just resellers who won’t be truly affected by your tests. Depending on your industry, it could be very loyal customers, in-store employees who order off the site, or another group that exhibits out-of-the-ordinary behavior.

Taylor Wilson

Especially in data sets with low sample sizes, outliers can mess up your whole day.

As Dr. Julia Engelmann, Head of Data Analytics at konversionsKRAFT, mentioned in a CXL blog post, “Almost every online shop has them, and usually they cause problems for the valid evaluation of a test: the bulk orderers.”

This isn’t a fringe problem. She shared a specific example of how including and excluding outliers can affect the results of a test, and, ultimately, the decision you make:

A problem outlier statistics can cause: They tend to be unaffected by smaller UI changes that do affect a more fickle mainstream population. Bulk orderers will push through smaller usability changes in a way that your average visitor may not.

This article outlines a case in which outliers skewed the results of a test. Upon further analysis, the outlier segment was 75% return visitors and much more engaged than the average visitor.

Think your data is immune to outliers? Maybe it is, but probably not—and, in any case, it’s best to know for sure. So how do you diagnosis a potential issue on your own? How do you detect outliers in your data?

## How to detect outlier statistics in data

Data visualization is a core discipline for analysts and optimizers, not just to better communicate results with executives, but to explore the data fully.

As such, outliers are often detected through graphical means, though you can also do so by a variety of statistical methods using your favorite tool. (Excel and R will be referenced heavily here, though SAS, Python, etc., all work).

### Boxplots and scatterplots

Two of the most common graphical ways of detecting data sets with outliers are the boxplot and the scatterplot. A boxplot is my favorite way.

You can see here that the blue circles are outliers, with the open circles representing mild outliers, and closed circles representing extreme outliers:

It’s really easy to analyze boxplots in R. Just use boxplot(x, horizontal = TRUE), where x is your data set. It generates something that looks like this:

Even better, you can use boxplot.stats(x) function, where x is your data set, to get summary stats that includes the list of outliers (\$out):

``````> boxplot.stats(rivers)
\$stats
[1] 135 310 425 680 1205

\$n
[1] 141

\$conf
[1] 375.7678 474.2322

\$out
[1]  1459 1450 1243 2348 3710 2315 2533 1306 1270 1885 1770``````

You can also see these in a scatter plot, though it’s a bit harder to tell where extreme and mild outliers are:

### Other graphical outlier detection options

A histogram can work as well:

You can also see outliers fairly easily in run charts, lag plots (a type of scatter plot), and line charts, depending on the type of data you’re working with.

Conversion expert Andrew Anderson also backs the value of graphs to determine the effect of outliers on data:

The graph is your friend. One of the reasons that I look for 7 days of consistent data is that it allows for normalization against non-normal actions, be it size or external influence.

The other thing is that if there are obvious non-normal action values, it is okay to normalize them to the average as long as it is done unilaterally and is done to not bias results.

This is only done if it is obviously out of normal line, and usually I will still run the test another 2–3 extra days just to make sure.

Andrew Anderson

(As to the latter point on non-normal distributions, we’ll go into that a bit later.)

But is there a statistical way of detecting outliers, apart from just eyeballing it on a chart? Indeed, there are many ways to do so (Wikipedia has a comprehensive article on outlier detection); the main two being a standard deviation approach or Tukey’s method.

In the latter, extreme outliers tend to lie more than three times the interquartile range (below the first quartile or above the third quartile), and mild outliers lie between 1.5 and three times the interquartile range (below the first quartile or above the third quartile).

It’s pretty easy to highlight outliers in Excel. While there’s no built-in function for outlier detection, you can find the quartile values and go from there.

## 5 ways to deal with outliers in data

Should an outlier be removed from analysis? The answer, though seemingly straightforward, isn’t so simple.

There are many strategies for dealing with outliers in data. Depending on the situation and data set, any could be the right or the wrong way. In addition, most major testing tools have strategies for dealing with outliers, but they usually differ in how they do so.

Because of that, it’s still important to do a custom analysis with regard to data sets with outliers, even if your testing tool has default parameters. Not only can you trust your testing data more, but sometimes analysis of outliers produces its own insights that help with optimization.

So let’s go over some common strategies:

### 1. Set up a filter in your testing tool

Even though this has a little cost, filtering out outliers is worth it. You often discover significant effects that are simply “hidden” by outliers.

According to Himanshu Sharma at OptimizeSmart, if you’re tracking revenue as a goal in your A/B testing tool, you should set up a code that filters out abnormally large orders from test results.

He says that you should look at past analytics data to secure an average web order, and to set up filters with that in mind. In his example, imagine that your website average order value in the last three months has been \$150. If so, any order above \$200 can be considered an outlier.

For there, it’s all about writing a bit of code to stop the tool from passing that value. The tl;dr is that you exclude values above a certain amount with code that looks something like this (for orders above \$200):

``````if(priceInCents &lt;20000){
window.optimizely = window.optimizely || [];
window.optimizely.push([‘trackEvent’,
‘orderComplete’, {‘revenue’: priceInCents}]);``````

### 2. Remove or change outliers during post-test analysis

Kevin Hillstrom, President of MineThatData, explains why he sometimes adjust outliers in tests:

On average, what a customer spends is not normally distributed.

If you have an average order value of \$100, most of your customers are spending \$70, \$80, \$90, or \$100, and you have a small number of customers spending \$200, \$300, \$800, \$1600, and one customer spending \$29,000. If you have 29,000 people in the test panel, and one person spends \$29,000, that’s \$1 per person in the test.

That’s how much that one order skews things.

Kevin Hillstrom

One way to account for this is simply to remove outliers, or trim your data set to exclude as many as you’d like.

#### Excel’s TRIMMEAN function

Trimming outliers is really easy to do in Excel—a simple `TRIMMEAN `function will do the trick. The first argument is the array you’d like to manipulate (Column A), and the second argument is by how much you’d like to trim the upper and lower extremities:

#### Trim outliers in R

Trimming values in R is super easy, too. It exists within the mean(function). So, say you have a mean that differs quite a bit from the median, it probably means you have some very large or small values skewing it.

In that case, you can trim off a certain percentage of the data on both the large and small side. In R, it’s just mean(x, trim = .05), where x is your data set and .05 can be any number of your choosing:

This process of using Trimmed Estimators is usually done to obtain a more robust statistic. The median is the most trimmed statistic, at 50% on both sides, which you can also do with the mean function in R—mean(x, trim = .5).

In optimization, most outliers are on the higher end because of bulk orderers. Given your knowledge of historical data, if you’d like to do a post-hoc trimming of values above a certain parameter, that’s easy to do in R.

If the name of my data set is “rivers,” I can do this given the knowledge that my data usually falls under 1210: rivers.low <- rivers[rivers<1210].

That creates a new variable consisting only of what I deem to be non-outlier values. From there, I can boxplot it, getting something like this:

There are fewer outlier values, though there are still a few. This is almost inevitable—no matter how many values you trim from the extremes.

#### Remove outliers beyond 3 standard deviations

You can also do this by removing values that are beyond three standard deviations from the mean. To do that, first extract the raw data from your testing tool. Optimizely reserves this ability for their enterprise customers (unless you ask support to help you).

Instead of taking real client data to demonstrate how to do this, I generated two random sequences of numbers with normal distributions, using `=NORMINV(RAND(),C1,D1),` where `C1` is mean and `D1` is SD, for reference.

In “variation 1,” though, I added a few very high outliers, making variation 1 a “statistically significant” winner:

Then, you can use conditional formatting to highlight those that are above three standard deviations and chop them off:

After you do, you have a different statistically significant winner:

My example is probably simpler than what you’ll deal with, but at least you can see how just a few high values can throw things off (and one possible solution to do with that). If you want to play around with outliers using this fake data, download the example spreadsheet with outliers in its data set.

### 3. Change the value of outliers

Much of the debate on how to deal with outliers in data comes down to the following question: Should you keep outliers, remove them, or change them to another variable?

Essentially, instead of removing outliers from the data, you change their values to something more representative of your data set. It’s a small but important distinction: When you trim data, the extreme values are discarded.

When you use winsorized estimators (i.e. change the values), extreme values are replaced by percentiles—the trimmed minimum and maximum.

Kevin Hillstrom mentioned in his podcast that he trims the top 1% or 5% of orders, depending on the business, and changes the value (e.g., \$29,000 to \$800). As he says, “You are allowed to adjust outliers.”

### 4. Consider the underlying distribution

Traditional methods to calculate confidence intervals assume that the data follows a normal distribution, but as with certain metrics like average revenue per visitor, that usually isn’t the way reality works.

In another section of Dr. Julia Engelmann’s wonderful article for our blog, she shared a graphic depicting this difference. The left graphic shows a perfect (theoretical) normal distribution. The number of orders fluctuates around a positive average value. In the example, most customers order five times. More or fewer orders arise less often.

The graphic to the right shows the bitter reality. Assuming an average conversion rate of 5%, some 95% of visitors don’t buy. Most buyers have probably placed one or two orders, and there are a few customers who order an extreme quantity.

The distribution on the right side is known as a right-skewed distribution.

Essentially, the problem comes in when we assume that a distribution is normal. In reality, we’re working with something like a right-skewed distribution. Confidence intervals can no longer be reliably calculated.

With your average ecommerce site, at least 90% of customers will not buy anything. Therefore, the proportion of “zeros” in the data is extreme, and deviations in general are enormous, including extremities because of bulk orders.

#### Use a different distribution test

In this case, it’s worth taking a look at the data using methods other than the t-test. (The Shapiro-Wilk test lets you test your data for normal distribution, by the way.) All of these were suggested in this article:

• Mann-Whitney U-Test. The Mann-Whitney U-Test is an alternative to the t-test when the data deviates greatly from the normal distribution.
• Robust statistics. Methods from robust statistics are used when the data is not normally distributed or distorted by outliers. Here, average values and variances are calculated such that they are not influenced by unusually high or low values—which I touched on with winsorization.
• Bootstrapping. This so-called non-parametric procedure works independently of any distribution assumption and provides reliable estimates for confidence levels and intervals.

At its core, it belongs to the resampling methods, which provide reliable estimates of the distribution of variables on the basis of the observed data through random sampling procedures.

### 5. Consider the value of mild outliers

As exemplified by revenue per visitor, the underlying distribution is often non-normal. It’s common for a few big buyers to skew the data set toward the extremes. When this is the case, outlier detection falls prey to predictable inaccuracies—it detects outliers far more often.

There’s a chance that, in your data analysis, you shouldn’t throw away outliers. Rather, you should segment them and analyze them more deeply. Which demographic, behavioral, or firmographic traits correlate with their purchasing behavior? And how can you run an experiment to tease out some causality there? You need to ask: what is an outlier for these statistics that we’re about to collect?

This is a question that runs deeper than simple A/B testing and is core to your customer acquisition, targeting, and segmentation efforts. I don’t want to go too deep here, but for various marketing reasons, analyzing your highest value cohorts can bring profound insights.

## No matter what, do something

In any case, it helps to have a plan in place. As Dan Begley-Groth once wrote:

In order for a test to be statistically valid, all rules of the testing game should be determined before the test begins. Otherwise, we potentially expose ourselves to a whirlpool of subjectivity mid-test.

Should a \$500 order only count if it was directly driven by attributable recommendations? Should all \$500+ orders count if there are an equal number on both sides? What if a side is still losing after including its \$500+ orders? Can they be included then?

By defining outlier thresholds prior to the test (for RichRelevance tests, three standard deviations from the mean) and establishing a methodology that removes them, both the random noise and subjectivity of A/B test interpretation is significantly reduced. This is key to minimizing headaches while managing A/B tests

Dan Begley-Groth

Whether you believe that outliers don’t have a strong effect (and choose to leave them as is) or whether you want to trim the top and bottom 25% of your data, the important thing is that you’ve thought it through and have a strategy.

Being data-driven means considering anomalies like this. To ignore them risks making decisions on faulty data.

## Conclusion

Outliers aren’t discussed often in testing, but, depending on your business and the metric you’re optimizing, they could affect your results.

One or two high values in a small sample size can totally skew a test, leading you to make a decision based on faulty data. No bueno.

For the most part, if your data is affected by these extreme cases, you can bound the input to a historical representative of your data that excludes outliers. That could be a number of items (>3) or a lower or upper bound on your order value.

Another way, perhaps better in the long run, is to export your post-test data and visualize it by various means. Determine the effect of data sets with outliers on a case-by-case basis. Then decide whether you want to remove, change, or keep outlier values. In this scenario, you’re questioning: what are the outliers in our statistics? Are they valuable or a pest?

Really, though, there are lots of ways to deal with outliers in data. There’s no quick fix that works across the board, which is why demand for good analysts continues to grow.

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