It’s a new semester! That means that I’ll be updating my websites a little more often. The first update is to the Statistics Help section of MattCHoward.com, in which I go over the basic rules of probability. It can be found here: Rules of Probability. If you have any questions or content that you’d like to see, please email me at MHoward@SouthAlabama.edu.
Tag: Methodology
Correlation Does NOT Equal Causation
Your variables may be related, but does one really cause the other?
Most readers have probably heard the phrase, “correlation does not equal causation.” Recently, however, I heard someone confess that they’ve always pretended to know the significance of this phrase, but they truly didn’t know what it meant. So, I thought that it’d be a good idea to make a post on the meaning behind “correlation ≠ causation.”
Imagine that you are the president of your own company. You notice one day that your highlypayed employees perform much better than your lowerpayed employees. To test whether this is true, you create a database that includes employee salaries and their performance ratings. What do you find? There is a strong correlation between employee pay and their performance ratings. Success! Based on this information, you decide to improve your employees’ performance by increasing their pay. You’re certain that this will improve their performance. . .right?
Not so fast. While there is a correlation between pay and performance, there may not be a causal relationship between the two – or, at least, such that pay directly influences performance. It is fully possible that increasing pay has little effect on performance. But why is there a correlation? Well, it is also possible that employees get raises due to their prior performance, as the organization has to provide benefits in order to keep good employees. Because of this, an employee’s high performance may not be due to their salary, but rather their salary is due to their prior high performance. This results in current performance and pay having a strong correlational relationship, but not a causal relationship such that pay predicts performance. In other words, current performance and pay may be correlated because they have a common antecedent (past performance).
This is the idea behind the phrase, “correlation does not equal causation.” Variables do not necessarily have a causal relationship just because they are correlated. Instead, many other types of underlying relationships could exist, such as both having a common antecedent.
Still don’t quite get it? Let’s use a different example. Prior research has shown that icecream sales and murder rates are strongly correlated, but does that mean that ice cream causes people to murder each other? Hopefully not. Instead, it is that warm weather (i.e. the summer) causes people to (a) buy ice cream (b) and be more aggressive. This results in both icecream sales and murder rates. Once again, these two variables are correlated because they have a common antecedent – not because there is a causal relationship between the two.
Hopefully you now understand why correlation does not equal causation. If you don’t, please check out one of my favorite websites: Spurious Correlations. This website is a collection of very significant correlations that almost assuredly do not have a causal relationship – thereby providing repeated examples of why correlation does not equal causation. If you do understand, beware of this fallacy in the future! Organizations can make disastrous decisions based on falsely assuming causality. Make sure that you are not one of these organizations!
Until next time, watch out for Statistical Bullshit! And email me at MHoward@SouthAlabama.edu with any questions, comments, or stories. I’d love to include your content on the website!
Bullshit Measurement
Are you measuring what you think you’re measuring? Could you be measuring something else entirely?
Accurate measurement of variables is essential for business success. Sometimes, it’s fairly easy to record these variables – sales, revenue, profit. Other times, it can be very very difficult. For example, let’s say that you want to hire employees that are smart and conscientious. How can we measure intelligence and conscientiousness?
Well, a good starting point is to develop a test or survey. Many intelligence tests exist with varying levels of sophistication and accuracy, and you could pay to give these tests to applicants. Many selfreport surveys also exist that can measure conscientiousness, and you could pay to give these tests to applicants, too. But what if you don’t want to use one of these existing measures? What’s the worst that could happen?
In this post, we won’t talk about the worst that could happen, but we’ll discuss a pretty bad outcome: when your measure inadvertently gauges the wrong construct, which could result in a lawsuit.
I should also note that this example comes from an actual consulting experience that I encountered. The names have been changed, but remember that these things actually happen in industry!
I was once hired along with a full team to review the new selection system of a trendy company. Let’s call them XCorp. XCorp wanted their selection to measure a construct that they invented: “the ideal XCorp employee.” They made a list of the ideal XCorp employee characteristics. It included the common constructs like intelligence and conscientiousness, but it also included some unorthodox constructs. These included hip, stylish, savvy, sleek and so fourth. XCorp argued that the ideal employee needed to appeal to any potential customers, and therefore needed to have these characteristics; however, my team was already doubtful about the business relevance of theses constructs.
Even more concerning, XCorp felt that their survey had to attract people to work for XCorp. For this reason, it couldn’t be a traditional survey. It had to be different and exciting. Once again, we were doubtful about how exciting a selection survey could be.
When we saw the survey to measure “the ideal XCorp employee,” we began to worry even more. The first question looked something like this:
What?
The text of the item read, “Using the scale, please indicate whether you are more like a sports car or a hybrid/electric car.”
…What?
Immediately, we asked XCorp what this item was meant to measure. Sure enough, they just said “the ideal XCorp employee.” We asked which subdimension, specifically, was the item meant to measure. As they couldn’t respond, we realized that they didn’t really have an idea. It seemed that they just put things in their survey that they thought would be a good idea without really thinking about the ramifications.
Do you think this item would help identify good employees? Well, we first have to ask what is the “correct” answer. According to XCorp, the correct answer was being more like a hybrid/electric car. So, anyone would indicated that they were more like a sports car got the item wrong. Do you think this is fair? More importantly, do you think those that feel more like a “hybrid/electric car” are necessarily better than those that feel more like a “sports car?” I would guess that the answer is probably not. There are probably many sports car people that are more intelligent, conscientious, hip, savvy, and so on when compared to hybrid/electric car people. Thus, this item probably fails to measure “the ideal XCorp employee.”
That item was bad, but it wasn’t the worst. The worst was probably the following item:
Once again, what?
The text of the item read, “Using the scale, please indicate whether you are more or less like Kanye West.”
Once again…what?
XCorp claimed that Kanye West was too narcissistic, and anyone who felt that they were like Kanye were not welcome at XCorp. Do you think that Kanye people are inherently worse than nonKanye people? Once again, I am guessing that the answer is probably not. Kanye people are probably just as good as nonKanye people, and perhaps even better in some regards (i.e. creative, hip, etc.). But can you think of anything else that this item might inadvertently measure? Let’s look at the graph below, which is similar to the actual results.
As some of you may have guessed, African Americans were much more likely to see themselves similar to Kanye than Caucasians. This makes sense, as Kanye himself is African American. Thus, this item partially measures the applicant’s ethnicity.
Remember when I said that those responding that they were more like Kanye were rated as worse applicants? If this survey went live, that would mean that African Americans would automatically be penalized, thereby resulting in adverse impact. This would almost assuredly result in a lawsuit, in which XCorp could not justifiably defend – or, at least, have a very hard time defending that the Kanye question actually represented job performance. This would have cost the company millions of dollars!
In the end, my team strongly recommended that the company should not use their selection survey, and should instead use a traditional survey. The company wasn’t happy, and we were never asked to work with the company again. But, they did guarantee that they would not use their selection system. While it wasn’t the most satisfying result, I was happy that we were able to stop another case of Statistical Bullshit!
If you have any questions or comments about this story, feel free to contact me at MHoward@SouthAlabama.edu . Also, feel free to contact me if you have any Statistical Bullshit stories of your own. I’d love to include them on StatisticalBullshit.com!
What is in a Mean? A Reader Story
Does your company make largestake decisions based on means alone? A reader tells the story.
I recently had a reader of StatisticalBullshit.com tell me a story regarding the post, “What is in a Mean?” This story is a perfect illustration of Statistical Bullshit in industry, and why you should be aware of these and similar issues. I have done my best to retell it below (with a few details changed to ensure anonymity). As always, feel free to email me at MHoward@SouthAlabama.edu if you have any questions, comments, or stories. I would love to include your email on StatisticalBullshit.com. Until next time, watch out for Statistical Bullshit!
I was hired as a consultant for a company that recently had recently become obsessed with performance management. The top management of the company was recently under the impression that their workteams were terribly inefficient, and somehow they decided that the teams’ leadership was to blame. The company had given survey after survey, analyzed the data, interpreted the data, implemented new changes, and continuously monitored performance; however, the workteams were still not performing at the standard that they had hoped.
So, I was brought in to help fix the problem. My first decision was to review the surveys that the organization was using to measure performance and related factors. The surveys were very simple, but they weren’t terrible. First, performance was measured by having a member of top management rate the outcome of the workteam. Next, the leader of the workteam was rated by team members on 11 different attributes. These included:
 Managed Time Effectively
 Communicated with Team Members
 Foresaw Problems
 Displayed Proper Leadership Characteristics
 Transformed Team Members into Better People
Overall, I thought it wasn’t bad, and my second decision was to ask about prior analyses. When they delivered the prior analyses, I was confused that they only provided mean calculations. I immediately went to the top management and asked for the rest. They exasperatedly proclaimed, “Why do you need anything else!? The means are right there!”
I was taken aback. What!? They only calculated the means? I asked, “What do you mean by that?”
They sent me a table very similar to the following:
Mean Rating (From 1 to 7 Scale) 

Managed Time Effectively 
6.3 
Communicated with Team Members 
5.9 
Foresaw Problems 
5.5 
Displayed Proper Leadership Characteristics 
6.1 
Transformed Team Members into Better People 
2.5 
“See! Our leaders are struggling with transforming team members into better people! This is obviously the problem, which is why we’ve made every leader enroll in mandatory transformation leadership courses.”
I immediately knew that this wasn’t right, but I needed a little time (and analyses) to make my case. I first calculated correlations of the related factors with team performance, and they looked like this:
Correlation with Team Performance 

Managed Time Effectively 
.24** 
Communicated with Team Members 
.32** 
Foresaw Problems 
.52** 
Displayed Proper Leadership Characteristics 
.17* 
Transformed Team Members into Better People 
.02 
* p < .05, ** p < .01
Aha! This could be the issue! While leaders could improve on transforming team members into better people, the data suggested that this factor did not have a significant effect on team performance. So, I then calculated a regression including all the related factors predicting team performance:
β 

Managed Time Effectively 
.170* 
Communicated with Team Members 
.082 
Foresaw Problems 
.389** 
Displayed Proper Leadership Characteristics 
.113 
Transformed Team Members into Better People 
.010 
* p < .05, ** p < .01
Again, the data suggested that transforming team members into better people did not have an effect on team performance. Instead, the strongest predictor was foreseeing problems. I lastly created a scatterplot of the relationship between foreseeing problems and team performance:
There is the problem! There were two groups of team leaders – those that could foresee problems and those that could not. Those that foresaw problems led teams with high performance, whereas those that could not foresee problems led teams with low performance. So, although the mean of foreseeing problems was not all that different from the other factors, it turned out to have the largest effect of them all. On the other hand, while transforming team members into better people had a mean that was much lower than the other factors, it did not have a significant effect at all.
With this information, I suggested that the organization should cut back on the transformational leadership training programs (after ensuring that they did not provide other benefits), and instead train leaders on how to anticipate problems. Through doing so, they could (a) save money (b) and finally reach the level of team performance that they had been wanting. I am unsure whether they implemented my recommendations, but I hope they learned a valuable lesson from my analyses:
Means should not be used to infer relationships between variables, and to always watch out for Statistical Bullshit – even if you accidentally do it yourself!
Note: The variables in this story have been changed to protect the identity of the reader. Please do not make management decisions based on these analyses.
Small Samples, Big Problems
Have you ever discussed statistical power or representative samples at work? Should you?
Often in business, we are restricted to relatively small samples. In fact, a recent publication in the Journal of Organizational Behavior suggest that the most common type of business is a microbusiness – often defined as a business with less than 10 employees (Brawley & Pury, 2017). As many readers already know, most all statistics require many more participants. For instance, the most common recommendation for a correlation analysis is a minimum of 30 participants, and more advanced statistics most often require even more participants – often in the 100s.
But what is really the harm in having a small sample size? Can the results really be that misleading? The answer is yes.
This post discusses two concerns of small samples: power and representativeness.
Power is the likelihood of a statistical analysis to discover a significant result if a significant result actually exists in the population…But what does that mean? Well, I’ll discuss this much more indepth in a later post, but sample size is an important component to calculating statistical significance. Even if an effect is extremely strong in the population, a statistical test using a small sample size will not identify that effect as statistically significant. Weird, right?
Let’s use this example: Imagine that we are studying pretty strong effect that has a population correlation of .40, such as the relationship between selfefficacy and job performance. To study this relationship, let’s say that we use a microbusiness – one with eight employees – and we measure selfefficacy and job performance with each employee. What is the likelihood that the resultant correlation between the two variables will be statistically significant, if we know the population correlation of the variables is .40? Well, the likelihood that the result will be statistically significant is only 15%! We would fail to reject the null more than every four out of every five times!
Crazy! This example demonstrates one important reason to have a large sample size – you cannot identify significant results even if they should be significant. To learn more about this phenomenon, I suggest reading more about statistical power (Cohen, 1992a, 1992b; Murphy et al., 2014) and playing with a sample size/power calculator (http://www.samplesize.net/correlationsamplesize/).
Next, let’s discuss having a representative sample. Even if we have more employees, let’s say 150, there is a chance that our sample is not representative of the population. If a sample is representative, it accurately reflects the members of the population. Often, we assume that a randomly selected sample is representative, but this is not always the case. Certain people may not volunteer to take your survey, and that may skew your results…But how bad can it be?
Well, let’s look at the selfefficacy and job performance example again with a correlation of .40. If we had a representative sample of 300 people, the result might look something like this:
Not too bad – the regression line shows a clear, increasing relationship. Now, let’s take 150 of these people and graph the results again:
Woah! Big difference! Now the correlation between the two is literally .00, and we only removed half of the participants. What happened?
As you guessed, I did not take a random subset of the 300 people. Instead, I selected only those that scored five or above on the selfefficacy measure, as you can see with the differing axis labels in the two charts. This resulted in the sample being nonrepresentative (because everyone with a selfefficacy score under five was missing), and thereby the result was greatly different than the entire set of 300 people.
But could this ever happen in business? Yes! Imagine that you are feeling down about your work performance and unable to do the most basic tasks. Then, you see an email about a job survey to measure selfefficacy and performance. Would you take it? Maybe, but a lot of people would just delete the email in order to avoid facing their lackluster selfperceptions, abilities, and performance.
Also, who would typically take those surveys anyways? The grumpy employees that just want to do their work and go home? Or the goodiegoodies that do whatever their boss asks? I’d guess the latter, and the samples may not be representative of all these employees.
And think about those satisfaction surveys at restaurants. Yes, people that really hated the service or really loved the service will complete them…but what about all the people in the middle? Have you ever completed a satisfaction survey when the service was just okay? I’m guessing not, which resulted in the results being nonrepresentative.
So, whenever you need to collect data, be sure to carefully consider your sample size – not only for statistical power, but also for representativeness. If you ignore these two aspects, then you could obtain results that are entirely misleading, and thereby implement policies that do nothing for your company – or worse!
Until next time, watch out for Statistical Bullshit! And email me at MHoward@SouthAlabama.edu if you have any questions, comments, or anything else!
References
Brawley, A. M., & Pury, C. L. (2017). Little things that count: A call for organizational research on microbusinesses. Journal of Organizational Behavior, 38, 917920.
Cohen, J. (1992a). Statistical power analysis. Current directions in psychological science, 1(3), 98101.
Cohen, J. (1992b). A power primer. Psychological bulletin, 112(1), 155.
Murphy, K. R., Myors, B., & Wolach, A. (2014). Statistical power analysis: A simple and general model for traditional and modern hypothesis tests. Routledge.
What is in a Mean?
When are mean comparisons appropriate? And when are they Statistical Bullshit?
This post is inspired by an interaction that I had while consulting. I was hired as a statistical analyst, and my duties included reviewing analyses that were already conducted internally. Most of the organization’s prior analyses were appropriate, but I noticed that certain assumptions were based on completely inappropriate mean comparisons. These assumptions led to needless practices that cost time and money – all because of Statistical Bullshit. Today, I want to teach you how to avoid these issues.
Let’s first discuss when mean comparisons are appropriate. Mean comparisons are appropriate if you (A) want to obtain a general understanding of a certain variable or (B) want to compare multiple groups on a certain outcome. In the case of A, you may be interested in determining the average amount of time that a certain product takes to make. From knowing this, you could then determine whether an employee is taking more or less time than the average to make the product. In the case of B, you may be interested in determining whether a certain group performed better than another group, such as those that went through a new training program vs. those that went through the old training program. The data from such a comparison may look something like this:
So, from this comparison, you may be able to suggest that the new training program is more effective than the old training program; however, you would need to run a ttest in be sure of this.
Beyond these two situations, there are several other scenarios in which mean comparisons are appropriate, but let’s instead discuss an example when mean comparisons are inappropriate.
Say that we wanted to determine the relationship between two variables. Let’s use satisfaction with pay (measured on a 1 to 7 Likert scale) and turnover intentions (also measured on a 1 to 7 Likert scale). As you probably already know, we could (and probably should) determine the relationship between these two variables by calculating a correlation. Imagine instead that you decided to calculate the mean of the two variables and the results looked like this:
Does this result indicate that there is a significant relationship between the two variables? In my prior consulting experience, the internal employee who ran a similar analysis believed this to be true. That is, the internal employee believed that two variables with similar means are significantly related; however, this couldn’t be further from the truth. Let’s look at the following examples to find out why.
Take the example that we just used – satisfaction with pay and turnover intentions. Which of the following scatterplots do you believe represents the data in the bar chart above?
Still don’t know? Here is a hint: The first chart represents a correlation of 1, the second represents a correlation of 1, the third represents a correlation of 0, and the fourth represents a correlation of 0. Any guesses?
Well, it was actually a trick question. Each figure could represent the data in the bar chart above, because the X and Y variables in each have a mean of 4.75…well, the last one is off by a few tenths, but you get my point.
So, if the means of two variables are equal, their relationship could still be anything – ranging from a large negative relationship, to a null relationship, to a large positive relationship. In other words, the means of two variables have nothing to do regarding their relationship.
But does it work the other way? That is, if the means of two variables are extremely different, could they still have a significant relationship?
Certainly! Let’s look at the following example using satisfaction with pay (still measured on a 1 to 7 Likert scale) and actual pay (measured in thousands of dollars).
As you can see, the difference in the means is so extreme that you can’t even see one bar! Now, let’s look at the following four scatterplots:
Seem a little familiar? As you guessed, the first represents a correlation of 1, the second represents a correlation of 1, the third represents a correlation of 0, and the fourth represents a correlation of 0. More importantly, each of these include a Y variable with a mean of 4.75 and an X variable with a mean of 47500. Although the means are extremely far apart, they have no influence on the relationship between two variables.
From these examples, it should be obvious that the mean of two variables has no influence on their relationship – no matter if the means are close together or far apart. Instead, it is the covariation between the pairings of the X and Y values that determine the significance of their relationship, which may be a future topic on StatisticalBullshit.com or even MattCHoward.com (especially if I get enough requests for it).
Now that you’ve read this post, what will you say if you are ever at work and someone tries to tell you that two variables are related because they have similar means? You should say STATISTICAL BULLSHIT! Then demand that they calculate a correlation instead…or a regression…or a structural equation model…or other things that we may cover one day.
That’s all for this post! Don’t forget to email any questions, comments, or stories. My email is MHoward@SouthAlabama.edu, and I try to reply ASAP. Until next time, watch out for Statistical Bullshit!
Regression Toward the Mean
Can you make a career on Statistical Bullshit?
Regression Toward the Mean is one of the most common types of Statistical Bullshit in industry. And, as the title quotation insinuates, some consultants have made an entire career swindling money from organizations through manipulating this statistical phenomenon. If you are currently in practice, or ever plan to be, read on to discover whether you are currently being swindled out of thousands – or possibly millions!
Businesses are always on a timeseries. That is, most organizations are not worried about the profit that they turned today, but rather the profit that they will turn tomorrow. For this reason, many types of statistics and methodologies applied in business are meant to analyze longitudinal trends in order to predict future results.
Let’s take perhaps the simplest timeseries design: a single variable measured on multiple occasions. In this example, let’s say that we are looking at overall company revenue in millions.
Month 
Revenue 
January 
10.5 
February 
10 
March 
11 
April 
12.5 
May 
10.5 
June 
10 
July 
12 
August 
11 
September 
11 
October 
5 
It seems that the average company revenue over the month was $11 million, but a severe drop occurred in October. What would you do if your company revenue looked something like this?
Most anyone would say panic and take extreme measures – and that is what most companies do. A company may replace the CEO, layoff a large number of workers, or immediately implement a new corporate strategy. Let’s say that a company does all three for our example, and the result looks like this:
Success! The new CEO is a genius! The layoffs worked! And the new corporate strategy is brilliant! Right? Well, maybe not.
The Regression Toward the Mean phenonmemon suggests that a timeseries dataset will revert back to its average after an extreme value. In other words, when an extreme high or lowvalue occurs, it is much more difficult to get any more extreme than it is to revert back to the average. So, in this instance, it is fully possible that the company’s actions successfully caused revenue to revert back to more normal values; however, it is perhaps just as likely that the revenue simply regressed back toward the mean naturally. So, the new changes (and money spent!) may have actually done very little or even nothing at all…but you can always be sure that the new CEO will take credit for it.
Let’s discuss another common example of Regression Toward the Mean in business. Imagine you are a floor manager at a factory, and your monthly number of dangerous incidents looks something like this.
Week 
Incidents 
January 
5 
February 
4 
March 
8 
April 
6 
May 
6 
June 
4 
July 
2 
August 
8 
September 
5 
October 
20 
Wow! Quite the spike in incidents! So, what do you do? Of course, you’d request for your CEO to bring in a safety expert to reduce the number of dangerous incidents, and I can guarantee that the results will look something like this:
Another success! The safety expert saved lives! You are brilliant! As you guessed, however, this may not be the case.
Once again, a Regression Toward the Mean effect may have occurred, and the number of safety incidents naturally reverted back to an average level. The money spent on the safety expert could have been used for other more fruitful purposes, but you can nevertheless take credit for saving your coworker’s lives.
Despite these two examples (and many many more that could be provided), not all instances of extreme values can be cured by waiting for the values to revert to more typical figures. Sometimes, an effect is actually occurring, and an intervention is truly needed to fix a problem. Without it, things could possibly get even worse.
So, what should you do when extreme values occur? Perform an intervention? Wait it out? In academia, the answer is simple. Most researchers have the luxury of collecting data from a control group that does not receive the intervention, and then comparing the data after a sufficient amount of time has passed. If the intervention group resulted in better outcomes than the control group, then the intervention was indeed a success. If the two groups have roughly equal outcomes, then the intervention had no effect.
Businesses do not have such luxuries. Decisions need to be made quickly and correctly – or else someone could lose their job (or their life!). For this reason, it is often common practice to go ahead and perform the intervention. If the values return to normal, then you seem like a genius. If they do not, then at least you tried. On the other hand, if you do nothing and the values return to normal, then you seem like a genius again. If they do not return to normal, however, then it seems like you ignored the severity of the issues. The table below summarizes this issue:
Values Remain Extreme 
Values Return to Normal 

Do Nothing 
You Ignored the Issue 
You Succeeded! 
Do Something 
You Tried 
You Succeeded! 
Long story short, you should probably make an attempt to fix the issue, although it may simply be Statistical Bullshit in the end.
Before concluding, one last question should be answered about Regression Toward the mean: How exactly can people make a career on it?
Well, imagine that you are a safety consultant, and you receive several consulting offers at once. You look at the companies, and they all seem to have a relatively stable number of incidents; however, you notice one that is going through a period of elevated incidents. Now that you know about Regression Toward the Mean, you know that you should take this company’s offer. Not only will they (probably) be willing to spend lots of money, but you (probably) need to do very little to reduce the incident rate. Even if your safety suggestions are bogus, you can still appear to be a competent safety consultant. Although it may sound crazy, I think you would be surprised how often this occurs in the realworld.
That is all for Regression Toward the Mean. Do you have your own Regression Toward the Mean story? Maybe a question? Feel free to email me at MHoward@SouthAlabama.edu. Until next time, watch out for Statistical Bullshit!