4.1: Measures of Central Tendency

Although drawing pictures of the data, as in Figure 5.1 is an excellent way to convey the “gist” of what the data is trying to tell you, it’s often extremely useful to try to condense the data into a few simple “summary” statistics. In most situations, the first thing that you’ll want to calculate is a measure of central tendency. That is, you would like to know something about the “average” or “middle” of your data. The most commonly used measures of central tendency are the mean, median, and mode.

The Mean

The mean of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values. The first five MLB margins were 2, 6, 14, 7, and 1, so the mean of these observations is just:

\[
\frac{2+6+14+7+1}{5}=\frac{30}{5}=6.00
\nonumber\]

Of course, this definition of the mean isn’t news to anyone: averages (i.e., means) are used so often in everyday life that this is pretty familiar stuff. However, since the concept of a mean is something that everyone already understands, we can use this as an excuse to start introducing some of the mathematical notation that statisticians use to describe this calculation and talk about how the calculations are done in SPSS.

The first piece of notation to introduce is N, which is used to refer to the number of observations that are being averaged (in this case N=5). Next, we need to attach a label to the observations themselves. Tradition ia to use X for this, and to use subscripts to indicate which observation we’re actually talking about. That is, X₁ refers to the first observation, X₂ to the second observation, and so on, all the way up to X_N for the last one. Or, to say the same thing in a slightly more abstract way, X_i  refers to the i-th observation.

***Just to make sure we’re clear on the notation, the following table lists the 5 observations in theWinMarginvariable, along with the mathematical symbol used to refer to it, and the actual value that the observation corresponds to:

The Observation	Its Symbol	The Observed Value
winning margin, game 1	X1	2 runs
winning margin, game 2	X2	6 runs
winning margin, game 3	X3	14 runs
winning margin, game 4	X4	7 runs
winning margin, game 5	X5	1 run

Okay, now let’s try to write a formula for the mean. By tradition, we use x̄ as the notation for the mean. So the calculation for the mean could be expressed using the following formula:

\[
\bar{X}=\frac{X_{1}+X_{2}+\ldots+X_{N-1}+X_{N}}{N}
\nonumber\]

This formula is entirely correct, but it’s terribly long, so we make use of the summation symbol ∑ to shorten it.⁶⁵ If I want to add up the first five observations, I could write out the sum the long way, X1+X2+X3+X4+X5 or I could use the summation symbol to shorten it to this:

\[
\sum_{i=1}^{5} X_{i}
\nonumber\]

Taken literally, this could be read as “the sum, taken over all i values from 1 to 5, of the value X_i”. But basically, what it means is “add up the first five observations”. In any case, we can use this notation to write out the formula for the mean, which looks like this:

\[
\bar{X}=\frac{1}{N} \sum_{i=1}^{N} X_{i}
\nonumber\]

In all honesty, I can’t imagine that all this mathematical notation helps clarify the concept of the mean at all. In fact, it’s really just a fancy way of writing out the same thing I said in words: add up all the values and then divide by the total number of items. However, that’s not really the reason I went into all that detail. My goal was to try to make sure that everyone reading this book is clear on the notation that we’ll be using throughout the book: \(\bar{X}\) for the mean, ∑ for the idea of summation, X_i for the ith observation, and N for the total number of observations. We’re going to be re-using these symbols a fair bit, so it’s important that you understand them well enough to be able to “read” the equations, and to be able to see that it’s just saying “add up lots of things and then divide by another thing”.

The Median

The second measure of central tendency that people use a lot is the median, and it’s even easier to describe than the mean. The median of a set of observations is just the middle value. As before let’s imagine we were interested only in the first 5 AFL winning margins: 2, 6, 14, 7, and 1. To figure out the median, we sort these numbers into ascending order:

1, 2, 6, 7, 14

From inspection, it’s obvious that the median value of these 5 observations is 6, since that’s the middle one in the sorted list (I’ve put it in bold to make it even more obvious). Easy stuff. But what should we do if we were interested in the first 6 games rather than the first 5? Since the sixth game in the season had a winning margin of 5 points, our sorted list is now

1, 2, 5, 6, 7, 14

and there are two middle numbers, 5 and 6. The median is defined as the average of those two numbers, which is of course 5.5. As before, it’s very tedious to do this by hand when you have a large set of data. To illustrate this, let's sort all 2,429 winning margins in SPSS. Now we know that there are 2,429 values, so the middle of this distribution would be the 1,215th value. Here is that value shown in SPSS data view:

The middle score is highlighted here. Not terribly difficult, but what if you have a million data points? Do you really want to scroll down to the 500,000th score in your data file? I'm pretty sure the answer to that is no.

In a bit I'll show you how to compute the central tendency values easily and quickly. But first...

Mean or median? What’s the difference?

Figure 5.2: An illustration of the difference between how the mean and the median should be interpreted. The mean is basically the “centre of gravity” of the data set: if you imagine that the histogram of the data is a solid object, then the point on which you could balance it (as if on a see-saw) is the mean. In contrast, the median is the middle observation. Half of the observations are smaller, and half of the observations are larger.

Knowing how to calculate means and medians is only a part of the story. You also need to understand what each one is saying about the data, and what that implies for when you should use each one. This is illustrated in Figure 5.2 the mean is kind of like the “center of gravity” of the data set, whereas the median is the “middle value” in the data. What this implies, as far as which one you should use, depends a little on what type of data you’ve got and what you’re trying to achieve. As a rough guide:

• If your data are nominal scale, you probably shouldn’t be using either the mean or the median. Both the mean and the median rely on the idea that the numbers assigned to values are meaningful. If the numbering scheme is arbitrary, then it’s probably best to use the mode instead.
• If your data are ordinal scale, you’re more likely to want to use the median than the mean. The median only makes use of the order information in your data (i.e., which numbers are bigger), but doesn’t depend on the precise numbers involved. That’s exactly the situation that applies when your data are ordinal scale. The mean, on the other hand, makes use of the precise numeric values assigned to the observations, so it’s not really appropriate for ordinal data.
• For interval and ratio scale data, either one is generally acceptable. Which one you pick depends a bit on what you’re trying to achieve. The mean has the advantage that it uses all the information in the data (which is useful when you don’t have a lot of data), but it’s very sensitive to extreme values, as we’ll see later.

Let’s expand on that last part a little. One consequence is that there are systematic differences between the mean and the median when the histogram is asymmetric (skewed; see Section 5.3). This is illustrated in Figure 5.2 notice that the median (right-hand side) is located closer to the “body” of the histogram, whereas the mean (left-hand side) gets dragged towards the “tail” (where the extreme values are). To give a concrete example, suppose Bob (income $50,000), Kate (income $60,000) and Jane (income $65,000) are sitting at a table: the average income at the table is $58,333 and the median income is $60,000. Then Bill sits down with them (income $100,000,000). The average income has now jumped to $25,043,750 but the median rises only to $62,500. If you’re interested in looking at the overall income at the table, the mean might be the right answer; but if you’re interested in what counts as a typical income at the table, the median would be a better choice here.

Trimmed mean

One of the fundamental rules of applied statistics is that the data are messy. Real life is never simple, and so the data sets that you obtain are never as straightforward as the statistical theory says.⁶⁸ This can have awkward consequences. To illustrate, consider this rather strange-looking data set:

−100,2,3,4,5,6,7,8,9,10

If you were to observe this in a real-life data set, you’d probably suspect that something funny was going on with the −100 value. It’s probably an outlier, a value that doesn’t really belong with the others. You might consider removing it from the data set entirely, and in this particular case, I’d probably agree with that course of action. In real life, however, you don’t always get such cut-and-dried examples. For instance, you might get this instead:

−15,2,3,4,5,6,7,8,9,12

The −15 looks a bit suspicious, but not anywhere near as much as that −100 did. In this case, it’s a little trickier. It might be a legitimate observation, it might not.

When faced with a situation where some of the most extreme-valued observations might not be quite trustworthy, the mean is not necessarily a good measure of central tendency. It is highly sensitive to one or two extreme values and is thus not considered to be a robust measure. One remedy that we’ve seen is to use the median. A more general solution is to use a “trimmed mean”. To calculate a trimmed mean, what you do is “discard” the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. The goal is to preserve the best characteristics of the mean and the median: just like a median, you aren’t highly influenced by extreme outliers, but like the mean, you “use” more than one of the observations. Generally, we describe a trimmed mean in terms of the percentage of observations on either side that are discarded. So, for instance, a 10% trimmed mean discards the largest 10% of the observations and the smallest 10% of the observations and then takes the mean of the remaining 80% of the observations. Not surprisingly, the 0% trimmed mean is just the regular mean, and the 50% trimmed mean is the median. In that sense, trimmed means provide a whole family of central tendency measures that span the range from the mean to the median.

Mode

The mode of a sample is very simple: it is the value that occurs most frequently. To illustrate the mode using the MLB data, let’s look at World Series winners from 1903 through 2021. Which team has won the most World Series? The data is located in WorldSeriesWinners.sav. Here are the first few lines of the data file:

When we use SPSS to create a frequency table (details to come), you get:

Clearly, the New York Yankees have won the most World Series titles, with 27. That would be the mode. If one makes a bar chart, it is even clearer where the mode is:

That's it! The mode is simply the most commonly occurring observation.

One last point to make with respect to the mode. While it’s generally true that the mode is most often calculated when you have nominal scale data (because means and medians are useless for those sorts of variables), there are some situations in which you really do want to know the mode of an ordinal, interval or ratio scale variable. For instance, let’s go back to thinking about ourMLB_GL2021.savvariable. This variable is clearly a ratio scale (if it’s not clear to you, it may help to re-read Section 2.2), and so in most situations, the mean or the median is the measure of central tendency that you want. But consider this scenario: a friend of yours is offering a bet. They pick a baseball game at random, and (without knowing who is playing) you have to guess the exact margin. If you guess correctly, you win $50. If you don’t, you lose $1. There are no consolation prizes for “almost” getting the right answer. You have to guess exactly the right margin^. For this bet, the mean and the median are completely useless to you. It is the mode that you should bet on. So, we calculate this mode of the WinMargin variable:

So this data from the 2021 MLB season would indicate you should bet on a 1-run margin, which happens to occur 28% of the time. Seems like a safe bet.