4.1 Central Tendency

Central tendency refers to the descriptive statistic that best represents the center of a data set, the particular value that all the other data seem to be gathering around—the “typical” score. Creating a visual representation of the distribution, as we did in Chapter 2, reveals its central tendency. The central tendency is usually at (or near) the highest point in the histogram or the polygon. The way that data cluster around their central tendency can be measured in three different ways: mean, median, and mode. For example, Figure 4-1 shows the histogram for the data on World Cup top finishes that we first considered in Chapter 2 (omitting scores for countries with no top finishes). Our guess is that their central tendency is just to the right of the tallest bar.

Figure 4-1

Estimating Central Tendency with Histograms Histograms and frequency polygons allow us to see the likely center of a sample’s distribution. The arrow points to our guess as to the center of the distribution of World Cup top finishes.

Mean, the Arithmetic Average

The mean is simple to calculate and is the gateway to understanding statistical formulas. The mean is such an important concept in statistics that we provide you with four distinct ways to think about it: verbally, arithmetically, visually, and symbolically (using statistical notation).

The Mean in Plain English

The most commonly reported measure of central tendency is the mean, the arithmetic average of a group of scores. The mean, often called the average, is used to represent the “typical” score in a distribution. Because this is different from the way we sometimes use the word average in everyday conversation, noting that someone is “just” average in athletic ability or that a movie was “only” average, we need to define the mean arithmetically.

The Mean in Plain Arithmetic

The mean is calculated by summing all the scores in a data set and then dividing this sum by the total number of scores. You likely have calculated means many times in your life.

Visual Representations of the Mean

Think of the mean as the visual point that perfectly balances two sides of a distribution. For example, the mean of 3.13 “top finishes” is represented visually as the point that perfectly balances that distribution, as you can see in the Figure 4-2 histogram.

Figure 4-2

The Mean as the Fulcrum of the Data The mean, 3.13, is the balancing point for all the scores for top finishes for the countries that competed in World Cup soccer tournaments. Mathematically, the scores always balance around the mean for any sample.

The Mean Expressed by Symbolic Notation

Symbolic notation may sound more difficult than it really is. After all, you just calculated a mean without symbolic notation and without a formula. Fortunately, we need to understand only a handful of symbols to express the ideas necessary to understand statistics.

Here are the several symbols that represent the mean. For the mean of a sample, statisticians typically use M or . In this text, we use M; many other texts also use M, but some use (pronounced “X bar”). For a population, statisticians use the Greek letter μ (pronounced “mew”) to symbolize the mean. (Latin letters such as M tend to refer to numbers based on samples; Greek letters such as μ tend to refer to numbers based on populations.) The numbers based on samples taken from a population are called statistics; M is a statistic. The numbers based on whole populations are called parameters; μ is a parameter. Table 4-1 summarizes how these terms are used, and Figure 4-3 presents a mnemonic, or memory aid, for these terms to help you grow accustomed to using them.

Table : TABLE 4-1. The Mean in Symbols The mean of a sample is an example of a statistic, whereas the mean of a population is an example of a parameter. The symbols we use depend on whether we are referring to the mean of a sample or of a population.

Number	Used for	Symbol	Pronounced
Statistic	Sample	M or	“M” or “X bar”
Parameter	Population	μ	“Mew”

Figure 4-3

Samples and Parameters Try using a mnemonic trick to remember the distinction between samples and parameters. The letter s means that numbers based on (s)amples are called (s)tatistics. The letter p means that numbers based on (p)opulations are called (p)arameters.

The formula to calculate the mean of a sample uses the symbol M on the left side of the equation; the right side describes how to perform the calculation. A single score is typically symbolized as X. We know that we’re summing all the scores—all the X’s—so the first step is to use the summation sign, Σ (pronounced “sigma”), to indicate that we’re summing a list of scores. As you might guess, the full expression for summing all the scores would be ΣX or “the sum of X.” This symbol combination instructs us to add up all of the X ’s in the sample.

Here are step-by-step instructions for constructing the equations:

Step 1: Add up all of the scores in the sample. In statistical notation, this is ΣX.

Step 2: Divide the total of all of the scores by the total number of scores. The total number of scores in a sample is typically represented by N. (Note that the capital letter N is typically used when we refer to the number of scores in the entire data set; if we break the sample down into smaller parts, as we’ll see in later chapters, we typically use the lowercase letter n.) The full equation would be:

Median, the Middle Score

The second most common measure of central tendency is the median. The median is the middle score of all the scores in a sample when the scores are arranged in ascending order. We can think of the median as the 50th percentile. The American Psychological Association (APA) suggests abbreviating “median” by writing “Mdn.” APA style, by the way, is used across the social and behavioral sciences, so this would be a good time to get used to it.

Mean versus Median The median is the part of the roadway that divides the directions in which vehicles are permitted to drive. It can be dangerous to confuse the mean and the median, especially when you are calculating the “middle” of the roadway!

David De Lossy/Photodisc Green/Getty Images

To determine the median, follow these steps:

Step 1: Line up all the scores in ascending order.

Step 2: Find the middle score. With an odd number of scores, there will be an actual middle score. With an even number of scores, there will be no actual middle score. In this case, calculate the mean of the two middle scores.

Don’t bother with a formula when a distribution has only a few data points. Just list the numbers from lowest to highest and note which score has the same number of scores above it and below it. The calculation is easy even when a distribution has many data points. Divide the number of scores (N) by 2 and add 1/2—that is, 0.5. That number is the ordinal position (rank) of the median, or middle score. As illustrated below, simply count that many places over from the start of the scores and report that number.

Mode, the Most Common Score

The mode is perhaps the easiest of the three measures of central tendency to calculate. The mode is the most common score of all the scores in a sample. It is readily picked out on a frequency table, a histogram, or a frequency polygon. Like the median, the APA style for the mode does not have a symbol—in fact, it doesn’t even have an abbreviation. When reporting modes, just write: “The mode is….”

EXAMPLE 4.5

Determine the mode for the World Cup data for the 16 countries. Remember that each score represents the number of that country’s top finishes in World Cup tournaments. The mode can be found either by searching the list of numbers for the most common score or by constructing a frequency table:

Did you get 2? If you didn’t, you might have made a common mistake. The mode is the score that occurs most frequently, not the frequency of that score. So, in the data set above, the score 2 occurs 6 times. The mode is 2, not 6.

The mode in this example is particularly easy to determine because it is easy to see that there is one most common score. Sometimes a data set has no specific mode. This is especially true when the scores are reported to several decimal places (and no number occurs twice). When there is no specific mode, we sometimes report the most common interval as the mode; for example, we may say that the mode on a statistics exam is 70–79 (1 interval in the full range of 0–100). When there is more than one mode, we report both, or all, of the most common scores. When a distribution of scores has one mode, we refer to it as unimodal. When a distribution has two modes, we call it bimodal. When a distribution has more than two modes, we call it multimodal. For example, the salaries of recent U.S. law school graduates are bimodal, as illustrated in Figure 4-4.

Figure 4-4

The Bimodal Salaries of Law School Graduates With a bimodal or multimodal distribution, neither the mean nor the median is representative of the data. In an example of good statistical journalism, the New York Times reported a bimodal distribution for the pay of law school graduates in the United States (Rampell, 2010). The reporter described the two modes as the $50,000 public service job or the $160,000 ‘Big Law’ job, and noted that very few people actually earn the mean of about $90,000.

As demonstrated in the law salary example, the mode can be used with scale data; however, it is more commonly used with nominal data. For example, the British newspaper the Guardian (Rogers, 2013) published maps based on census data that showed how residents of England and Wales typically commute to work. They reported that 3.5% work at home; 10.5% use public transportation; 40.4% commute by car; 0.5% ride a motorcycle; 1.9% ride a bicycle; 0.3% take a taxi; and 6.9% walk. In this data set, the modal commute is by car. (Note that this doesn’t add up to 100 because the remaining people are not employed.)

How Outliers Affect Measures of Central Tendency

The mean is not the best measure of central tendency when the data are skewed by one or a few statistical outliers—extreme scores that are either very high or very low in comparison with the other scores. To demonstrate the effect of outliers on the mean, as well as the median’s resistance to the effect of outliers, let’s take a look at the Mundi Index estimates, which compare how many physicians there are per 1000 people in different countries (a statistic called physician density). We’ll focus on the top five.

EXAMPLE 4.6

To get a sense of overall physician density, we might want to calculate a measure of central tendency for these five countries. We’ll use the formula to get a little more practice with the symbols of statistics.

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In Figure 4-5, weights are placed to represent each of the scores in the sample, and they demonstrate an important feature of the mean. Like a seesaw, the mean is the point at which all the other scores are perfectly balanced. The physician density scores range from 4.87 to 47.35 and demonstrate the problem with using the mean when there is an outlier: The mean is not typical for any of the five countries in this sample, even though the seesaw is perfectly balanced if we put its fulcrum at the mean of 14.09.

Figure 4-5

Outliers and the Mean With the physician density data, the mean of 14.09 is far above the lowest four scores yet well below the highest. San Marino’s score, an outlier, pulls the mean higher, even among the other countries with a high physician density. With such an extreme outlier, the mean does not do a good job of representing the story that these data tell.

We can’t help but notice that the tiny country of San Marino (an enclave within Italy; population 33,000) is very different from the other countries. Four of the scores are between 4.87 and 6.4 physicians per 1000 people, a much smaller range. But San Marino enjoys 47.35 physicians per 1000 people. When there is an outlier like San Marino, it is important to consider what this score does to the mean, especially if we have only a few observations.

EXAMPLE 4.7

When we eliminate San Marino’s score, the data are now 6.40, 6.04, 5.81, and 4.87, and the new mean is:

The new mean of these scores, 5.78, is a good deal lower than the original mean of the scores that included San Marino’s outlier. We see from Figure 4-6 that this new mean, like the original mean, marks the point at which all other scores are perfectly balanced around it. However, this new mean is a little more representative of the scores: 5.78 is a more typical score for these four countries. Pay attention to outliers; they usually tell an interesting story!

Figure 4-6

The Mean Without the Outlier When the outlier—San Marino—is omitted from the physician density data, the mean becomes more representative of the actual scores in the sample.

Which Measure of Central Tendency Is Best?

Celebrity Outliers Reports of the cost of a typical home depend on whether the mean or the median is reported. In early 2013, performer Beyoncé bought a $5.9 million home in Houston, Texas, for her mother, Tina Knowles. The 23,562 square foot mansion has eight bedrooms and 12 bathrooms! According to the real estate Web site trulia.com, the median home price in Houston around this time was $88,693 and the mean was $318,377. Beyoncé’s purchase was an outlier for Houston; such a purchase explains why the mean is much higher than the median for home prices in most cities. Extreme outlier purchases boost the mean, but do not affect the median.

Rex/Rex USA

Different measures of central tendency can lead to different conclusions, but when a decision needs to be made, the choice is usually between the mean and the median. The mean usually wins, especially when there are many observations—but not always. When distributions are skewed by outliers such as San Marino, for example, or when there are only a handful of available observations, the median may provide a better sense of a distribution’s central tendency.

The mode is generally used in three situations: (1) when one particular score dominates a distribution; (2) when the distribution is bimodal or multimodal; and (3) when the data are nominal. When you are uncertain as to which measure is the best indicator of central tendency, report all three.

Central tendency communicates an enormous amount of information with a single number, so it is not surprising that measures of central tendency are among the most widely reported of descriptive statistics. Unfortunately, consumers can be tricked by reports that use the mean instead of the median. When you hear a radio report about the central tendency of housing prices, for example, first notice whether it is reporting an average (mean) or a median. You’ll see an example of the housing market and a celebrity outlier in the photograph above.