Section 3.1: Measures of Center

A measure of center, or measure of central tendency, is a number that gives the sense of what is typical in the population. For example, you might want to know the typical starting salary for nurses in California. Or you might want to estimate the typical number of calories in a fast-food meal. Or you might want to describe the average student at your college or employee at your work. You can answer these questions by collecting data and identifying the most common or typical observation.

How we interpret what it means to be typical is how we arrive at three different measures of center. You may already be familiar with them, they are the mean, median, and mode. The first, the arithmetic mean, is probably most familiar. Arithmetic mean is the formal name for the average that you have likely used.

But before we begin looking at measures of center, we need to recall the difference between populations and samples. Recall that a parameter is a measure of all individuals in a population. A statistic is a measurement that is taken from sample data. A sample, you will recall, is a subset of the population. Typically, we use the measurements from the sample (the statistic) to estimate the measure of the population (the parameter). In this chapter it will be extremely important that we identify whether we are dealing with a population or a sample because the notations and sometimes even the formulas we use will be different. The following definition was presented in section 1.1.

The Mean

The arithmetic mean is the proper name for what we usually call the mean, often used synonymously with the word average. In your classes, when you add up your test scores and divide by the total you are computing your average. Now you know it has a fancier name. It also has two formulas, depending on whether you are dealing with population data or sample data.

These formulas look complicated, but you already know how to use them in practice.

Table 1: Student Exam Scores
Name	Score
Derek	70
Marshawn	95
Isabel	80
Sophia	83
Juan	82

Solution:The population mean is calculated as follows:

$$\begin{array}{cc}\mathrm{\mu }& =\frac{x_1+x_2+x_3+\dots +x_N}{N}\\ & =\frac{70+95+80+83+82}{5}\\ & =\frac{410}{5}\\ & =82\end{array}$$

So, the mean, or typical value for this data set is 82. That suggests that the “average grade” on this exam was an 82. As you look at the data, there were three scores right around the mean, the 80, 82, and 83. There was one score that was higher, and one that was lower. In this sense, the mean is doing a good job of finding the center of the data. If we look at the dot plot of the data in figure 3.2, we can see that the value 82 is right in the middle.

Figure 3.2: Dot plot for data in Table 1.

Table 2: Updated Student Exam Scores
Name	Score
Derek	70
Marshawn	95
Isabel	80
Sophia	83
Juan	82
Jackson	10
Carol	0

Solution:The population mean is now:

$$\begin{array}{cc}\mathrm{\mu }& =\frac{x_1+x_2+x_3+\dots +x_7}{7}\\ & =\frac{70+95+80+83+82+10+0}{7}\\ & =\frac{420}{7}\\ & =60\end{array}$$

As we see, the mean grade has been calculated to be 60. That is a considerable drop from the original mean that was 82 in example 1. The population mean 82 was typical of the first 5 scores in the data set, but does the number 60 represent a typical score for the 7 students on this test? Checking the data, we note there were no scores close to 60. Look at the dot plot for the data in figure 3.3. Do you think the mean value of 60 is typical of the data?

Figure 3.3: Dot plot for data in Tables 2. Does the mean still represent a typical value?

From the dot plot, it appears that the mean is in the middle, as it should be, but it doesn’t represent the typical observation anymore. Over half of the scores were 80 or higher, and most were 70 or higher. The typical grade letter grade was a B, and no exam score was near 60. Figure 3.4 shows how the inclusion of the extremely low data values caused the mean to drop in such a way that it no longer represents a typical score on the exam.

Figure 3.4: The addition of the extremely low observations drags the mean away from the center.

Example 2 shows that the mean is sensitive to extreme observations, meaning that observations that are far from the rest will pull the mean in their direction. That makes the mean an unreliable measurement in the cases where there are extremely high or extremely low data values. In the example you can see that the scores of 0 and 10 dragged the mean so much lower that it failed to be a good measure of a typical exam score.

This happens because the value of every observation is included in the calculation of the arithmetic mean. Every data value carries weight in the computation, so every data value has influence over the value of the mean. As a result, even one or two extreme values can ruin the mean’s ability to measure the center of the data. In the cases where we have extreme values, we need another way to look at what it means to be typical. That’s where the median comes into play.

The Median

A second way to identify a typical value is to arrange the data in order, from smallest to largest, then find the value or values that are right in the middle of the list. This is called finding the median. Like the mean, the formulas can be confusing, but the idea is straight-forward.

Figure 3.5: The median divides the data into two groups, with about half the data above and half below.

Solution:For the data in Table 1, we first need to arrange the data in ascending order.

70, 80, 82, 83, 95

Next, we find the value that is in the middle, or at the halfway point.

We can see that the score 82 is in the middle, so 82 is the median, M = 82. The mean for the data in Table 1 was also $\bar{x}=82$. Since the mean and median agree, we would consider both to be reliable measures of the center of the data.

Next, let’s look at the data in Table 2. Again, we first arrange the data in ascending order.

0,10,70,80,82,83,95

Looking for the middle, we find that 80 has three scores lower and three scores higher. So, the median is M = 80. The mean of this data set was $\bar{x}=60$. In the case of the second data set, the median does a much better job of identifying the center of the data. See figure 3.6. The red triangle represents the median in the illustration. Notice that it is closer to more of the data than the green triangle representing the mean. When the data have extreme observations, the median does a better job of identifying the typical observations.

Figure 3.6: The mean and median on the dot plot for Table 2 data.

In the last example, the presence of the extremely low scores did not affect the median as much as the mean. The median dropped from 82 to 80, still representing the typical values reasonably well, while the mean became unreliable. This is generally true when there are extreme observations. We have already noted that the mean is sensitive to extreme observations, and now we see that the median is resistant to extreme observations.

The reason why the median is resistant to extreme observations is that it does not include each data value in the computation. It only includes the order, or rank, of the data from smallest to largest, not the actual values. Consider how the mean and median would be affected if the professor had input a typo on the highest score as 195 instead of 95 as it was in Table 1.

Original (correct) data, arranged in ascending order:

Data with typo, arranged in ascending order:

The change in the last data entry did not affect the median, but it did affect the mean. The median doesn’t care how large the largest score is, only that it was the largest. The median treats 95 and 195 the same as simply “the highest score”. The mean, however, is influenced by the values of the 95 and 195. When the mean is calculated including the typo, it is dramatically changed. Again, we emphasize that the mean is sensitive to extreme observations, while the median is resistant to them.

Finding the median is not difficult if you have a small and odd-numbered set of data, like in the last examples. But what do you do if there is not one value in the middle? What if the data set has an even number, or thousands of observations? It’s not always easy to spot the middle value in a data set. Consider example 4.

Solution:Step one, as before, is to arrange the data in ascending order.

101114161919

Here, there is no observation in the middle. There are two observations lower than 14, and three that are higher. So, 14 is not in the middle, and so it is not the median.

What about the data value 16? It isn’t in the middle either.

The problem is that we have an even number of observations in the data. When this happens, the median is the mean (the average) of the two observations that are in the middle. In this case, the two observations in the middle are 14 and 16. The median is $M=\frac{14+16}{2}=\frac{30}{2}=15$.

The process outlined above is important if you are trying to find the median by looking at a data set that has an even number of observations. The first thing you will need to know is whether the data set has an odd or even number of observations.

Solution:(a)The exam scores from Table 1 are {70, 95, 80, 83, 82}

Step 1:Put the data in ascending order.

7080828395

Step 2:There are N = 5 (we use n = 5 in the formula) observations. The number 5 is odd. The locator formula gives the location of the median at

$$\begin{array}{cc}\frac{n+1}{2}& =\frac{5+1}{2}=\frac{6}{2}=3.\end{array}$$

So, the third item in the list is the median. That is, the median is 82. This agrees with the answer we found in example 1.

(b)The quiz scores are {14,11,19,16,10,19}

Step 1:Put the data in ascending order.

101114161919

Step 2:There are n = 6 observations. The number 6 is even. The locator formula gives the location of the median at

$$\begin{array}{cc}\frac{n}{2}& =\frac{6}{2}=3\end{array}$$

averaged with the item at location

$$\begin{array}{cc}\frac{n}{2}+1& =3+1=4.\end{array}$$

So, the median will be the average of the third item in the list with the fourth item.

Again, the median is
$$\begin{array}{cc}M& =\frac{14+16}{2}=\frac{30}{2}=15\end{array}$$

which is the value that is exactly between the third and fourth observations. This is precisely what we calculated in example 4.

Solution:We can calculate the mean and median by hand.

There are n = 24 observations. First, we calculate the sample mean.

$$\begin{array}{cc}\bar{x}& =\frac{x_1+x_2+x_3+\dots +x_n}{n}\\ & =\frac{2+3+\dots +5+1}{24}\\ & =\frac{75}{24}\\ & =3.125\end{array}$$

The student received 3.125 spam calls per day, on average (usually, we will round this off to 3.1). To find the median, we need to arrange the data in order from smallest to largest. That can be complicated when the data set is large. (It is preferable to use technology to sort large amounts of data!)

011111122222334445556677

Since the data set is even with n = 24, the median will be the average of the 12^th and 13^th observation in the list.

So, the median is
$$\begin{array}{cc}M& =\frac{12\mathrm{th}+13\mathrm{th}}{2}=\frac{2+3}{2}=\frac{5}{2}=2.5.\end{array}$$

That means that she gets fewer than 2.5 spam calls per day half the time, and more than 2.5 spam calls per day half the time.

Using STATBELT Utility, first type the data into an empty column and name it “Spam Calls.” Then, we follow Descriptive Statistics → Basic Statistics, and check “Spam Calls” under “Columns to include.”

The STATBELT Utility output gives more information than we need. But you can see the mean and median are two of the summary statistics that are provided. You can also see the number of observations, n, is given. The other statistics will be discussed later as we progress through chapter 3.

So, it appears that she typically gets around 2 or 3 spam calls per day, perhaps more. We can say this with some certainty because the mean and median are close together. To determine which of the mean or median is a better measure of center in this situation, we need to consider the shape of the distribution. But first, there is one more measure of center to consider.

The Mode

Solution:In example 1 in section 2.2 we created the frequency table for the number of spam calls distribution.

Table 4: Frequency and Table
Number of Calls per Day	Frequency
0	1
1	6
2	5
3	2
4	3
5	3
6	2
7	2
Total	24

Since the number of calls with the highest frequency was 1 call per day, which occurred 6 times, the mode number of spam calls is 1.

Now we have seen three methods to determine a “center” or “typical” observation in the data. Which one is the best to use? That depends on the type of data and the distribution.

The Shape of the Distribution and Measures of Center

Figure 3.7 shows the dot plot for spam phone calls along with the three measures of center that we have computed. The mode is 1 (represented with a blue triangle), the median is 2.5 (represented with a red triangle), and the mean is 3.1 (represented with a green triangle). Which of these is the best representative of what is typical?

Figure 3.7: The mean, median, and mode shown on the dot plot for Table 3 data.

The mode is the value on the x-axis that corresponds to the “highest peak” of the dot plot. Even though it is the most common observation, the mode = 1 does not look to be in the “center” of the data. It seems very far to the left side of the distribution with almost all the observations larger than 1. Further, it appears that the data are skewed to the right as there are extreme observations of 6 and 7 on the far-right side. Because the mean is sensitive to the extremes, it is pulled to the right by them, perhaps too far to the right. The median is resistant to the extremes, and it is further to the right than the mode. The median looks to be the most “in the middle of the data” of the three measures.

Mode = 1 is too low.

Mean = 3.1 is probably a little too high because the distribution is skewed.

Median = 2.5 looks to be just right.

If this sounds like Goldilocks and the Three Bears to you, you are correct. The median plays the role of Baby Bear in this story. When data are skewed, the median is just right.

The relationship we see above is the usual pattern for data distributions that are skewed to the right. In fact, if data are skewed in either direction, we will use the median as the best measure of center.

What if the data are not skewed? How will the mean, median, and mode compare?

Solution:First copy or type the data into STATBELT Utility, or load the sample data. Then, select Descriptive Statistics → Basic Statistics and check “Age at inauguration” under “Columns to include.” Under “Statistics to show,” click “Select none,” then check Mean, Median, and Mode. Click “Show results in a new window.”

The mean is 55.5 when rounded to one decimal place. The median is 55. That means that about half of the ages are lower than 55, and about half are higher. STATBELT Utility lists two modes, 51 and 54. Looking back at the data in Table 5 we see that the data is bimodal since each of 51 and 54 appear 5 times. Figure 3.8 shows the histogram we made by hand in section 2.2 with the three measures of center added. All three measures of center are close together. See figure 3.8.

Figure 3.8: The mean, median, and mode shown on the histogram for table 5 data.

The tallest bar of the histogram represents the class 50-54. Both modes are in that class (the mode at 51 is shown faded). The mode at 54 is very close to the median at 55 and the mean at 55.5. This histogram is bell-shaped with a very slight skew. The slight skew causes the mean to be slightly larger than the median, but not by much. The mean, median, and mode are consistently near 55, so the best measure of the typical age is around 55 years.

In general, when data are bell-shaped, the mean, median, and mode will be close together. When we have a choice, we will choose to use the mean. This is also true when data are uniform, and in fact for any symmetric distribution. Symmetric means that the distribution is the same on each side, like a mirror image. When that happens, any extreme values on the left are balanced by extreme values on the right, and the mean remains in the middle. Although there is nothing wrong with the median, we prefer to use the mean whenever possible, because most of the advanced procedures we will see later in the class will involve the mean. We can simply do more interesting analysis using the mean that we can using the median, at least for now.

Figure 3.9 shows how the measures of center are positioned in relation to the shape of the distribution. For instance, if the data are skewed left, the mean is less than the median. If it is skewed right, the mean is larger than the median. If it is bell-shaped, they should be roughly equal.

Figure 3.9: The mean, median, and mode in relation to the shape of the data distribution.

Tip!

Which of the Measures of Center is Best?

Now we will summarize the three measures, and when they are best used.

As we have seen, when data are skewed, the median does the best job at describing what is typical. If the data are bell-shaped, we prefer to use the mean. The only time we use the mode is when we want to describe the most typical observation from qualitative data. When we are dealing with qualitative data, there are no numbers in which to add to compute a mean, nor to arrange from smallest to largest to find a median. Mean and median are only useful for quantitative data. As a result, the mode is the only option for qualitative data. See figure 3.10.

Figure 3.10: Flowchart for deciding which measure of center to use.

Solution:Since the data is qualitative, you cannot find a mean or median. The only option is to find use mode. In this case, pepperoni was the category selected with the highest frequency (12 times), so the mode is pepperoni. We would say that the most typical pizza selected was pepperoni.

Solution:Supply and demand tells us that there are more lower to moderately priced homes sold each year than there are expensive ones. But there can be extremely expensive homes, in the multiple millions of dollars, that are purchased by the few that are very wealthy. That means that the frequency for lower to moderately priced homes is high, while the frequency for very expensive homes is smaller, and smaller still for the most expensive homes.

Thus, homes are skewed to the right. As a result, the median is the best measure of center for home prices. Since the data are skewed right, the mean is also pulled to the right, and so it is larger than the median (see figure 3.9). In fact, when you hear about home prices in the news, they always report the median price, meaning that approximately 50% of all home sales are lower, and 50% are higher than the median price they give. Now you know why they don’t tell you the average price (which would be even higher and likely discourage many first-time home buyers).

Solution:In example 6, section 2.2, we used STATBELT Utility to make the histogram shown in figure 3.11. First, we copied the data into a new column. Next, we selected Graphs → Histogram and chose the average poverty column for “Column.” We unchecked “Choose start and bin width automatically,” typed 5 for “Start at” and 2.5 for “Bin width,” and selected “Relative frequency” for “Bar height.” Then we clicked “Open graph in a new window.” The histogram appears to be close to bell shaped, with possibly a very slight skew to the right. Since it is bell-shaped, we would use the mean rather than use the median to best represent a typical average poverty rate. But because of the slight skew, we would expect the mean to be very slightly larger than the median.

Figure 3.11: Histogram for average poverty rates in the US, looks bell-shaped.

To compute the mean and median, we select Descriptive Statistics → Basic Statistics, then check the column under “Columns to include.” Then click “Show results in a new window.” This gives us more than we need, but the mean and median are easy to locate in the output table.

The mean is 12.03%, rounded to two decimal places (the original data was given to one decimal place), and the median is 11.5%. As expected, the mean is a little bit higher than the median due to the slight skew. By and large, the two values are very close due to the bell-shape of the distribution.

Recall that the goal of this chapter is to be able to describe the data we collect in terms of shape, center, and dispersion. We can now describe shape and center. Dispersion, or spread of the data, will be discussed in section 3.2.

Figure 3.12: Histograms, mean, and median for high school GPA and college GPA.

Solution:High School Grade Point Average: This distribution is skewed to the left. Since it is skewed, the median, M = 3.43, is the best representative for the center. Approximately half of high school students had a GPA below 3.43, and about half had a GPA that was above 3.43. The mean is lower than the median, as you can see in figure 3.12.

College Grade Point Average: This distribution appears to be bell-shaped. As such, the mean $\bar{x}=3.12$ is the better measure of center. The average college student has a grade point average of 3.12. The median is reasonably close to the mean because the data are close to bell-shaped.

The shape of the distributions suggests that there are many high school students who earn a 4.0 grade point average, but not as many earn a 4.0 at college. The average high school student had a mean GPA that was 0.19 grade points higher than their college GPA because 3.31 – 3.12 = 0.19. The difference in the medians was 0.42, or about half of a grade. So, the average high school student can expect a slight drop in their grade point average as they transition to college. Of course, that describes the general trend, and individuals will have varying results.

Photo by John Schnobrich (opens in a new tab) on Unsplash (opens in a new tab)