Explore the measures of central tendency. Learn more about mean, median, and mode and how they are used in the field of psychology. At the end, test your knowledge with a short quiz.
What Is Central Tendency?
Think about how you describe a single piece of numerical data. This is usually done in terms of its value. For example, in order to describe the number 2, you might put up two fingers or you might say 2 = 1 + 1. How would you describe a group of data? It would not be beneficial to use your fingers in this instance. Nor is it beneficial to simply add the data together. However, you can describe a group of data in a single value by using measures of central tendency.
So, what exactly is a measure of central tendency? A measure of central tendency is a single value that describes the way in which a group of data cluster around a central value. To put in other words, it is a way to describe the center of a data set. There are three measures of central tendency: the mean, the median, and the mode.
Why Is Central Tendency Important?
Central tendency is very useful in psychology. It lets us know what is normal or 'average' for a set of data. It also condenses the data set down to one representative value, which is useful when you are working with large amounts of data. Could you imagine how difficult it would be to describe the central location of a 1000-item data set if you had to consider every number individually?
Central tendency also allows you to compare one data set to another. For example, let's say you have a sample of girls and a sample of boys, and you are interested in comparing their heights. By calculating the average height for each sample, you could easily draw comparisons between the girls and boys.
Central tendency is also useful when you want to compare one piece of data to the entire data set. Let's say you received a 60% on your last psychology quiz, which is usually in the D range. You go around and talk to your classmates and find out that the average score on the quiz was 43%. In this instance, your score was significantly higher than those of your classmates. Since your teacher grades on a curve, your 60% becomes an A. Had you not known about the measures of central tendency, you probably would have been really upset by your grade and assumed that you bombed the test.
Three Measures of Central Tendency
Let's talk more about the different measures of central tendency. You are probably already familiar with the mean, or average. The mean is calculated in two steps:
Add the data together to find the sum
Take the sum of the data and divide it by the total number of data
Now let's see how this is done using the height example from earlier. Let's say you have a sample of ten girls and nine boys.
The girls' heights in inches are: 60, 72, 61, 66, 63, 66, 59, 64, 71, 68.
Here are the steps to calculate the mean height for the girls:
First, you add the data together: 60 + 72 + 61 + 66 + 63 + 66 + 59 + 64 + 71 + 68 = 650. Then, you take the sum of the data (650) and divide it by the total number of data (10 girls): 650 / 10 = 65. The average height for the girls in the sample is 65 inches. If you look at the data, you can see that 65 is a good representation of the data set because 65 lands right around the middle of the data set.
The mean is the preferred measure of central tendency because it considers all of the values in the data set. However, the mean is not without limitations. In order to calculate the mean, data must be numerical. You cannot use the mean when you are working with nominal data, which is data on characteristics like gender, appearance, and race. For example, there is no way that you can calculate the mean of the girls' eye colors. The mean is also very sensitive to outliers, which are numbers that are much higher or much lower than the rest of the data set and thus, it should not be used when outliers are present.
To illustrate this point, let's look at what happens to the mean when we change 68 to 680. Again, we add the data together: 60 + 72 + 61 + 66 + 63 + 66 + 59 + 64 + 71 + 680 = 1262. Then we take the sum of the data (1262) and divide it by the total number of data (10 girls): 1262 / 10 = 126.2. The mean height (in inches) for the sample of girls is now 126.2. This number is not a good estimate of the central height for the girls. This number is almost twice as high as the height of most of the girls!
However, we can still use other measures of central tendency even when there are outliers. In the scenario above, where a girl who is 680 inches is an outlier, we could use the median. But first, let's explore how to find the median.
The median is the value that cuts the data set in half. If you have an odd number of data, then it's the value that's right in the middle. Let's practice the boys' heights since there are nine boys. There are two steps to finding the median in a sample with an odd number of data:
List the data in numerical order
Locate the value in the middle of the list
Now let's find the median height for our sample of boys. The boys' heights in inches are: 66, 78, 79, 69, 77, 79, 73, 74, 62. So, first we list the data in numerical order: 62, 66, 69, 73, 74, 77, 78, 79, 79. Then, we locate the value in the middle of the list: 62, 66, 69, 73, 74, 77, 78, 79, 79. In a data set that consists of nine items, the datum in the fifth place is the median. The median height for the boys is 74 inches.
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But there are extra steps that need to be taken when working with an even number of data. In order to find the median of an even set, you must find the mean of the two middle numbers. Let's calculate the median height for the girls. Keep in mind that there are ten girls in this sample, which is an even number.
We'll start again by listing the data in numerical order: 59, 60, 61, 63, 64, 66, 66, 68, 71, 72. Now, we locate the two values in the middle of the list: 59, 60, 61, 63, 64, 66, 66, 68, 71, 72. Then, we find the mean of the two middle numbers: 64 + 66 = 130;130 / 2 = 65. The median height for the girls is 65 inches.
Let's see how the median changes once we change 68 to 680 like we did with the mean. List the data in numerical order: 59, 60, 61, 63, 64, 66, 66, 71, 72, 680. Locate the two values in the middle of the list: 59, 60, 61, 63, 64, 66, 66, 71, 72, 680. Find the mean of the two middle numbers: 64 + 66 = 130; 130 / 2 = 65.
You will notice that in both cases, the median height for girls was 65 inches. The median was not changed by adding an outlier. This is because the median is only calculated by using the values in the middle, which makes it extremely useful when working with skewed data. The median, like the mean, cannot be used to describe nominal data.
These graphs visualize why the median is the most useful for of central tendency when there are outliers. The graph on the left is of positively skewed data. You can see here that the mean is close to the right end. However, most of the data is on the left. The median is at a more central location than the mean and the mode.
The graph on the right is of negatively skewed data. The mean is close to the left tail end. You can see here that the median is at a more central location than the mean and the mode.
Our last measure of central tendency is the mode. The mode is the only measure of central tendency that can be used with both nominal and numerical data. The mode is defined as the value that appears most often. If every value only appears once, there is no mode. Let's find the mode for the girls' heights.
The heights are: 60, 72, 61, 66, 63, 66, 59, 64, 71, 68. The number 66 appears twice. All of the other numbers only appear once. Therefore, the mode is 66 inches.
Now let's look at the boys' height. The heights are: 66, 78, 79, 69, 77, 79, 73, 74, 62. The number 79 appears more times than any of the other numbers in the list. The mode height for the boys is 79 inches.
Let's say you are also interested in the most common shirt size of the girls. The sizes of the shirts for each girl is: small, small, medium, extra-large, extra-large, small, large, small, large, medium. Four girls wear a small, one girl wears a medium, two wear a large, and three wear an extra-large. The mode shirt size for the girls would be small, since small occurs more frequently than any other size.
Measures of central tendency are used to describe what is normal for a set of data. Mean, median, and mode are the three measures of central tendency. The mean and median can only be used for numerical data; however, the mean is more sensitive to outliers than the median. The mode can be used with both numerical and nominal data. So remember the next time that you get a grade that you do not like, look around at your classmates' scores to see if you can calculate the mean. Then see where your score falls in relation to your classmates. You may find you didn't do as badly as you think.
When you are done, you should be able to:
State what a measure of central tendency is used for
Name and describe the three measures of central tendency
Recite how to calculate each measure of central tendency and when each is most appropriate
In the following sets of data, find the mean, median, and mode, and then explain which measure is most appropriate given the data and circumstances.
Data Set #1
Your class recently took a test on the latest chapter in your book. Your teacher writes the different scores on the whiteboard so you can see how your score compares to others from your class. Your score is a 79. How does your score compare to those of your classmates? Find the mean, median, and mode with the following set of data. Then explain which measure of central tendency seems to be the most accurate.
You decide to take an 11-day vacation to Berlin, Germany. Since you are interested in weather, you decide to keep track of the high temperature each day while you are there. At the end of the trip, you look at your data. Find the mean, median, and mode with the following set of data. Then explain which measure of central tendency seems to be the most accurate.
85, 91, 84, 87, 88, 95, 79, 88, 86, 89, 82
Data Set #1
Explanation: In this case, either the median or the mode would be the most accurate measure. The mode seems to be a bit lower than the other two measures. Although this score occurred the most often, it does not provide an accurate measure of how the class did as a whole.
Data Set #2
Explanation: Since the mean, median, and mode are quite similar and there aren't any major outliers, all three of these measures would be suitable and appropriate.
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