Measures of central tendency are numbers that describe what is average or typical within a distribution of data. There are three main measures of central tendency: mean, median, and mode. While they are all measures of central tendency, each is calculated differently and measures something different from the others. The mean is the most common measure of central tendency used by researchers and people in all kinds of professions. It is the measure of central tendency that is also referred to as the average. A researcher can use the mean to describe the data distribution of variables measured as intervals or ratios. These are variables that include numerically corresponding categories or ranges (like race, class, gender, or level of education), as well as variables measured numerically from a scale that begins with zero (like household income or the number of children within a family). A mean is very easy to calculate. One simply has to add all the data values or "scores" and then divide this sum by the total number of scores in the distribution of data. For example, if five families have 0, 2, 2, 3, and 5 children respectively, the mean number of children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five households have an average of 2.4 children. The median is the value at the middle of a distribution of data when those data are organized from the lowest to the highest value. This measure of central tendency can be calculated for variables that are measured with ordinal, interval or ratio scales. Calculating the median is also rather simple. Let’s suppose we have the following list of numbers: 5, 7, 10, 43, 2, 69, 31, 6, 22. First, we must arrange the numbers in order from lowest to highest. The result is this: 2, 5, 6, 7, 10, 22, 31, 43, 69. The median is 10 because it is the exact middle number. There are four numbers below 10 and four numbers above 10. If your data distribution has an even number of cases which means that there is no exact middle, you simply adjust the data range slightly in order to calculate the median. For example, if we add the number 87 to the end of our list of numbers above, we have 10 total numbers in our distribution, so there is no single middle number. In this case, one takes the average of the scores for the two middle numbers. In our new list, the two middle numbers are 10 and 22. So, we take the average of those two numbers: (10 + 22) /2 = 16. Our median is now 16. The mode is the measure of central tendency that identifies the category or score that occurs the most frequently within the distribution of data. In other words, it is the most common score or the score that appears the highest number of times in a distribution. The mode can be calculated for any type of data, including those measured as nominal variables, or by name. For example, let’s say we are looking at pets owned by 100 families and the distribution looks like this: Animal Number of families that own it
The mode here is "dog" since more families own a dog than any other animal. Note that the mode is always expressed as the category or score, not the frequency of that score. For instance, in the above example, the mode is "dog," not 60, which is the number of times dog appears. Some distributions do not have a mode at all. This happens when each category has the same frequency. Other distributions might have more than one mode. For example, when a distribution has two scores or categories with the same highest frequency, it is often referred to as "bimodal."
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Learning Outcomes
 Let’s begin by trying to find the most “typical” value of a data set. Note that we just used the word “typical” although in many cases you might think of using the word “average.” We need to be careful with the word “average” as it means different things to different people in different contexts. One of the most common uses of the word “average” is what mathematicians and statisticians call the arithmetic mean, or just plain old mean for short. “Arithmetic mean” sounds rather fancy, but you have likely calculated a mean many times without realizing it; the mean is what most people think of when they use the word “average.”
The mean of a set of data is the sum of the data values divided by the number of values.
Marci’s exam scores for her last math class were 79, 86, 82, and 94. What would the mean of these values would be? The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below. 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6 What is the mean number of TD passes? Both examples are described further in the following video.
The price of a jar of peanut butter at 5 stores was $3.29, $3.59, $3.79, $3.75, and $3.99. Find the mean price.
The one hundred families in a particular neighborhood are asked their annual household income, to the nearest $5 thousand dollars. The results are summarized in a frequency table below.
What is the mean average income in this neighborhood? Extending off the last example, suppose a new family moves into the neighborhood example that has a household income of $5 million ($5000 thousand). What is the new mean of this neighborhood’s income? Both situations are explained further in this video. While 83.1 thousand dollars ($83,069) is the correct mean household income, it no longer represents a “typical” value. Imagine the data values on a see-saw or balance scale. The mean is the value that keeps the data in balance, like in the picture below. If we graph our household data, the $5 million data value is so far out to the right that the mean has to adjust up to keep things in balance. For this reason, when working with data that have outliers – values far outside the primary grouping – it is common to use a different measure of center, the median.
The median of a set of data is the value in the middle when the data is in order.
Returning to the football touchdown data, we would start by listing the data in order. Luckily, it was already in decreasing order, so we can work with it without needing to reorder it first. 37 33 33 32 29 28 28 23 22 22 22 21 21 21 20 20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6 What is the median TD value? Find the median of these quiz scores: 5 10 8 6 4 8 2 5 7 7 Learn more about these median examples in this video.
The price of a jar of peanut butter at 5 stores was $3.29, $3.59, $3.79, $3.75, and $3.99. Find the median price.
Let us return now to our original household income data
What is the mean of this neighborhood’s household income? If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median. As we discovered in the last example, the 51st value is $35 thousand. Notice that the new neighbor did not affect the median in this case. The median is not swayed as much by outliers as the mean is. View more about the median of this neighborhood’s household incomes here. In addition to the mean and the median, there is one other common measurement of the “typical” value of a data set: the mode.
The mode is the element of the data set that occurs most frequently. The mode is fairly useless with data like weights or heights where there are a large number of possible values. The mode is most commonly used for categorical data, for which median and mean cannot be computed.
In our vehicle color survey earlier in this section, we collected the data
Which color is the mode? Mode in this example is explained by the video here. It is possible for a data set to have more than one mode if several categories have the same frequency, or no modes if each every category occurs only once.
Reviewers were asked to rate a product on a scale of 1 to 5. Find
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What measure of central tendency is calculated by adding all the values and dividing sum by the number of values?
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