Mean, median, and mode

Definitions & Key takeaways

Mean, median, and mode are all measures of "central tendency" - they tell something about the distribution of data points in a set.

The mean is simply the average of all the data points in a set. The median is the middle value in a set (if you ordered the values from smallest to largest, the median would be the value that falls in the middle). And mode is just the value that appears most often in a set.

All three measures are useful for different things. The mean is good for finding out what an average value is, the median is good for identifying outliers (values that are far from the rest of the data), and the mode is good for getting a sense of how clustered or spread out the values in a set are.

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In statistics, it’s often helpful to know the central point of a set of data, because it gives a pretty good idea about the whole data set.
It’s like a one number summary of the data. What’s the number of words on a page in a book?
About 250. Of course it depends on the book, but that’s the one number summary.
The mean, median and mode are the most commonly used ways to measure this central point. Let’s start with the mean, which is also called the average.
You can calculate the mean by adding up each value in a data set and then dividing by the total number of data points. Let’s look at an example.
Let’s say 7 students took a test on biostatistics and out of 100 possible points, one student got 17, another got 19, two got 20, two more got 21 and the last student got 22.
The mean score would be the total number of points they all got added up together divided by the number of students which is 7.
So that’s: 17+19+20+20+21+21+227=1407= 20. To show this as a formula, we can say that the mean, written as X with a bar over it, is the total sum of the individual data points X1, X2, ......., Xn, divided by n, which is the number of data points.
A mean test score of 20, quickly tells us that overall, these students didn’t do well on this test. But, the problem with the mean, is that it can be influenced by an extreme value called an outlier.
Let’s say that another student comes along and get a perfect 100 out of 100 on the test. That means that the average is now: 17+19+20+20+21+21+22+100 = 240 divided by 8, which is 30.
This one number summary isn’t a very good summary because 7 out of 8 students scored below 23, and the only reason that the average is so high is because of this one high-scoring student.
In this case, 100 is an outlier, and any data set with an outlier is called skewed data. To calculate the central point when there may be outliers, you can use the median.
The median is the value which divides a set of data into two halves so that half of the values are greater than the median and half are less than the median.
To find the median in our test scores example, you have to first arrange the test scores in order of magnitude like this: Then you have to find the midpoint.
If there’s an odd number of values, then there will be a “middle” number, but if there’s an even number of values like in this dataset, there are two middlemost values- 20 and 21.
So to get the median, we take the average of these two values (20+21)/2 = 20.5 is the median for this dataset. Notice that an extreme value like 100 doesn’t affected the median; and that the median of 20.5 gives a pretty accurate one number summary of the data.
Like the median, the mode is another kind of statistical average which does not get affected by any extreme value. The mode is the value that occurs most frequently in a set of data, and if there are two modes it’s bimodal, and with three modes it’s trimodal.
So using the same example of test scores - 17 19 20 20 21 21 22 100 - the most frequently occurring values are 20 and 21.
So, this is a bimodal dataset with two mode values of 20 and 21. Now, keep in mind that in some data sets all of the values are unique and that means that there’s no mode at all - it’s amodal.
That’s why the median is most commonly used as the central point when reporting data. Alright, as a quick recap, the central point of a dataset is the one number summary of a data set.
The mean or average can be influenced by extreme values, whereas the median and mode are not as affected by extreme values.
Between the median and mode, the median is usually preferred because some data sets are amodal.