A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. These measures indicate where most values in a distribution fall and are also referred to as the central location of a distribution. You can think of it as the tendency of data to cluster around a middle value. In statistics, the three most common measures of central tendency are the mean, median, and mode. Each of these measures calculates the location of the central point using a different method.
Choosing the best measure of central tendency depends on the type of data you have. In this post, I explore these measures of central tendency, show you how to calculate them, and how to determine which one is best for your data.
Locating the Center of Your Data
Most articles that you’ll read about the mean, median, and mode focus on how you calculate each one. I’m going to take a slightly different approach to start out. My philosophy throughout my blog is to help you intuitively grasp statistics by focusing on concepts. Consequently, I’m going to start by illustrating the central point of several datasets graphically—so you understand the goal. Then, we’ll move on to choosing the best measure of central tendency for your data and the calculations.
The three distributions below represent different data conditions. In each distribution, look for the region where the most common values fall. Even though the shapes and type of data are different, you can find that central location. That’s the area in the distribution where the most common values are located.
As the graphs highlight, you can see where most values tend to occur. That’s the concept. Measures of central tendency represent this idea with a value. Coming up, you’ll learn that as the distribution and kind of data changes, so does the best measure of central tendency. Consequently, you need to know the type of data you have, and graph it, before choosing a measure of central tendency!
Related posts: Guide to Data Types and How to Graph Them
The central tendency of a distribution represents one characteristic of a distribution. Another aspect is the variability around that central value. While measures of variability is the topic of a different article (link below), this property describes how far away the data points tend to fall from the center. The graph below shows how distributions with the same central tendency (mean = 100) can actually be quite different. The panel on the left displays a distribution that is tightly clustered around the mean, while the distribution on the right is more spread out. It is crucial to understand that the central tendency summarizes only one aspect of a distribution and that it provides an incomplete picture by itself.
Related post: Measures of Variability: Range, Interquartile Range, Variance, and Standard Deviation
Mean
The mean is the arithmetic average, and it is probably the measure of central tendency that you are most familiar. Calculating the mean is very simple. You just add up all of the values and divide by the number of observations in your dataset.
The calculation of the mean incorporates all values in the data. If you change any value, the mean changes. However, the mean doesn’t always locate the center of the data accurately. Observe the histograms below where I display the mean in the distributions.
In a symmetric distribution, the mean locates the center accurately.
However, in a skewed distribution, the mean can miss the mark. In the histogram above, it is starting to fall outside the central area. This problem occurs because outliers have a substantial impact on the mean. Extreme values in an extended tail pull the mean away from the center. As the distribution becomes more skewed, the mean is drawn further away from the center. Consequently, it’s best to use the mean as a measure of the central tendency when you have a symmetric distribution.
When to use the mean: Symmetric distribution, Continuous data
Related post: Using Histograms to Understand Your Data
Median
The median is the middle value. It is the value that splits the dataset in half. To find the median, order your data from smallest to largest, and then find the data point that has an equal amount of values above it and below it. The method for locating the median varies slightly depending on whether your dataset has an even or odd number of values. I’ll show you how to find the median for both cases. In the examples below, I use whole numbers for simplicity, but you can have decimal places.
In the dataset with the odd number of observations, notice how the number 12 has six values above it and six below it. Therefore, 12 is the median of this dataset.
When there is an even number of values, you count in to the two innermost values and then take the average. The average of 27 and 29 is 28. Consequently, 28 is the median of this dataset.
Outliers and skewed data have a smaller effect on the median. To understand why, imagine we have the Median dataset below and find that the median is 46. However, we discover data entry errors and need to change four values, which are shaded in the Median Fixed dataset. We’ll make them all significantly higher so that we now have a skewed distribution with large outliers.
As you can see, the median doesn’t change at all. It is still 46. Unlike the mean, the median value doesn’t depend on all the values in the dataset. Consequently, when some of the values are more extreme, the effect on the median is smaller. Of course, with other types of changes, the median can change. When you have a skewed distribution, the median is a better measure of central tendency than the mean.
Comparing the mean and median
Now, let’s test the median on the symmetrical and skewed distributions to see how it performs, and I’ll include the mean on the histograms so we can make comparisons.
In a symmetric distribution, the mean and median both find the center accurately. They are approximately equal.
In a skewed distribution, the outliers in the tail pull the mean away from the center towards the longer tail. For this example, the mean and median differ by over 9000, and the median better represents the central tendency for the distribution.
These data are based on the U.S. household income for 2006. Income is the classic example of when to use the median because it tends to be skewed. The median indicates that half of all incomes fall below 27581, and half are above it. For these data, the mean overestimates where most household incomes fall.
When to use the median: Skewed distribution, Continuous data, Ordinal data
Mode
The mode is the value that occurs the most frequently in your data set. On a bar chart, the mode is the highest bar. If the data have multiple values that are tied for occurring the most frequently, you have a multimodal distribution. If no value repeats, the data do not have a mode.
In the dataset below, the value 5 occurs most frequently, which makes it the mode. These data might represent a 5-point Likert scale.
Typically, you use the mode with categorical, ordinal, and discrete data. In fact, the mode is the only measure of central tendency that you can use with categorical data—such as the most preferred flavor of ice cream. However, with categorical data, there isn’t a central value because you can’t order the groups. With ordinal and discrete data, the mode can be a value that is not in the center. Again, the mode represents the most common value.
In the graph of service quality, Very Satisfied is the mode of this distribution because it is the most common value in the data. Notice how it is at the extreme end of the distribution. I’m sure the service providers are pleased with these results!
Finding the mode for continuous data
In the continuous data below, no values repeat, which means there is no mode. With continuous data, it is unlikely that two or more values will be exactly equal because there are an infinite number of values between any two values.
When you are working with the raw continuous data, don’t be surprised if there is no mode. However, you can find the mode for continuous data by locating the maximum value on a probability distribution plot. If you can identify a probability distribution that fits your data, find the peak value and use it as the mode.
The probability distribution plot displays a lognormal distribution that has a mode of 16700. This distribution corresponds to the U.S. household income example in the median section.
When to use the mode: Categorical data, Ordinal data, Count data, Probability Distributions
Which is Best—the Mean, Median, or Mode?
When you have a symmetrical distribution for continuous data, the mean, median, and mode are equal. In this case, analysts tend to use the mean because it includes all of the data in the calculations. However, if you have a skewed distribution, the median is often the best measure of central tendency.
When you have ordinal data, the median or mode is usually the best choice. For categorical data, you have to use the mode.
In cases where you are deciding between the mean and median as the better measure of central tendency, you are also determining which types of statistical hypothesis tests are appropriate for your data—if that is your ultimate goal. I have written an article that discusses when to use parametric (mean) and nonparametric (median) hypothesis tests along with the advantages and disadvantages of each type.
If you’re learning about statistics and like the approach I use in my blog, check out my Introduction to Statistics eBook!
It was so clear and understandable. I appreciate. Was really helpful. Thanks.
Jim,
This is a wonderful article. I would only suggest that you consider paraphrasing this portion by possibly qualifying it: ” … Unlike the mean, the median value doesn’t depend on all the values in the dataset. ..”.
Technically, the median depends on all the values in the dataset. This is why your explanation of its calculation reads, “To find the median, order your data from smallest to largest, and then find the data point that has an equal amount of values above it and below it”.
Regards.
Hi Ooko,
Thanks for writing. I understand what you’re saying. However, there’s a large difference between how the median uses values compared to the mean. For the median, while you do sort from smallest to largest, the values above and below the median are literally just placeholders. For example, you can take any value that is above the median and change it to any other value that is above the median, and the median won’t change at all. You can do the same with values below the median. Conversely, with mean, you make a change to any value, and it affects the mean. Consequently, the median value does not depend on all the values in the dataset. You can literally change them and not affect the median. I show an example of that in this post.
Hi
This is my first time to read an article by you
It is clear wonderfull article and delivered it smoothy. I could see your proffession and love to statistic.
I kept foraging the web for explanations on these topics and i didnt find any article as simple yet so informative and understandable.
Thank you for your kind words. They make my day! I’m also happy to hear that it was helpful!
I really appreciate that! You made very clear to me!
Abena that so funny of you. i pray your lecturer wont read your comment.
Hi Jim. I would like to acknowledge for for the wonderful blog you have written. Most of the blogs/tutorials cover the basic stuff and the important stuff( like when to use which measure) are all scattered up. This blog helped me to collate everything at one place. Thank you very much!
Thank you so much, very good explination.
What are the things to include when presenting about descriptive quantitative data analysis
Thank you very much
Thank you very much.
Bom dia Jim,
Meu segundo dia de leitura do seu post. Parabéns! Estou aprendendo muito. Comprei o seu livro sobre regressão linear para meus trabalhos com ciência de dados.
Muito obrigado.
Awesome @Jim Frost!. The histogram blog was the exact answer I was searching for. Thanks again for putting together!.
Jim, Thank so much! My lecturer can never explain the difference between mean and median.
Dear sir,
Thank you so much for your blog. It was so easy to understand. As addition can you explain the properties of good measures of central tendency?
Many thanks for the helpful article. You have given a clear explanation of the central tendency measures, and a guide where best to use each of them.
Great resource!. What about the central tendency data with double hump distribution where two data set in which the first data set has lower hump frequency higher and second data has the second one high. How the median or mean helps the central tendency? Is there any other method for such scenario?
Hi Shams,
Thank you! I’m glad my blog has been helpful!
The technical name for a double hump distribution is a bimodal distribution. More generally, a distribution with more than one peak is a multimodal distribution.
When you find a multimodal distribution, consider whether underlying subpopulations are producing it. For example, the heights of men and women have different means. It’s almost a bimodal distribution. In these cases, you might want to graph the separate distributions for each subgroup and identify each group’s central tendency.
In other cases, subgroups don’t explain the multiple peaks. It’s just the natural shape of the distribution. In those cases, graphs become extra important because no measure of central tendency will convey the true nature of the distribution.
For more information about multimodal distributions, please see my blog post about histograms.
thanks very helpful indeed
Thank you so much for this article. I am in the second week of my first attempt at taking a Statistics class! This was so very helpful.
Thanks Jim!
Quite simple and informative.
Looking ahead to read more of your articles.
Very informative…
Hi Jim,
thanks for your blog. I found this searching for information, and so far it’s been the easiest to read and understand! I’m attempting to do a course on using and interpreting data in schools and I get so confused with it all! When someone asks about the relationship between the mean and the median, what are they asking for?
Hi Carol,
Thanks! I’m glad to hear it’s been helpful! I strive to make my blog as easy to understand as possible.
That is a bit of a vague question, but I hope the context in which it’s being asked helps.
What they might be asking for is a description of how the mean and median tend to be approximate equal for symmetric distributions. As the distribution becomes more skewed, the difference between the mean and median increases, with the mean being pulled towards the long tail.
Maybe that’s what they’re asking for? I hope this helps!
Hllo sir ..
I am ur biggest fan .
Gve ur posts on Distributions of Probability
…hope I wll get ths soon
Hi Khursheed, I’ve already written that post. You can find it here: Understanding Probability Distributions.
Jim , It has been a while since my last stat class and i needed a refresh on those basic unfortunately a mooc the cost hundereds of dollars could not help . Your articles helped my greatly and i love the intuitive approach . I am going to go through all your articles , Please do keep writing more
A very nice and to the point data!!
Thank you, Sakshi!!
why it is called a measure of central tendencies?
Hi Christian,
In many distributions, there are values that are more likely and less likely to occur. A measure of central tendency identifies where values are more likely to occur–or where they *tend* to occur. Hence, “tendency.”
Central is more applicable to the mean and median. Both of these measures identify a central point in the distribution. This central point is where the values are more likely to occur.
As we saw in the post with categorical data, there is no central value. Consequently, central doesn’t really apply for the mode. But, we still use the terminology.
So nicely described..Its worth.
Thank you, Manas!
Jim, how do you handle data spread when not using the mean?
Hi, you must be psychic! I’m writing a post about different measures of variability right now! If you’re not using the mean because your data are skewed, I find that using the median for the central tendency and interquartile range (IQR) for the variability goes together nicely. The median splits that data in half and the IQR tells you where the middle half of the data fall. The wider the IQR, the greater the spread the data spread. You can also use percentiles to determine the spread for other proportions. For example, 95% of the data fall between the 2.5th and 97.5th percentiles.
Thanks for that response Jim. I have one more quick question. I would think that the mode for continuous data would be important when it comes to distributions that have two (bimodal) or more (multi-modal) peaks. In these cases, where one has more than one center of tendency, it would seem to me that the mode measure of central tendency becomes the more important piece of information than either the mean or median. Would this be accurate? Or is the answer, “It depends”? 🙂
Hi Chuck! Apologies for the delay in getting back to you. I’ve been on vacation!
I agree with what you say about multimodal continuous distributions. In fact, if you have a multimodal distribution, it’s often crucial that you make that determination. Suppose that you use a histogram to display the distribution of body heights. You notice that there are two peaks. There are at least three important issues here.
1) If you are trying to identify the best probability distribution for your data, you won’t succeed!
2) You also know that there is something else of interest for you to learn about your data. For our example, the two peaks might indicate separate measures of central tendencies for males and females. You can then better understand your data and how to analyze it.
3) As you mention, the mean and median are less meaningful for the single multimodal distribution. You’ll probably want to identify the subpopulations (if they exist) and change your analysis.
Graphing is always important for understanding your data. In this case, you do want to know about multimodal distribution because it affects how you interpret the measure of central tendency and could very well change how you analyze your data. It can actually point you to understanding something new about your data. In the example about the heights, we learned that males and females each have their own distribution. Gender is a relevant variable in our analysis. That’s a fairly obvious example. However, in other cases, it might lead you to something that you didn’t already consider. It’s a bit like being a detective and looking for clues!
Thanks for the great question and good insight!
Hi Jim,
Yet another helpful article! I did have two questions:
1) It seems like a good tie-in to this article would be one that describes box plots and how to understand the information that they provide. Is there an article that you’ve written on box plots that could be linked to this?
2) Unless I’m mistaken, the central tendency of a distribution and variability around that central value tie into the concepts of accuracy and precision. Any chance that you could speak to those concepts in a future article?
Hi Chuck,
Thank you very much! Those are both great ideas too.
I definitely plan to write a more comprehensive post about how the various aspects of distributions work together–which would be a natural place to show box plots. I haven’t written that yet but it is on my list of things to write about this spring.
As for accuracy and precision, we definitely have very specific definitions for those terms in statistics. While in everyday English they are often considered synonyms, in statistics they’re very different. And, you’re correct, they do tie into those two concepts. These terms often come up in measurement system analysis.
If you measure parts repeatedly and the average or central tendency of the measurements are unbiased (on target on average), you have an accurate measurement system. However, if the measurements are biased (systematically too high or too low), your measurement system is inaccurate.
If you measure the same part multiple times and the variability between measurements is low, your measurement system is precise. However, if the measurements vary quite a bit, your system is imprecise.
You can have any combination of accuracy and precision. Accurate and precise. Accurate but not precise. Not accurate but precise. Neither accurate nor precise.
Very informative
Thank you, John!
Thnks a lot …..