When to Use Mean, Median, or Mode

Arithmetics mathematical distributions mathematics

The mean*, mode and median are the most well-known and widely used arithmetic averages. But what exactly are each used for? While all of them are measures of central tendency in a distribution of values – obtained from different kinds of measurements or observations – the question of when to use mean, median or mode is not properly understood even by seasoned practitioners and statisticians.

If you’ve ever wondered when to use mean, median, or mode, it’s time to answer these questions.

What is Mean, Median, Mode?

The mean, median, and mode are all summary statistics describing a property of a set of values. Each of them may be called an “average” depending on the context, but typically – unless otherwise specified – “average” refers to the arithmetic mean. In this article “average” is used to refer to all three as a group.

All of the three averages are lossy, meaning that information is inevitably lost in the process of compressing a whole distribution into a single number.

The mean, median, and mode are lossy but often convenient summaries of the central tendencies of a set of values.

Mean

The arithmetic mean is defined mathematically as the sum of all values divided by their count. It has the useful property of minimizing the mean squared error, meaning that the mean value is such that it is the point of balance of the distribution. It exactly balances the sums of distances of all numbers from itself. In this sense it is the single best predictor of the set, so if you were tasked with guessing what a randomly drawn value from the distribution is, choosing the mean would minimize your error in either under or overestimating the true value.

Median

The median, on the other hand, is the value which splits the distribution in two equal parts. The number of members of the set to the left of the median equals the number of members of the set to the right of it. It has the property of minimizing the average distance between itself and each number of that set.

Mode

The mode is the value which occurs the most times out of all the values of a set. If you want to know which is the most frequent value, the mode is the answer. It should be noted that a distribution might have multiple modes if many values occur with the same frequency.

For a very in-depth explanation of the three concepts, including simulations, consider this article.

When to Use Mean, Median, or Mode? – The Textbook Approach

Generally, most articles and textbooks on mathematics and statistics recommend using the median for most purposes. Some even go as far as saying it is “the best average” due to it being least affected by adding new values to a data set. For example, adding the number 100 to the set [1,2,3,4,5,6,7] only shifts the median from 4 to 4.5 while the mean jumps from 4 to 16.

The mode is less often recommended even though it is even less affected by adding new values, on average, since a distribution can have multiple modes.

The arithmetic mean is only recommended as a good average to use if a distribution is normal or close to normally distributed. Its use is discouraged in the presence of high skewness. This appears to be mainly on the grounds of how often the mean lies in the range where the majority of values in a distribution are, as well as how it responds to new values further from where the majority of values lie. Fancier wording is that the mean is very sensitive to outliers hence one should use an average which is “more robust”, like the median.


You can use our mean, median, and mode calculator for quick estimation of these summary statistics of a data set.


Examined closely, these general recommendations amount to only using the mean when it is identical or nearly identical to the median. Therefore, most textbooks could have just as well said that one should use the median in every case.

This attitude seems to stem from a mistaken implicit assumption that when looking at an average, most people are interested in where most values lie, instead of what each of these averages can convey. Some interpret measures of central tendency as representing values “typical” for the distribution, but no such quality is present in the description of the mean or the median, and it applies to the mode only in the narrowest sense. Such qualities only arise when the distribution is normal-like, or at least symmetrical around the mean, and this may or may not apply to any given case.

The Proper Way to Decide When to Use the Mean, Median, or Mode

The textbook recommended approach often “forgets” that what is of primary interest is whether the chosen statistic corresponds to the question posed to the data or not. Sensitivity to outliers and other characteristics are of secondary concern, if at all.

Hence, a proper choice of descriptive statistic should be informed only by the question asked of the data it summarizes. It does not matter how robust to “outliers” or how close to the majority of the values it is.

The best way to understand when to use mean, median, or mode is by looking at a real-life scenario. For instance, let’s assume an analyst is examining a country’s car market and works with a distribution of prices paid by new car owners in a given period, say, a year. For the purpose of illustration, assume the following dataset of forty sales in total, with the following prices:

Car sales ordered by sale price
Car sales ordered by sale price

Plotting the distribution of prices results in the following histogram:

Histogram (distribution) of sale prices
Histogram (distribution) of sale prices

Here, the mean is $47,368, the median is $35,500 and the mode is $30,000. They all fall within the first bin of the histogram in the graph above.

When the mean answers the question

If an analyst wants to determine the price which is the closest to all the prices paid, then the only statistic which answers that question is the mean. It is often what one is interested in since the mean reflects many different changes to buying behavior, be it more purchases of cheaper cars, purchases of fewer but more expensive cars, etc. Comparisons of means across periods, segments, markets, etc. are therefore quite informative.

Neither the median nor the mode can provide the answer to the above question, nor would they reflect changes in the underlying distribution in that same responsive way.

When to use the median

In another inquiry using the same data, one might be interested in what is the price point that half of the consumers are not able or willing to pay above. This would be answered by the median and no other average will do.

When to choose the mode

If, instead, one is interested in the price most commonly paid for a new car, then neither the arithmetic mean nor the median can answer that question, only the mode does.

Note that in the above example we have the same data, but different questions and the choice of the average to use was guided entirely by the question posed, and not by the distribution of the data which is of course one and the same.

When to use the mean, the median, or the mode depends entirely on which of them (if any) answers the question being examined with the data.

The above example should make it obvious that there is no such thing as the “best average” or “best measure of central tendency”. There are measures fitting the question, a.k.a. appropriate averages, and unfitting or inappropriate ones.

When Neither Average Does The Job

Examining just the mean, median, or mode may be sufficient in many situations. However, since these summary statistics are lossy, in some inquiries neither of them would contain sufficient information to answer the question of interest.

When that is the case, one can resort to using quantiles or frequency cut-offs such as percentiles. For example, one might be interested in changes of the mean price in each quintile or in the price that 95% of buyers are paying below ($250,000 in this data).

Average sale price by quintile
Average sale price by quintile

Sometimes the price range of 90% of new car sales might be what’s of interest. In this case that is between $27,000 and $200,000. The upper and lower values of the quantiles can also be useful:

Price ranges by quintile, bottom to top
Price ranges by quintile, bottom to top

In a year over year analysis a comparison of the whole distribution of current year sales versus last year’s sales might be the most informative, but the outcome of such comparisons is difficult to communicate.

Takeaways

After all this, how does one answer the question “When to use the mean, median, and mode“?

The proper average to choose is informed entirely by the question being asked of the data under analysis. The interpretation of the chosen average should match the question at hand. The mean minimizes the average distance to all points, the median splits the set in two halves, and the mode (or modes) is the most frequently occurring value. If the question corresponds to one of these qualities, the appropriate average is self-evident.

Switching from an appropriate average to an inappropriate one based on considerations such as how influenced by “outliers” it is or how close it is to the majority of the values in a distribution makes no sense. The temptation to introduce such considerations hints at an inadequate transformation of the business question.

Sometimes an inquiry cannot be satisfactorily answered by any average, be it the mean, median, or mode, due to their inherently lossy nature. Quantile ranges and their means, as well as percentiles, while also lossy, provide a different balance between interpretability and information retained that may often be appropriate. In yet other situations frequency ranges or even whole distribution plots might be required, depending on the task at hand.

* The article will use mean as a shorthand for “arithmetic mean”. Other means such as the geometric mean and the harmonic mean are not subject of the current piece.

This entry was posted in Mathematics and tagged , , , , , . By Georgi Georgiev