We somehow want to summarize a bunch of values giving a single number (technical term: a statistic). That neccesarily means to ignore (usually) a lot of information that is in the data. We are willing to pay that price for having a simple, easy-to-communicate number that highlights (or focuses on) a particular aspect of the data.

To file an extreme case: consider the data is the win in a game of chance that you played for a while. Say, in 20% of the cases you won 100 (dollars, say), in 45% cases you lost 20, and in the remaining 35% of the cases you lost 10. If you want to report your winnings in this game giving a single number to your friend, you can say, for instance:

i) the mean win: 7.5

This is what your friend should expect to win, based on your experience/data. It tells us that the game may not be fair, with an advantage to win (one could perform a statistical test, testing the data under the hypothesis of a fair game and see how (un-)likely the data is under that hypothesis).

ii) the median win: -10 (a loss, actually)

This tells your friend that in most cases you lost at least 10.

iii) the modal win: -20 (also a loss)

This tells your friend that in most cases lost just 20.

Mode is used with categorical data (the most frequent category).

Median is used with ordinal data, and is always relevant.

Mean is interesting, easy to compute but not always relevant to describe a set of data.

https://en.wikipedia.org/wiki/Central_tendency

For instance:

- the mode is interesting to know what is the most frequent iris colour in a population.

- the mean is interesting to know what is the average grain weight of your sample.

- and the median will be probably more interesting to describe the "central" wages in the research institute you work for, or even at the world scale. Indeed, the median will identify the wage which divides the workers as follow: i) 50% of the workers earn less than the median, ii) 50% of the workers earn more than the median. The arithmetic mean is strongly influenced by extreme situations.

https://www.theguardian.com/inequality/2017/nov/14/worlds-richest-wealth-credit-suisse

For the central tendency alternatives, see the Wikipedia link.

https://en.wikipedia.org/wiki/Central_tendency

But central tendency is generally not a sufficient description of your data, I hope you also include dispersion indication (number of modes, interquartile range, standard deviation).

https://en.wikipedia.org/wiki/Statistical_dispersion

We somehow want to summarize a bunch of values giving a single number (technical term: a statistic). That neccesarily means to ignore (usually) a lot of information that is in the data. We are willing to pay that price for having a simple, easy-to-communicate number that highlights (or focuses on) a particular aspect of the data.

To file an extreme case: consider the data is the win in a game of chance that you played for a while. Say, in 20% of the cases you won 100 (dollars, say), in 45% cases you lost 20, and in the remaining 35% of the cases you lost 10. If you want to report your winnings in this game giving a single number to your friend, you can say, for instance:

i) the mean win: 7.5

This is what your friend should expect to win, based on your experience/data. It tells us that the game may not be fair, with an advantage to win (one could perform a statistical test, testing the data under the hypothesis of a fair game and see how (un-)likely the data is under that hypothesis).

ii) the median win: -10 (a loss, actually)

This tells your friend that in most cases you lost at least 10.

iii) the modal win: -20 (also a loss)

This tells your friend that in most cases lost just 20.

For my money, the mean is most meaningful for data with a symmetric distribution. The mean is moved by skew and outliers, making it a less robust statistic to these effects. That is why we often use the median for skewed data like income. The mean is also meaningful only for interval or ratio data (not ordinal data unless you make assumptions about the spacing between the categories).

The median is robust to skew and outliers. It can also be used for ordinal data.

The mode is rarely used in practice. As @FrançoisCollin points out, it could be used to describe the category with the highest frequency, but I think normally you would just call this e.g. "the category with the highest proportion".

There are a lot of other measures of central tendency. I'll just include some off the top of my head. These might be used in specific situations or as robust statistics.

• trimmed mean and winsorized mean - means that remove extreme values to be more robust to outliers
• M-estimators for central tendency (incl. Huber) - robust for skewed distributions and outliers
• geometric mean - best for log-normal data. somewhat robust to skew and outliers without ignoring them. In the U.S., regulations for bacteria in water are based on the geometric mean. Also for average annual return for investments.
• harmonic mean - used for averaging rates.

Source: http://rcompanion.org/handbook/C_02.html .

@Salvatore,

i) Interesting, IMHO the mode is a far, far better representation of a "typical income" than the median...

ii) the category with the highest frequency is the mode. That's the definition.Wikipedia writes "The mode of a set of data values is the value that appears most often", citing the Encyclopedia of Mathematics. For a categorical variable, the values are categories (or are defined as belonging to a particular category). Having 5x "yes" and 3x "no", then "yes" is the mode.

@Jochen,

I think a problem I have with using mode for numeric data is that it requires the data to be binned in a reasonable manner. Case 1, if there are few data points, it might be that one comes out the mode just by chance even though it doesn't reflect the central tendency of the data well. Case 2, if there are many data points, but they aren't binned. For example, if you were reporting income to the nearest dollar it might be unlikely that there are two values exactly the same even for a reasonably large number of observations. Again, any observation could be the mode if there just happened to be two or more observations with the same value. Furthermore, the mode could change depending on the way the data are binned.

For categorical data, I just don't see the term "mode" used much. There's nothing theoretical here, or important. I think if the question is "Which of these five television shows is your favorite?", I don't see someone saying, "The mode response was 'Game of Thrones'". I see someone saying "The most common repsonse was 'Game of Throne', [with 25% of responses]". That's all.

I think the idea of using the mode for something like income is interesting. I did some googling and couldn't find anything reporting income in this way. Consider U.S. household income in 2010 (https://upload.wikimedia.org/wikipedia/commons/0/0d/Distribution_of_Annual_Household_Income_in_the_United_States_2010.png). The reported median is $50,000. But the mode is clearly $15,000 - $19,999. Which is a better measure of e.g. "typical"? I'm not sure. I think I would continue to use quantiles as indicated on the plot (median, 75th percentile, 90th percentile). But it does make a compelling case to indicate the mode category as well.

With all due respect to all of the respondent, I believe the correct conclusion is for the researchers to use the central tendency that most accurately depicts the data. Mean does not depict the “story” in nominal data or in abnormal distribution (skewed, with drastic outliers). One must be honest in their representation of the story and not mislead through misuse of central tendency.

Regards,

Christopher

very good information regarding statistics

I think you got that conclusion exactly wrong. The mean is best for unskewed data with no outliers. The median can tolerate skew and outliers. And there are many other statistics you can use to describe the location of the data. We discussed the mode, and I listed others that are useful.

Thank you Salvatore S. Mangiafico;

"The mean is best for unskewed data with no outliers." and

"The median can tolerate skew and outliers."

My question is what would be much better and more adaptive than the mean and median for a data set of n elements?

Best regards.

There is nothing "more adaptive". A single number can represent a single aspect of the data. To provide more information about the data you need to give more than one number. You could provide several quantiles, so that the reader could reconstruct the approximate distribution. Or, you could provide a distribution model together with the values of the parameters that specify the model (possibly the most condensed way to give the most complete info about the distribution). You will only have no information loss when you report all the n values.

The last advice sounds a bit strange - but actually there shouldn't be any problem today. We use electronic media to measure and to store data, we write manuscripts electronically and we submit them electronically to journals, that usually allow submitting any amount of supplementary material - also electronically. There is no reason not to submit the original data (that is, all n values). It will be less improtant then how these values were summarized in the text or in a figure of the main article, as the reader can easily have a look at the data itself.

There is an additional problem, which is one of reporting.

When you say "the average bed stay was 13 days" what do people understand by this?

In fact, people in general either cannot explain the mean at all or, worse, explain it wrong. For this reason, it's not a good idea to use it if you intend to communicate anything about your data to people without a statistical background.

The median is easy to explain. And, furthermore, you don't have to most of the time. If you say "half of the patients were discharged in 4 days or less", you don't even have to use a technical term (median) so people won't even get that chilly feeling of disempowerment that they usually get when they hear a stats word.

And the mode, as Salvatore points out, is really only suited to coarse grained data. And again, you hear it all the time in plain language:

"The commonest cause of dysentery is rotavirus". It's amusing that most people using the mode don't realise they are doing so because they can express their conclusion in plain language (as opposed to "the modal cause is rotavirus").

Jochen's example is thought-provoking. Here, the expectation is that you gain 7·5 dollars for each play, but half of the time you lose 10, and the commonest outcome is to lose 10. Which statistic do you need? In this case, because we are talking about long term expectation, it's the mean, isn't it? The others describe the individual trials, while the mean summarises the process. Cunning, Jochen!

@Ronán,

it's always a pleasure reading your posts.

I just want to comment that one can well give a mode of the distribution of a continuous variable. Note that the famous normal distribution is said to be a uni-modal distribution (because it has a single, defined mode, which here coincides with the mean and with the median). Even for "empirical distributions" one can use a kernel density estimate to find the modes.

@Jochen Wilhelm

The data vary according to certain parameters, but they come in sets of 8, and there could be occurrence of zeros, an example,

2 -10 -9 -8 -9 3 4 2

They are all centering constants.Each will be used for different type of data distributions. For normally distributed data Mean will be an ideal measure of central value. where as for a skewed data Median will be an ideal measure of central value(ordinal data) . Mode less commonly used and for even nominal data mode can be established.

The five common measures of central tendency are

arithmetic mean, geometric mean, harmonic mean, median and mode.

The term "mean" usually refers to the arithmetic mean.

Besides these five, there are more measures of central tendency. Some of them are

root mean square, squared mean, square root mean, cubic mean, cube root mean, generalized p mean, generalized pth root mean, mid range, weighted mean etc.

There are lots of measures of average. Commonly used five averages are arithmetic mean, geometric mean and harmonic mean, median and mode. The term "mean", usually refers to arithmetic mean.

Besides these five there are many measures of average.

Depending upon the type of data and upon the nature of what is wanted to determine, measure of average is to be selected.

Besides mean, median and mode there are many measures of average that can be used as measures of central tendency of data.

For the other measures of average, go through the following papers:

1. Dhritikesh Chakrabarty (2018): “Derivation of Some Formulations of Average from One Technique of Construction of Mean”, American Journal of Mathematical and Computational Sciences, 3(3), 62 – 68. Available in http://www.aascit.org/journal/ajmcs.

2. Dhritikesh Chakrabarty (2018): “One Generalized Definition of Average: Derivation of Formulations of Various Means”, Journal of Environmental Science, Computer Science and Engineering & Technology, Section C, (E-ISSN : 2278 – 179 X), 7(3), 212 – 225, Also available in www.jecet.org .