# What is the difference between “mean value” and “average”?

Wikipedia explains:

For a data set, the mean is the sum of the values divided by the number of values.

This definition however corresponds to what I call "average" (at least that's what I remember learning). Yet Wikipedia once more quotes:

There are other statistical measures that use samples that some people confuse with averages - including 'median' and 'mode'.

Now that's confusing. Are "mean value" and "average" different from one another? If so how?

• The mean you described (the arithmetic mean) is what people typically mean when they say mean and, yes, that is the same as average. The only ambiguity that can occur is when someone is using a different type of mean, such as the geometric mean or the harmonic mean, but I think it is implicit from your question that you were talking about the arithmetic mean. – Macro Aug 10 '11 at 15:43
• For more information about means, the different kinds that exist, & how they are related to each other, see this excellent CV question: Which "mean" to use and when? – gung - Reinstate Monica Oct 30 '12 at 18:59
• Mean, or Expected Value - is a theoretical property of a certain probability. Average is the observed/measured outcome of a certain sample. If a measured average diverge too much from the expected mean, it's a sign that the underlying probability assumption, or one of its properties, is wrong. This is the main distinction between the terms that statisticians use. – David Refaeli Jan 6 '18 at 10:35

### Mean versus average

• The mean most commonly refers to the arithmetic mean, but may refer to some other form of mean, such as harmonic or geometric (see the Wikipedia article). Thus, when used without qualification, I think most people would assume that "mean" refers to the arithmetic mean.
• Average has many meanings, some of which are much less mathematical than the term "mean". Even within the context of numerical summaries, "average" can refer to a broad range of measures of central tendency.
• Thus, the arithmetic mean is one type of average. Arguably, when used without qualification the average of a numeric variable often is meant to refer to the arithmetic mean.

### Side point

• It is interesting to observe that Excel uses the sloppier but more accessible name of AVERAGE() for its arithmetic mean function, where R uses mean().
• You can end up with bizarre conversations like this: "So we look at the average return..." "Which average do you mean? A median? A weighted average?" "The mean return." "Oh, okay"... and it seems like everyone understood each other, ... except that the first person may actually be talking about the geometric mean of the returns. I've seen it happen. – Glen_b -Reinstate Monica Mar 5 '13 at 23:15
• Which of this denotes $\mu$ ? – user486983 Jan 22 '19 at 23:31
• @Isa The symbol $\mu$ is generally used to refer to the arithmetic mean of the population ($\bar{x}$, on the other hand, is used to refer to the arithmetic mean of the sample). – Acccumulation Jul 19 '19 at 17:36
• @Acccumulation thanks – user486983 Jul 19 '19 at 19:47

There are several "averages." Just think of this trick question: "What is the probability that the next person you meet has more than the average number of arms?"

The "mean" or "arithmetic mean" or "arithmetic average" is one average that you learned in the past. But the median (the value with half the observations greater and half less than it), the mode (the most common value), the geometric mean (multiply the values then take the nth root), the harmonic mean (the reciprocal of the mean of the reciprocals of the data), and others all fall under the general term "average."

Mean and average are generally used interchangeably (although I've seen them used as referring to population versus empirical).

They, like median and mode, are measures of central tendency, but in many cases, the other two are different.

The mean you described (the arithmetic mean) is what people typically intend when they say "mean" and, yes, that is the same as average. The only ambiguity that can occur is when someone is using a different type of mean, such as the geometric mean or the harmonic mean, but I think it is implicit from your question that you were talking about the arithmetic mean

I see "average" and "mean" used mostly as synonyms. One author who draws a clear distinction is Donald Wheeler, in "Advanced Topics in Statistical Quality Control." He declares that the "average" is a statistic determined by some arithmetic procedure, whereas "mean" is a parameter, specifying location of a distribution. By way of example, he writes that one could calculate an "average" telephone number, which would be meaningless (pun?). The average (a statistic) is an unbiased estimate of the mean (a parameter).

• This is needs to be a much higher rated answer. Although the average is a unbiased estimate of the mean only in Asymptopia. 'Mean' can be used to refer to the expected value of any population. 'Average' is statistic an empirical function of the population which is sometimes, but not always a good point estimate of the mean. For certain distributions (e.g. lognormal) there may be a better point estimate of the mean which is not the average. – Dalton Hance Sep 20 '16 at 17:36