# Is considering a specific distribution necessary before computing an average?

Many times, I compute averages of variables without considering distributions at all, and I use those computed averages to represent measures of variables in my data without mentioning specific distributions.

For a specific example, I compute the average purchase count for ecommerce users' purchases but I don't think about a specific distribution for that average.

So I don't know what I am doing is right or wrong because I compute an average without considering a specific distribution, so I have no idea about what distribution the average belongs to.

Do I have to think about a specific distribution before computing an average of a variable? or I can just compute the average use it without considering any distribution?

• An average (e.g. arithmetic mean) is just the output of an equation. It has meaning regardless of how the data is distributed. The distribution of the data is usually important when you are considering the use-case. For example, simulating speeds around the average would require a distribution that is positive. Or if you are doing a hypothesis test, then the distribution of the data will inform you how to do the test. Commented Jul 18 at 6:45

You do not need to know the distribution of the average to use it. Suppose the observations are assumed to be independent and identically distributed. In that case, the estimated expected value of the next observation equals the mean, which is the only usage of the mean without other information.

Sometimes, the reliability of the mean needs to be considered. Different samples from the population will generate different mean, which gives information about the population mean. In this case, the distribution of the mean is useful for further use, like computing the likelihood of a positive mean, or you need to use this mean to price a product.

The mean (average) is a descriptive statistic, just like the variance (standard deviation), skewness, or the coefficients of a linear regression, etc. They are just the mathematical results of applying an equation to the data sample(s). As such they do not require any assumption on the distribution. These observed statistics depend only on the sample, and not on the population from which the sample came, and hence do not care about the population’s distribution.
The only assumption for computing observed statistics is that your data is i.i.d. Independent (a value in your sample is not influenced by the other values: otherwise, e.g. a mean is not meaningful, as its value will depend on the first sample you took, and may not converge). And Identically Distributed (otherwise, you are literally taking the mean of apples and oranges; again, not a meaningful description of the data).
Now, when you are trying to make inferences about the population based on the sample (and its observed statistics), you may need to make assumptions about the population’s distribution. For example, if you not only compute the mean, but also a 95% CI around it (this will require normality of the sampling distribution of the mean), or CI’s around the regression coefficients (normality of the residuals – errors), etc. Same for deciding if 2 means are equal, or statistically significantly different (normality of the samples).
Note that there are some techniques which do not require any assumption about the parent population distribution (distribution free techniques), such as e.g. the bootstrap. There are also other techniques which rely on a given distribution, but that population distribution is guarantied by definition (e.g. binomial CI of a proportion: that proportion does, by definition follow a binomial distribution – no need to assume, or specify this).
(NOTE: I said earlier that observed statistics are valid regardless of the parent population’s distribution, as long as your sample is i.i.d. This is technically not quite true; that particular statistic needs to “exist”. E.g. the Cauchy distribution is such a pathological distributions for which the mean and variance are undefined; so trying to compute them is not meaningful. However, in practice -applied statistics-, I would not worry about such a situation, which you have almost no chance of encountering).

• The only assumption for computing observed statistics is that your data is i.i.d. Independent - I don't agree with this. I think you're confusing the application of the statistic, with the statistic itself. You can calculate the sample mean on anything - a simple random sample, a time series, clustered data, etc. It's the application in which you need to consider things like independence. Commented Jul 19 at 6:26
• @Alex, but if not i.i.d, that statistic does not mean what you think it does. E.g. the arithmetic average can indeed be computed even if not i.i.d, but it iwill not be the expectation of the population, it will not be the "center" of the population, etc. It is computable, but it has no usable meaning. If you average non-i.d. values, what is the meaning of this value? How much information does it carry? Commented Jul 19 at 16:55
• That depends on the context of your data. For example, if you do a clustered survey design, your data is not independent. The way you estimate the mean is still exactly the same, but the way you calculate the variance on the mean is what changes. Commented Jul 22 at 23:05

The other answers are correct about the validity of computing the mean. However when you "use those computed averages to represent measures of variables", there is a caveat.

Let's say the variable you are measuring is heavily bimodal. Then, you compute the mean and standard deviation and represent it as a bar plot. The calculation is perfectly correct, but misleading: giving only the mean of a bimodal distribution is essentially a lie. You should prefer a violin plot that shows the bimodality.

Similarly, if your variable is strongly skewed, the mean can be misleading, maybe the median and a boxplot would be better suited.

So, for representation and interpretation purpose, the mean is fully relevant if your distributions are roughly normal (not too skewed, bimodal, etc). You should always look at the full distribution, and think about the best representation for your dataset.