# Mean or median when data interpretation

When doing results treatment, sometimes the mean values agree more or less with the median, but sometimes not. Thus using one or other value may change the final results. Then, is there any criteria that could be followed to know when it is better to use one or the other?

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Suppose you measure heights of grown male persons. Let $X = \{x_1,\ldots,x_n\}$ be your ordered measurements (i.e. $\forall x \in \{1,\ldots,n-1\} : x_i \leq x_{i+1}$).

The mean is

$mean(X) := \frac{1}{n} \sum_{i=1}^n x_i$

and the median is

$median(X) := \begin{cases} x_{(n+1)/2} & n \text{ odd} \\ \frac{1}{2}(x_{n/2} + x_{n/2 + 1}) & n \text{ even} \end{cases}$

So in general the mean incorporates all measurements (even single distortions) while the median is more resilient against distortions. A general rule when to use which is hard to tell I guess. Depends on what data you have. Sometimes it might be even senseful to combine both ideas: First strip off the lower and upper quarter and then just calculate the mean of the two inner quarters. It just depends...

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Generally no. There is no best statistical measure of data.

The median can be useful when for example comparing salaries, because a few high-income people will increase the average and make it look like the socio-economic status of the employees is higher than it really is.

The mean (average) value is often good for normal distributed data, but not always.

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(i) General approach + (ii) Exercise to grasp the difference:

(i) Just play with them, keeping the definition in mind to understand what their relation is and what they singularly mean and then try to give an interpretation of the usefulness of each in a particular problem (I learned statistics applying interpretation rules, but one day I realised (hope correctly) that statistics is math, which is not real life, so every index does not tell you any truth about reality, its up to you to give a reasonable interpretation).

(ii) I think a good example for intuition of their difference can be:
let N={1,2,3} be a set of three individuals, and their respective weights be $W=\{90,10,20\}$, the mean $\mu_W$ of the weights is $\mu_W=40$ but what is the median $m_W$ ???

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