Where is $x=\frac{a+b+c+d}{4}$ in $0<a<b<c<d<e$?

Question

Where is the exact position of $x=\frac{a+b+c+d}{4}$ in $0<a<b<c<d<e$?

The real problem

I have a box-and-whisker plot. It shows

• $L$ : the min value.
• $Q_1$ : the lower quartile.
• $Q_2$ : the median (a.k.a the second quartile).
• $Q_3$ : the upper quartile.
• $R$ : the max value.

I think we cannot find the exact position of the mean $\bar{x}$ but I am interested in finding the tight bound in which the mean can be.

I guess the bound (it might not be so tight) is

$$ \frac{L+Q_1+Q_2+Q_3}{4}\leq \bar{x} \leq \frac{Q_1+Q_2+Q_3+R}{4} $$

That seems to be useless as well.

Answer 1

You can’t tell; the most you can say is that x is between a and d. Also i don’t see how point e is significant.

One way to think about what it means geometrically is to take the center of gravity of the points; imagine putting 4 equal sized weights at a b c d on a number line.

Answer 2

It is not possible to say. It depends on the actual values of $a,b,c,d$. Depending on the dispersion of $a,b,c,d$, the average $(a+b+c+d)/4$ can be anywhere between $a$ and $d$.