Graph median, mode, range

Question

The question is:

A scientist collects data. He determines both the mean of the data and the median of the data are equal to 7 and the data are symmetrical about this value. He starts to create a bar graph shown, but does not finish the graph.

If the range of the data is 8 and the maximum value of the data is 11, then how many data points fall above the value 7?

The answer is 14, but I don't know how this is the answer.

Answer 1

Use the fact that the data are supposed to be symmetric about $7$. So fill in stuff to make it symmetric, completing what the scientist did not get to finish. The symmetry automatically makes the mean and median equal to $7$.

For symmetry, we need a black bar of height $7$ at Value $8$, to balance the black bar of height $7$ at Value $6$. We also need a black bar of height $4$ at Value $9$, to balance the black bar of height $4$ at Value $5$.

We also need a couple of short black bars on the left, to balance the short black bars at Values $10$ and $11$. These short black bars are at Values $4$ and $3$ respectively.

We already have range $8$, So there is no more stuff to fill in. Now add up the heights of the black bars to the right of Value $7$. We get $14$.

Answer 2

The range is $8$, max $11$, and it's symmetric around $7$. So we know we have at least:

1. one $3$ (giving us a range of $8$, or alternately because there is an $11$ and they symmetry)
2. two $4$'s (from the symmetry, as there are two $10$'s)
3. four $9$'s (from the symmetry, as there are four $5$'s)
4. and seven $8$'s. (from the symmetry, as there are sevel $6$'s).

Checking, it turns out that this gives the correct range, median, mean, and symmetry, so we see fourteen (seven $8$'s, four $9$'s, two $10$'s, one $11$) above $7$.