Estimating users z-score given avg, min, max for various tests Given a series of tests, where we are given one users score, the overall minimum, the overall maximum, and the overall average, how would I estimate the user's z-score on total score (ie. sum of all of the users tests scores compared to the average sum of all test taker's test scores)? 
We cannot assume that the lowest scoring person from one test was the lowest scoring in the next test, but I think it is fair to assume that people generally stay within some score bands (although if this can be done without that assumption, that would be better). We can also assume that the group of test takers is the same from test to test.
My intuition tells me that this seems to be some sort of application of Monte Carlo methods, but I can't seem to figure out how to actually do this.
Some example data:
+-----------+------------+------------+------------+------------+--------+
| test_name | usr_score  |    high    |    avg     |    low     | weight |
+-----------+------------+------------+------------+------------+--------+
| Test_1    | 0.94615385 | 1          | 0.92307692 | 0.65384615 |     26 |
| Test_2    | 0.71621622 | 0.95945946 | 0.79459459 | 0.74074074 |     37 |
| Test_3    | 1          | 1          | 0.92222222 | 0.7037037  |     27 |
| Test_4    | 0.85135135 | 0.97297297 | 0.85675676 | 0.66756757 |     37 |
| Test_5    | 0.83333333 | 1          | 0.76666667 | 0          |      6 |
| Test_6    | 1          | 1          | 0.92857143 | 0.66666667 |     21 |
+-----------+------------+------------+------------+------------+--------+

Given this data, we know the user's total score is 135.6 (usr_score $*$ weight). Similarly, the average score is 134.1, the maximum score one test-taker may have is 151.6, and the minimum score one test-taker may have is 102.1, although it is unlikely that one person has either the minimum or maximum score as one person probably didn't always score the best/worst. I'd like to calculate the z-score of the 134.1, but am unsure as to how to do that without the standard deviation.
 A: Suppose there are 100 students in all and that test scores are
normal. Then the range $W$ of scores has $E(W) \approx 5\sigma,$
where $\sigma$ is the population standard deviation of scores.
I estimated this mean from a simulation in R, as shown below.
[In R, range returns min and max, so one needs diff to get
the actual range.]
w = replicate(10^5, diff(range(rnorm(100))))
mean(w)
[1] 5.015726

So it might be reasonable to assume that individual students
have a standard deviation that is about $1/5$ of the range of
the 100 students.
Group size matters in such approximations. If the group/class
has only about 30 students, then 4 is a better divisor than 5.
w = replicate(10^5, diff(range(rnorm(30))))
mean(w)
[1] 4.083182

This method probably wouldn't work well for tests where the best students
have scores that are close to the maximum possible score. Then
the SD of best students might be much smaller than is typical
of the population.
Your response to @norvia's question leads me to believe you might
have data that would permit a regression approach. If so, you
might regress SD on range (max - min) and average scores across the whole group and see
if that helps to predict SD for an individual student in the group from
his/her average.
If you're trying to get an individual's SD from the range of 
only $4$ tests during the term, then divide the range by $2.$
w = replicate(10^5, diff(range(rnorm(4))))
mean(w)
[1] 2.059537

For very small normal samples, $S$ is well estimated as a fraction of $W.$

