What constraints does Std Deviation, Mean and Median put on the data? I know there is no correct answer to this problem, but it seems a very common question among students who receive their own grade and want to be able to understand what it means.
They know (with example data):


*

*lower bound: 0 

*upper bound: 100 

*Std Dev: 23.32 

*Mean: 87.07 Median: 95

*Their own grade: 98


And they want to visualize the data in some way, in order to understand, did I crush it, fail hard, or pretty much do the same as the rest of the class.
I understand that I can not render a graph fro the data
But I did make some fake data that matches this data.
Student score
1   87.07
2   95
3   95
4   95
5   95
6   95
7   95
8   95
9   0
10  90
11  92
12  80
13  95
14  98
15  98

avg: 87.00466667
Median: 95
Std Dev: 23.67277561
My question is, what constraints does standard deviation put on the data?  Can that help me understand how many students may have failing grades for instance?  
I think I completely understand that Median is the middle, so to have a very high median like the sample data would mean most students are scoring very high.
I think I understand Mean also, the average of the students is also pretty high.
Is that correct?
 A: Data does satisfy certain constraints involving the sample standard deviation and the sample mean. So, if $x_1$ is the minimum data point and $x_n$ is the largest (order statistics), then the general version of Samuelson's inequality states $$\bar{x}-\sqrt{{n-k} \over k} \ s \leq x_k \leq \bar{x}+\sqrt{{k-1} \over {n-k+1}} \ s,$$ where $n$ is the number of data points and $s^2$ is calculated using a denominator of $n.$ 
So, for example, the minimum data point, $x_1,$ must be at least as large as  $\bar{x}-\sqrt{n-1} \ s$
This can help you put a bound on how many students failed. Using your first data set (and assuming $n=15$) we find $x_7 \geq 62.14$ and $x_8 \geq 65.26.$ Therefore, if 65 is a failing grade, at most 7 students failed. 
You have the additional constraint of scores being in the range from 0 to 100 so you can find a better bound than this. 
A: One constraint is definitely that mean, variance, etc. consider a dataset  to be interval data. Hence by taking the mean you assume that the difference between a student scoring 60 and a student scoring 80 is the same as between a student scoring 80 and a student scoring 100.
