# 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

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?

• How many students are there? Is it 15 as in your fake data? – soakley Sep 15 '16 at 13:46
• – Tim Sep 15 '16 at 13:50
• Are you not able to show them a boxplot of the scores? This allows them to easily visualize where their score lies without showing the scores themselves. – dsaxton Sep 15 '16 at 14:16
• yes, there are 15 students in my fake data. I only made that up to see if I could mimic the mean/media/std.dev – nycynik Sep 15 '16 at 14:28
• If I wanted to make a box plot, how would I calculate the Q1 and Q3 numbers? support.microsoft.com/en-us/kb/155130 – nycynik Sep 15 '16 at 14:34

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.

• Why did you pick x7 and x8? Is that because it is the middle of the results? – nycynik Sep 15 '16 at 21:24
• I picked the threshold of 65 first. Then I found the two data points that give us a bound on how many students could have failed based on that threshold. I made some other calculations, but I am only showing you the two that matter. If I had chosen a different threshold, then I would have presented results for whatever data points were appropriate to that value. – soakley Sep 16 '16 at 13:28

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.

• I'm not sure what that means exactly. Do you mean difference as in mathematical difference? 80-60=20, and 100-80=20? – nycynik Sep 15 '16 at 14:31
• I think you don't mean what you say. The property you are invoking defines interval data. not ordinal data. Also, the question is muddier as many people average ordinal data (what is a grade point average, otherwise?) and think it's defensible, whereas many measurement theorists argue against that. Some discussion at e.g. stats.stackexchange.com/questions/67551/… – Nick Cox Sep 15 '16 at 16:38
• I just confused the nomenclature and edited it. Let's assume that on a scale from 0 to 100 you pass with 60. It makes a huge difference if half of the students scores 70 and the other half scores 80 points or if half of the students gets 50 and the other half scores 100 points. As in the latter case half of the students fail while in the first case all students pass the course. Maybe a percentage of people above below a certain value, e.g. "pass or better" would be also useful inference from the data. – Ferdi Sep 16 '16 at 6:41