Question

I know this question is difficult to understand but I need a mathematical equation for javascript.

I am having dynamic number from 0 to 7.

Now I am having 5 records:

a = from 0 to 7,
b = from 0 to 7,
c = from 0 to 7,
d = from 0 to 7,
e = from 0 to 7

Now i need an equation from which i can find possible output of a,b,c,d and e that makes average 5.

Like for average = 5:
1. a = 5, b = 5, c = 5, d = 5, e = 5
2. a = 2, b = 7, c = 6, d = 5, e = 5

User will enter desired average and i need to throw possible outputs to make filled average.

Answer 1

This seems to be a problem related to Integer Programming. It can be solved relatively efficiently using a Dynamic Programming strategy that maintains an invariant for smaller sub problems and merges these sub problems into a final solution. Here’s a high-level algorithm that gets you to your goal:

1) Start with a random seed number, x_1 between 0..7

2) Now do the following three times (i = 2, 3, 4) :

3) Find the minimally possible next number. To get an average of 5 for your five numbers, your target sum for all five is 25. So you need to ensure that you don’t deny yourself the ability to reach 25 at all. So let min_i+1 determine the smallest number of 0..7 so that the sum of all x_i so far plus min_i+1 * (5-i) >= 25.

Example: Say i=2 and your numbers so far were 4 and 5. Then the minimally allowed third number min_3 is 6. The sum so far is 9 and 5-i = 3. 9 + 3*6 = 27 >= 25 and 9 + 3*5 = 24 < 25. This means if you chose 5 as min_3, you would no longer be able to reach your goal of a sum of 25.

4) Choose x_i+1 as a random number between min_i+1 and 7.

5) Finally, choose x_5 as 25 – sum of x_1..x_5. Output x_1 to x_5.

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Example for the entire algorithm:

Let x_1 = 6.

=> min_2 = 5, sum = 6

Let x_2 = 5.

=> min_3 = 5, sum = 11

Let x_3 = 7.

=> min_4 = 4, sum = 18

Let x_4 = 5

=> sum = 23

=> x_5 = 2 (=25 – 23)

Output: [6, 5, 7, 5, 2] which indeed has an average of 5.

Now this algorithm will produce very biased output, if you want to make it appear more randomly, simply serve a permutation of the final result, e.g. in our example you could output [5, 7, 2, 5, 6] instead.

It should be no problem to implement this in Javascript and it can be easily adapted to suit other possible ranges for x_i and the total average. Just be sure to modify the target sum as n * avg if your final sample size should be n with an average of avg.