What is the variance of a discrete uniform distribution on {1,2,3,4,5,6}?

• A. 7
• B. 6
• C. 35/12
• D. 1

Answer: (C)

Variance tells us how far the values tend to be from the average value. For a discrete uniform distribution that takes each integer from 1 to 6 with equal probability, the mean is the middle value: (1 + 6) / 2 = 3.5. The variance comes from averaging the squared differences from that mean: (3.5−1)², (3.5−2)², (3.5−3)², (3.5−4)², (3.5−5)², (3.5−6)². Those are 6.25, 2.25, 0.25, 0.25, 2.25, 6.25. Sum them to 17.5 and divide by 6, giving 17.5/6 = 35/12 ≈ 2.9167.

A quick formula check for a discrete uniform on n consecutive integers is (n² − 1)/12. With n = 6, you get (36 − 1)/12 = 35/12, matching the calculation above.

So the variance is 35/12. The other numbers don’t fit the actual spread of these six equally likely values.