To figure out the probabilities exactly, it is necessary to make the assumption that the distribution is normal. Given normality and the formula for the standard error of the mean, the probability that the mean of 5 students is over 580 can be calculated in a manner almost identical to that used in calculating the area under portions of the normal curve.

Since the question involves the probability of a mean of 5 numbers being over 580, it is necessary to know the distribution of means of 5 numbers. But that is simply the sampling distribution of the mean with an N of 5. The mean of the sampling distribution of the mean is μ (500 in this example) and the standard deviation is = 100/2.236 = 44.72. The sampling distribution of the mean is shown below.

The area to the left of 580 is shaded. What proportion of the curve is below 580? Since 580 is 80 points above the mean and the standard deviation is 44.72, 580 is 80/44.72 = 1.79 standard deviations above the mean.