The empirical rule says that 95 of the population is within

The empirical rule says that 95% of the population is within 2 standard deviations of the mean, but when I find the z-scores that mark off the middle 95% of the standard normal distribution I calculate -1.96 and 1.96. Is this a contradiction? Why or why not? In other words why are the normal distribution calculators not agreeing with the empirical rule?

Solution

As you can see, -2 and 2 are close to -1.96 and 1.96.

The empirical rule is just an approximation (for practical purposes), so that explains the fine difference.

Anyway, they agree (they do not contradict), we just have a round off error here (for practical purposes).

[Note: The more exact area of P(-2<z<2) = 0.9545, which rounds to 0.95 to 2 decimal places.]