Question

The normal distributions for three different SAT examinations is displayed above what can you conclude about the standard deviations of the three exams?

Answer 1

Comparing the standard deviations of each distribution, you have blue < red < yellow.

For any normally distributed random variable X with mean µ and standard deviation σ, we have the property that

Pr[|X - µ| ≤ σ] ≈ 68%

In other words, roughly 68% of a normal distribution lies within 1 standard deviation of the mean. This means the plot of the density of a distribution with a smaller standard deviation forms a tighter peak (like the blue curve), whereas a larger standard deviation would be indicated by a more dispersed, flattened curve (like the yellow one).