Deviation and Mean

Question

The math scores for an exam for the state of Georgia are normally distributed with a mean of 400 and a standard deviation of 20. About what percent of the test takers received scores between 460 and 500?

Answer 1

Hi Baklie!

A normally distributed set of scores means that the scores follow a unique distribution pattern. A simple definition is that the normal distribution is symmetrical and is centered mostly around the mean. The left and right "tails" are much smaller compared to the center, meaning less of your sample size is represented as you deviate farther from the mean. This gives the distribution the classic bell curve shape.

We know that the mean of this distribution is µ = 400, and the standard deviation is σ = 20. Standard deviation (SD) is a measure of how the average value from a distribution varies from the mean. So here, a score of 460 would be 3 standard deviations to the right of the mean of 400:

400 (average) + 3 × 20 (SD) = 460

A score of 500 would be 5 standard deviations to the right of the mean:

400 (average) + 5 × 20 (SD) = 500

Since the data is said to be normally distributed, there's a special rule about the distribution of data. According to the 68-95-99.7 rule, 1 SD to the right includes about 34% of the data. 1 SD to the left and right includes 34% both ways, or 68%, hence the name. 2 SDs to the left and right includes 95% of the data.

Going 3 SDs to the left and right includes 99.7% of the data, leaving 0.3% of the data on the tails (or 0.15% on the left and 0.15% on the right). So a test score that is 3 or more SDs above the mean would represent 0.15% of the normally distributed data.

We can assume that 5 standard deviations in both directions includes 100% the data, or that 0% of the data is represented beyond 5 standard deviations.

Therefore, about 0.15% of test takers scored between 460 and 500 on the exam for the state of Georgia.

Hope that helps!