Standard Deviation

Question

Suppose a normally distributed set of data with 7100 observations has a mean of 188 and a standard deviation of 13. Use the 68-95-99.7 Rule to determine the number of observations in the data set expected to be belowa value of 227. Round your result to the nearest single observation.

Answer 1

We will begin by calculating the z-score. The formula for this is z = (x – x̅)/s, where x is the data value (227), the mean, x̅ , is 188, and s is the standard deviation of 13. Therefore, z = (227 – 188)/13 = 39/13 = 3. In symbols this means we want P(Z <= 3), the probability or percent of the normal distribution that is less than or equal to z = 3. Note: with the normal distribution, and other continuous distributions, P(Z < 3) and P(Z <=3) mean the same thing.

From the 68-95-99.7 Rule we know from z = –3 to z = 3 contains 99.7% of all data values. Conversely, further than 3 standard deviations are 100% – 99.7% = 0.3% of all data values. This is split equally into the left tail (smaller than z = –3) and right tail (bigger than z = 3). Thus, 0.3%/2 = 0.15% in each of these tails.

Returning to the end of the first paragraph to find P(z <= 3) we need to add the 99.7% which is from z = –3 to z = 3 and the 0.15% that is to the left of z = –3. Thus, the total percentage is 99.7% + 0.15% = 99.85%. Since there are total of 7100 observations we can multiply it by the percentage to find the expected amount below 227: 0.985(7100) = 7089.35, which we round down to 7089.

Answer 2

1. Find the Z score: z = (X - mean) / SD = (227 - 188) / 13 = 3
2. P(z < 3) = .5 + .997 / 2 = .9985 or 99.85%
3. .9985 * 7100 = 7089.35 ~~ 7089 observation

Explanation

By 68 - 95 - 99.7 Rule, 3 SD away gives us 99.7 %. However, this does not include the left tail (left side of z = -3).

We need to include the left tail since we are looking for the below value of 227.

Hence, we use half of 99.7 (right side of the mean ~ 227) and the left side (below the mean)

=> .5 (left side of mean) + .997/2 (right side of the mean ~ 227) = .9985.

=> .9987 * 7100 = 7089.35 ~~ 7089 (nearest single observation)