The empirical rule, also known as the 68-95-99.7 rule, states that for a bell-shaped distribution, approximately:
68% of the data falls within one standard deviation of the mean
95% of the data falls within two standard deviations of the mean
99.7% of the data falls within three standard deviations of the mean
Using this rule, we can answer the following questions:
A. What percentage of people have an IQ score between 85 and 115?
To find the percentage of people with an IQ score between 85 and 115, we need to find how many standard deviations away from the mean these scores are:
An IQ score of 85 is (85 - 100) / 15 = -1 standard deviation below the mean.
An IQ score of 115 is (115 - 100) / 15 = +1 standard deviation above the mean.
Therefore, the percentage of people with an IQ score between 85 and 115 is approximately the percentage of data that falls within one standard deviation of the mean, which is 68%.
B. What percentage of people have an IQ score less than 70 or greater than 130?
To find the percentage of people with an IQ score less than 70 or greater than 130, we need to find how many standard deviations away from the mean these scores are:
An IQ score of 70 is (70 - 100) / 15 = -2 standard deviations below the mean.
An IQ score of 130 is (130 - 100) / 15 = +2 standard deviations above the mean.
Therefore, the percentage of people with an IQ score less than 70 or greater than 130 is approximately the percentage of data that falls outside of two standard deviations from the mean, which is 5% + 5% = 10%.
C. What percentage of people have an IQ score greater than 130?
To find the percentage of people with an IQ score greater than 130, we need to find how many standard deviations away from the mean this score is:
An IQ score of 130 is (130 - 100) / 15 = +2 standard deviations above the mean.
Therefore, the percentage of people with an IQ score greater than 130 is approximately the percentage of data that falls outside of two standard deviations from the mean, which is 5%. However, this only gives us the percentage of people with an IQ score greater than 130 but less than infinity. To get the percentage of people with an IQ score greater than 130, we need to subtract the percentage of people with an IQ score less than or equal to 130, which is 50% + 34% + 5% = 89%. Therefore, the percentage of people with an IQ score greater than 130 is approximately 11%.