Mastering The Empirical Rule For Normal Distribution Estimation And Percentile Calculation In Math

Use the empirical rule to estimate the proportion of values in a specified interval or to find the value that corresponds to a given percentile in a Normal distribution.

68-95-99.7 rule. 1sd, 2sd, or 3sd away from mean, respectively.

The empirical rule, also known as the 68-95-99.7 rule, can be used to estimate the proportion of values in a specified interval or to find the value that corresponds to a given percentile in a Normal distribution. The rule states that for any Normal distribution:

– About 68% of the observations fall within one standard deviation of the mean.
– About 95% of the observations fall within two standard deviations of the mean.
– About 99.7% of the observations fall within three standard deviations of the mean.

To estimate the proportion of values in a specified interval, we can first standardize the data to a z-score using the formula:

z = (x – mu) / sigma

where x is the observed value, mu is the mean of the distribution, and sigma is the standard deviation of the distribution. Once we have the z-score, we can use a Normal distribution table or calculator to find the proportion of values between two z-scores.

For example, suppose we have a Normal distribution with a mean of 50 and a standard deviation of 10. We want to find the proportion of values between 30 and 70. First, we standardize the values:

z1 = (30 – 50) / 10 = -2
z2 = (70 – 50) / 10 = 2

Using a Normal distribution table or calculator, we find that the proportion of values between -2 and 2 is approximately 0.954. Therefore, we can estimate that about 95.4% of the observations fall between 30 and 70 in this Normal distribution.

To find the value that corresponds to a given percentile, we can use the reverse process. First, we find the corresponding z-score for the percentile using a Normal distribution table or calculator. Then, we use the formula:

x = mu + z * sigma

where x is the value, mu is the mean of the distribution, sigma is the standard deviation of the distribution, and z is the standardized score.

For example, suppose we have a Normal distribution with a mean of 100 and a standard deviation of 15. We want to find the value that corresponds to the 90th percentile. First, we find the z-score for the 90th percentile using a Normal distribution table or calculator, which is approximately 1.28. Then, we use the formula:

x = 100 + 1.28 * 15 = 119.2

Therefore, we can estimate that the value that corresponds to the 90th percentile in this Normal distribution is 119.2.

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