Normal Distribution
S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Types of Distributions
Skewed Left
Normal
Skewed Right
Standard Deviation
Standard deviation is a measure of how much the values in the data set vary from the mean.
The Empircal Rule
Normal distributions have a special property called "The Empirical Rule". The rule states that by using the mean and the standard deviation, we can find the approximate percentage of data values falling an any interval of the range.
Empirical Rule
Within one standard deviation, we can approximate 68% of the data. Within two standard deviations, we can approximate 95% of the data. Within three standard deviations, we can approximate 99.7% of the data
WARNING: When completing assignments involving the empirical rule, there are charts that have slight variations with the numbers that they use.
Example #1
The lifespans of gorillas in a particular zoo are normally distributed The average gorilla lives 20.8 years; the standard deviation is 3.1 years. Use the empirical to estimate the probability of a gorilla living longer than 23.9 years.
Step 1: Draw graph
Start with drawing the graph, and showing where the mean, or the average, is.
Step 2: Add in the standard deviations
From where you marked the average, move away one standard deviation to both the left and the right, and mark those places. Do the same for the second and third standard deviations.
To find the first mark to the right, you have to add one standard deviation to the average. The find the next mark to the right, you add another standard deviation.
To find the marks to the left, you have to subtract one standard deviation from the average. Then to find the next mark to the left, you subtract another standard deviation.
One Standard Deviation
Two Standard Deviations
Three Standard Deviations
Example #2
The lifespans of lions in a particular zoo are normally distributed. The average lion lives 12.6 years; the standard deviation is 2.4 years. Find the percentage of lions that live less than 15 years.
Step 1: Draw graph
Start with drawing the graph, and showing where the mean, or the average, is.
Step 2: Add in the standard deviations
From where you marked the average, move away one standard deviation to both the left and the right, and mark those places. Do the same for the second and third standard deviations.
To find the first mark to the right, you have to add one standard deviation to the average. The find the next mark to the right, you add another standard deviation.
To find the marks to the left, you have to subtract one standard deviation from the average. Then to find the next mark to the left, you subtract another standard deviation.
One Standard Deviation
Two Standard Deviations
Three Standard Deviations
Step 3: Find percentages
Compare the empirical rule and your graph. Since we are looking for the percentages of lions that live less than 15 years, we have to find the percentages that are less than 15 years old. 15 years is less than. 15 is the first standard deviation to the right so we are going to count everything before that.
34 + 13.5 + 2.35 + 0.15 = 64.85
This means that 64.85% of lions live less than 15 years old.
