1. Unit 2
Section 2.4 – Day 2

2. The Empirical Rule (Normal Rule)
Section 2.4
The Empirical Rule (Normal Rule)
Applies when the distribution is bell-shaped (or what is called normal).
Approximately 68% of the data falls within 1 standard deviation of the mean.
Approximately 95% of the data falls within 2 standard deviation of the mean.
Approximately 99.7% of the data falls within 3 standard deviation of the mean.

3. Section 2.4
Empirical Rule

4. Section 2.4
Example 1:
In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20 – 29) was 64.2 inches, with a sample standard deviation of 2.9 inches. Estimate the percent of women whose heights are between 58.4 inches and inches. Use a diagram to help you organize your data.

5. Section 2.4
The Empirical Rule only applies to distributions that are bell shaped (or normal distributions). Chebyshev’s Theorem allows us to determine similar information from ANY distribution.
Chebyshev’s theorem – the proportion (or percentage) of values from a data set that will fall within k standard deviations of the mean will be at least 1-1/k2, where k is a number greater than 1.
This can be applied to any data set regardless of its distribution or shape
This means at least three-fourths (or 75%) of the data values will fall within 2 standard deviations of the mean.
This means at least eight-ninths (or 89.9%) of the data values will fall within 3 standard deviations of the mean.

6. Section 2.4
Example 2:
You are conducting a survey on the number of siblings each of students in your grade have. The sample contains 40 students, the mean number of siblings is 2, and the standard deviation is 1 sibling. Using Chebyshev’s Theorem, determine at least how many of the households have 0 to 4 pets.

7. Section 2.4
Coefficient of Variation – the standard deviation divided by the mean.
Notation: CV
The result is expressed as a percentage.

8. Section 2.4
Example 3:
The mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two.

9. Finding the Standard Deviation for Grouped Data
Section 2.4
Finding the Standard Deviation for Grouped Data
Formula:

10. Finding the Variance and Standard Deviation for Grouped Data
Section 2.4
Finding the Variance and Standard Deviation for Grouped Data
Make a table as shown
Find the midpoints of each class and place them in column C.
Multiply the frequency by the midpoint for each class, and place the product in column D.
Multiply the frequency by the square of the midpoint for each class, and place the product in column E.
Find the sums of column B, D, and E. A B C D E
Class
Frequency f
Midpoint Xm
f*Xm
f*Xm2

11. Section 2.4 To find the Variance:
Take the Sum of E, subtract away the quantity of the Sum of D squared divided by the Sum of B.
Then, divide your value by the Sum of B minus one.
To find the Standard Deviation, take the square root of the variance.

12. Finding the Variance and Standard Deviation: Grouped Data
Section 2.4
Finding the Variance and Standard Deviation: Grouped Data
Find the variance and standard deviation for the grouped data below:
Class
Frequency f
Midpoint Xm
f*Xm
f*Xm2
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5
35.5 – 40.5

13. Section 2.4
Homework
Pg (29 – 32, 35, 37, 39, 43)