1. Modeling Continuous Variables
Lecture 24
Section 6.3.1
Fri, Feb 27, 2004

2. Models
A mathematical model is an abstraction and, therefore, a simplification of a real situation.
Real situations are usually much to complicated to deal with directly.
For example, we might replace a jagged histogram with a smooth curve that has a simple equation.

3. Models No mathematical model is perfect.
A mathematical model is useful (and powerful) to the extent that it is a faithful representation of reality.
Similarly, to the extent that is it not faithful to reality, it can lead to false conclusions about the situation it is supposed to model.

4. Example of a Model
Using a random number generator to simulate how a pair of rolled dice will land.
Use the TI-83 and
Enter randInt(2, 12), or
Enter randInt(1, 6) twice and add the results.
Which is right?

5. Histograms and Area
If a histogram is drawn appropriately, then frequency is represented by area.
Consider a histogram of the following test scores.
Grade
Frequency
60 – 69 3
70 – 79 8
80 – 89 9
90 – 99 5

6. Histograms and Area
Frequency 10 8 6 4 2
Grade 60 70 80 90
100

7. Histograms and Area
In the histogram, we may replace the frequency with the proportion (of the total).
Grade
Frequency
Proportion
60 – 69 3
0.12
70 – 79 8
0.32
80 – 89 9
0.36
90 – 99 5
0.20

8. Histograms and Area Proportion 0.40 0.30 0.20 0.10 Grade 60 70 80 9060 70 80 90
100

9. Histograms and Area Proportion 0.40 0.30 0.20 0.10 Grade 60 70 80 9060 70 80 90
100

10. Histograms and Area
Furthermore, we may divide the proportions by the width of the classes to get the density.
Grade
Frequency
Proportion
Density
60 – 69 3
0.12
0.012
70 – 79 8
0.32
0.032
80 – 89 9
0.36
0.036
90 – 99 5
0.20
0.020

11. Histograms and Area Density 0.040 0.030 0.020 0.010 Grade 60 70 80 9060 70 80 90
100

12. Histograms and Area
The final histogram has the special property that the proportion can be found by computing the area of the rectangle.
For example, what proportion of the grades are less than 80?
Compute: (10 x 0.012) + (10 x 0.032)
= = 0.44 = 44%.

13. Density Functions
This is the fundamental property that connects histograms with the graphs of continuous models that we use to approximate them, namely:
The area under the curve between two numbers a and b on the x-axis represents the proportion of values that lie between a and b in the distribution.

14. Density Functions
The area under the curve between a and b is the proportion. a b

15. Density Functions
The area under the curve between a and b is the proportion. a b

16. Density Functions
The area under the curve between a and b is the proportion. a b
Area = Proportion

17. Density Functions
A consequence of this is that the total area under the curve must be 1, representing a proportion of 100%. a b

18. Density Functions
A consequence of this is that the total area under the curve must be 1, representing a proportion of 100%.
100% a b

19. The Normal Distribution
The normal distribution is the statistician’s name for the bell curve.
It is a density function in the shape of a bell (sort of).
Symmetric.
Unimodal.
Extends over the entire real line.
Main part lies with 3 of the mean.

20. The Normal Distribution
The curve has a bell shape (with infinitely long tails).

21. The Normal Distribution
The mean  is located at the peak. 

22. The Normal Distribution
The width of the “main” part of the curve is 6 standard deviations.  

23. The Normal Distribution
The area under the entire curve is 1.
Area = 1 

24. The Normal Distribution
The normal distribution with mean  and standard deviation  is denoted N(, ).
For example, if X is a variable whose distribution is normal with mean 30 and standard deviation 5, then we say that “X is N(30, 5).”

25. Various Normal Distributions1 2 3 4 5 6 7 8

26. Various Normal Distributions1 2 3 4 5 6 7 8

27. Various Normal Distributions1 2 3 4 5 6 7 8

28. Various Normal Distributions1 2 3 4 5 6 7 8