1. Normal Distribution
Introduction

2. Compare to Discrete Variables
No. of Doctor’s Visits During the Year
No. of Patients P No. of Visits

3. Histograms
The height of each bar represents the probability of that event
If each bar is one unit in width, then the area also equals the probability
The total area under all the bars has to add to 1.

5. Continuous Variables
Patient’s Weight Frequency

6. But… Can take on any value
Can make the weight intervals as small as we want: every 10 lbs or 5 or 1, or … 0.5, 0.1, 0.001
Histogram: As the intervals get smaller, the bars decrease in width

8. Line Graph
Completely continuous, no width at all. Just connected points

9. Line Graph

10. Infinitesimally Small Intervals
Then really just points on a smooth curve.
We can also have n, the number of cases, increase to infinity.
The total probability is still one.

11. Infinitesimally Small Intervals
Smoother Curve
Area under the curve = 1.

12. Probabilities Can no longer read the probability of a single event.
In a continuous distribution, can only measure the probability of a value falling within some range

13. Probability Within a Range
Probability of a value falling within the range is equal to the area under the curve.

14. Bad News
To calculate the area under the curve we would need to use calculus
But not so bad news, others have done the calculations and set up tables for us
Applause

15. Diversity of Continuous Distributions
Lots of different distributions
Lots of different shaped curves
Would need lots of different tables, however….

16. The Most Important Distribution
Introducing the Normal Distribution
“Bell-Shaped Curve”
What are its characteristics?

17. Normal Distribution First described in 1754.
A lot of the relevant math done by Carl Gauss, therefore “Gaussian Curve”

18. Properties Symmetrical about the mean
Mean, Median & Mode are all equal
Asymptotic, height never reaches zero.
What’s the total area under the curve?

19. Ranges & Probabilities
50% of all values fall above the mean & 50% below it.
All probabilities depend on how far the values lie from the mean
Distance measured in number of standard deviations from the mean

20. Probabilities related to S.D. One S.D. on either side of the mean
Area =

21. Other Distances
1 S.D. on either side of the mean includes 68% of the cases
2 S.D. on either side of the mean includes 95% of the cases
3 S.D. on either side of the mean includes 99.7% of the cases

22. Many Different Normal Distributions
Determined by their mean and standard deviation
Mean gives location. Standard Deviation gives shape – more or less dispersed.

23. Proportions remain Same
Relationships between probability and standard deviation are the same in all Normal Distributions
However in order to use the tables provided, we have to convert to the “Standard Normal Distribution”

24. The Standard Normal Distribution
Mean = Standard Deviation = 1.

25. Z-values
Converts values in any normal distribution to the standard normal distribution.
It’s a way to express the distance from the mean in units of S.D.
Z = X – X Compare this to 18 eggs.
s.d How many dozen?

26. From Z find Probabilities
Use Table A-3. Gives areas in the upper tail of the S.N.D.
What is the area above Z = 1.28? Go to the Table.
Go to 1.2 in Left-hand column & across to 0.08
A = 0.10.
The probability that a value will fall above Z = 1.28 is 10%
S.N.D.
mean = 0. S.D. = 1

27. Test It Let’s look up the ones we already know.
Range = 1 S.D. on either side of the mean
Z = 1. Find 1.0 in the right hand column
Go across to 0.00
Reads So area in the tail is 1.59.
What’s the area between 1.59 and the mean?

28. If Area above z = 1 is 0.159, what is the area between Z and the mean?
Always draw the N.D.
If Area above z = 1 is 0.159, what is the area between Z and the mean?
A = = 0.341
We need to add an equal area on the other side of the mean.
Total shaded area = 0.682

29. You Try It
What is the probability that a value will fall within 2 s.d. of the mean?
Draw the N.D
Look up area that corresponds to Z = 2.
A = 0.023
Find the area between mean & Z = 2.
0.500 – = 0.477
Double it. A = 0.954

30. Try the Reverse
I want to find the value above which 10% of the population falls.
This time, area = 0.100
Look in body of table for 0.100
Read across and up. Z = 1.28
Would have to use the formula for Z in reverse in order to get the value for X

31. To convert to X, have to know mean & S.D.
Finding X
Z = X – X
S.D.
1.28 = X – X
S.D. * X = X
To convert to X, have to know mean & S.D.

32. Example
Weights of 40-yr old women are normally distributed with a mean of 150 and an S.D. of 10.
What is the value above which the highest 10% of weights falls?
X = 1.28 * = 202

33. Application
Studying a progressive neurological disorder. At autopsy, we weigh the brains. Find the wts are normally distributed with a mean of 1100 grams and an S.D. of 100 g.
Find the probability that one of the brains weighs less than 850 g.

34. Draw the N.D. Z = (800 – 1100)/100 = -3 P(X<800) = Area = 0.0001

35. The End
For Now

36. More Ranges
The cholesterol levels for a certain population are approximately normally distributed with a mean of 200 mg/100 ml & an S.D. of 20 mg/100 ml.
Find the probabilities for an individual picked at random to have cholesterol levels in the following ranges

37. Mean = 200 mg/100ml S.D. = 20 mg/100 ml A. Between 180 & 200
B. Greater than 225
C. Between 190 & 210

38. So the area is from the mean to one S.D.
Mean = 200 mg/100ml S.D. = 20 mg/100 ml
A. Between 180 & 200
Z1 = 0. Z2 = (180 – 200)/20 = -1
So the area is from the mean to one S.D.
If it was both sides, would be Since
only one side = P = 0.32.

39. Mean = 200 mg/100ml S.D. = 20 mg/100 ml Z = (225 – 200)/20 = 1.25
B. Greater than 225
Z = (225 – 200)/20 = 1.25
Look it up. Area = 0.106
P(X>225) = 0.106

40. Z2 = 0.5. Symmetrical. So Z2 to the mean is also 0.191.
C. Between 190 & 210
Mean = 200 mg/100ml S.D. = 20 mg/100 ml
Z1 = (190 – 200)/20 = -0.5 Look up = But that is the tail. What is Z = 0.5 to mean? – = 0.191
Z2 = Symmetrical. So Z2 to the mean is also
P = 2 times = 0.382