1. Properties of the Normal Distribution
7.1
Properties of the
Normal Distribution

2. Between numbers
Sometimes we want to model a random variable that is equally likely between two limits
Examples
Choose a random time … the number of seconds past the minute is random number in the interval from 0 to 60
Observe a tire rolling at a high rate of speed … choose a random time … the angle of the tire valve to the vertical is a random number in the interval from 0 to 360

3. Uniform Probability Distribution
When “every number” is equally likely in an interval, this is a uniform probability distribution
Any specific number has a zero probability of occurring
The mathematically correct way to phrase this is that any two intervals of equal length have the same probability

4. Example For the seconds after the minute example
Every interval of length 3 has probability 3/60
The chance that it will be between 14.4 and 17.3 seconds after the minute is 3/60
The chance that it will be between 31.2 and 34.2 seconds after the minute is 3/60
The chance that it will be between 47.9 and 50.9 seconds after the minute is 3/60

5. Example Continued For the seconds after the minute example
Every individual value has probability 0
The chance that it will be exactly 13.4 seconds after the minute (i.e. between 13.4 and 13.4) is 0/60 or 0
The chance that it will be exactly 31.2 seconds after the minute (i.e. between 31.2 and 31.2) is 0/60 or 0
The chance that it will be exactly 47.9 seconds after the minute (i.e. between 47.9 and 47.9) is 0/60 or 0
We need to use a different way to specify the probabilities

6. Probability Density Function
A probability density function is an equation used to specify and compute probabilities of a continuous random variable
This equation must have two properties
The total area under the graph of the equation is equal to 1 (the total probability is 1)
The equation is always greater than or equal to zero (probabilities are always greater than or equal to zero)

7. Probability
This function method is used to represent the probabilities for a continuous random variable
For the probability of X between two numbers
Compute the area under the curve between the two numbers
That is the probability

8. Probability The probability of being between 4 and 8 The probability
From 4 (here)
The probability
To 8 (here)

9. Probability An interpretation of the probability density function is
The random variable is more likely to be in those regions where the function is larger
The random variable is less likely to be in those regions where the function is smaller
The random variable is never in those regions where the function is zero

10. Graph of Probability
A graph showing where the random variable has more likely and less likely values
More likely values
Less likely values

11. Time Example The time example … uniform between 0 and 60
All values between 0 and 60 are equally likely, thus the equation must have the same value between 0 and 60

12. Time Example Continued
The time example … uniform between 0 and 60
Values outside 0 and 60 are impossible, thus the equation must be zero outside 0 to 60

13. Time Example Continued
The time example … uniform between 0 and 60
Because the total area must be one, and the width of the rectangle is 60, the height must be 1/60
1/60

14. Time Example Continued
The time example … uniform between 0 and 60
The probability that the variable is between two numbers is the area under the curve between them
1/60

15. Normal Curve
The normal curve has a very specific bell shaped distribution
The normal curve looks like

16. Normal
A normally distributed random variable, or a variable with a normal probability distribution, is a random variable that has a relative frequency histogram in the shape of a normal curve
This curve is also called the normal density curve (a particular probability density function)

17. Inflection Point
In drawing the normal curve, the mean μ and the standard deviation σ have specific roles
The mean μ is the center of the curve
The values (μ – σ) and (μ + σ) are the inflection points of the curve

18. Different Normal?
There are normal curves for each combination of μ and σ
The curves look different, but the same too
Different values of μ shift the curve left and right
Different values of σ shift the curve up and down

19. Normal
Two normal curves with different means (but the same standard deviation)
The curves are shifted left and right

20. Normal
Two normal curves with different standard deviations (but the same mean)
The curves are shifted up and down

21. “Normal” Properties Properties of the normal density curve
The curve is symmetric about the mean
The mean = median = mode, and this is the highest point of the curve
The curve has inflection points at (μ – σ) and (μ + σ)
The total area under the curve is equal to 1
The area under the curve to the left of the mean is equal to the area under the curve to the right of the mean

22. Normal Properties Properties of the normal density curve
As x increases, the curve goes to zero … as x decreases, the curve goes to zero
In addition, the empirical rule holds
The area within 1 standard deviation of the mean is approximately 0.68
The area within 2 standard deviations of the mean is approximately 0.95
The area within 3 standard deviations of the mean is approximately 0.997

23. Normal Properties The curve is symmetric about the mean
Because the curve is symmetric
The mean = median = mode = μ
The area under the curve to the left of the mean is equal to the area under the curve to the right of the mean

24. Normal Properties The highest point of the curve is at x = μ
This can be seen in a previous chart
The total area is equal to 1
This is a complex calculation
But it is true

25. Normal Properties
It has inflection points (where the concavity changes) at (μ – σ) and (μ + σ)
This can be seen in a previous chart
As x increases, the curve goes to zero … as x decreases, the curve goes to zero
This is clear from the chart also

26. Normal Properties The empirical rule is true
Approximately 68% of the values lie between (μ – σ) and (μ + σ)
Approximately 95% of the values lie between (μ – 2σ) and (μ + 2σ)
Approximately 99.7% of the values lie between (μ – 3σ) and (μ + 3σ)

27. Reminder…
An illustration of the Empirical Rule

28. Normal Curve
The equation of the normal curve with mean μ and standard deviation σ is
This is a complicated formula, but we will never need to calculate it (thankfully)

29. Area under the curve
When we model a distribution with a normal probability distribution, we use the area under the normal curve to
Approximate the areas of the histogram being modeled
Approximate probabilities that are too detailed to be computed from just the histogram

30. Example
Assume that the distribution of giraffe weights has μ = 2200 pounds and σ = 200 pounds

31. Example
What is an interpretation of the area under the curve to the left of 2100?

32. Example
It is the proportion of giraffes that weigh pounds and less

33. How? How do we calculate the areas under a normal curve?
If we need a table for every combination of μ and σ, this would rapidly become unmanageable
We would like to be able to compute these probabilities using just one table
The solution is to use the standard normal random variable

34. How?
The standard normal random variable is the specific normal random variable that has
μ = 0 and
σ = 1
We can relate general normal random variables to the standard normal random variable using a Z- score calculation

35. Z- Score
If X is a general normal random variable with mean μ and standard deviation σ then
is a standard normal random variable
This equation connects general normal random variables with the standard normal random variable
We only need a standard normal table

36. Example Continued
The area to the left of 2100 for a normal curve with mean 2200 and standard deviation 200

37. Z-Score To compute the corresponding value of Z, we use the Z-score
Thus the value of X = 2100 corresponds to a value of Z = – 0.5
In the next section, we will learn how to use this to find the probability!!!

38. Summary
Normal probability distributions can be used to model data that have bell shaped distributions
Normal probability distributions are specified by their means and standard deviations
Areas under the curve of general normal probability distributions can be related to areas under the curve of the standard normal probability distribution

39. The birth weights of full-term babies are normally distributed with mean 3,400 grams and standard deviation 505 grams
Draw a normal curve with the parameters labeled
Shade the region that represents the proportions of full-term babies who weigh more than 4,410 grams
Suppose that the area under the curve to the right of x = 4,410 is Provide two interpretations of this result.