1. Normal distribution, Central limit theory, and z scores
7/28/2018
Normal distribution, Central limit theory, and z scores
PA330
Draw a standard normal curve and its st dev units on the blackboard.

2. Review - Problem 1.3 - frequency distribution

3. Problem 1.3 - frequency polygon

4. Mean, problem 2.7

5. Problem 2.7 Group median (lecture method)
There are a total of 1720 people. Therefore, the median person is #860.
This person is in the interval “15-19”.
The midpoint of this interval is 17.
Therefore, the median age is 17.

6. Problem 2.7 Group median (textbook method)
There are a total of 1720 people. Therefore, the median person is #860.
This person is in the interval “15-19”. The 190th person in that interval.
The width of the interval is “5”.
The median is /194 of the 4th class.
Or, /194(5) = = 19.89

7. Which weapon, on average, comes closest to the target?
Problem 3.7
Which weapon, on average, comes closest to the target?
Which weapon is most reliable?
Which should be selected and why?

8. Problem 3.14

9. Frequency Distributions Shape
Modality
The number of peaks in the curve
Skewness
An asymmetry in a distribution where values are shifted to one extreme or the other.
Kurtosis
The degree of peakedness in the curve

10. Frequency Distributions Modality
Unimodal
Bimodal
Multi-modal

11. Frequency Distributions Skewness
Right Skewed (Positively Skewed)
Left Skewed (Negatively Skewed)

13. Right (or positively) skewed Note - long tail to right

14. Left (or negatively) skewed Note: long tail to left

15. Kurtosis
Platykurtic
Fewer values near the mean, more in the tails. Appears flatter.
Leptokurtic
More values near the mean, fewer in the tails. Appears taller.
Mesokurtic
Evenly distributed values. The normal distribution is mesokurtic.

16. The normal distribution
The normal distribution, or normal curve, is a family of distributions that have the same general shape.
This group of distributions differs in how “spread out” their values occur but each share some common characteristics.
Overhead/handout

17. Characteristics of the normal distribution
Curve is perfectly symmetrical
The mean divides the curve exactly in half. Each half is a mirror image of the other.
Scores above and below mean are equally like to occur.
So, half of the area under the curve is above (or right of) the mean and half is below (or left of) the mean. In other words, half of the cases fall above and half below the mean.
The mean, median, and mode are equal and occur at the highest point of the curve.

18. Characteristics of the normal distribution
Most values fall close to the mean.
As values fall further from the mean, towards the tails of the distribution, their probability of occurrence decreases.
The curve is asymptotic. The tails of the curve get closer and closer to the X axis but never touch it.
A fixed proportion of the values lie between the mean and any other point.
Thus, we can compute the probability of any particular value occurring.

19. Differences in normal curves
Two parameters affect the shape and location on the X axis of a normal curve.
mean
standard deviation

20. Differences in normal curves 2 curves with same standard deviation but different means.

21. Differences in normal curves 2 curves with same mean but different standard deviations.

22. Differences in normal curves 2 curves with same mean but different standard deviations.

23. Standard normal curve
The standard normal curve is a normal distribution whose
mean = 0
standard deviation = 1
See handout/overhead

24. Portion of values falling under normal curve
For all normal distributions, the distance between the mean and one unit of standard deviation includes approximately 34% of the observations.
This is true independent of the mean and the standard deviation.
Approximately 14% of cases fall between 1 st. dev and 2 st. dev.
Approximately 2% of cases fall between 2 st. dev and 3 st. dev.

25. Using the Normal Distribution
We can use this information in the following fashion
The P(0.0  Xi  1.0) = .3413
Thus 68% of the Xis will fall between 1
Thus 96% of the Xis will fall between 2

26. Example You make a 95 on a methods exam. How do you feel?
What if the highest possible score was 200?
What if the highest grade in the class was 101?
What if the average grade was 98?
What if the lowest grade was 95?
Raw numbers by themselves don’t tell us enough. You have to know the distribution.

27. Z scores
One of the ways we can take a raw number and interpret it (relative to the mean and standard deviation of the distribution) is to convert it to a “standard normal variable” or “z score.”
A z score simply converts a raw score into the number of standard deviation units from the mean that it represents.
Z scores can only be computed for data that is normally distributed.

28. Formula

29. Z scores
A z score represents a specific point on the x axis of a normal curve.
If the z score is positive, that point is above (to right of) the mean.
If the z score is negative, that point is below (to left of) the mean.
Using a normal distribution table, given any z score, we can determine the percentage of cases falling above/below that score AND the probability of that score occurring.

30. Example - you and your father have the same IQ score of 132
Example - you and your father have the same IQ score of He took the test in 1968; you in Did you score the same? Are you equally intelligent?
In 1968:
In 1998

31. To compare, compute a z score
1968
So, dad was smarter than 97.72% of the population in 1968.
(50% %)
1998
So, you are smarter than 86.43% of the population in 1998.
(50% %)