Understanding Normal Distribution and Probability With Z-Scores

1. Frequency Distributions FSE 200

2. Why Probability? • Basis for the normal curve • Provides basis for understanding probability of a possible outcome • Basis for determining the degree of confidence that an outcome is “true” • Example: Are changes in student scores due to a particular intervention that took place or by chance alone?

3. The Normal Curve (a.k.a. the Bell-Shaped Curve) • Visual representation of a distribution of scores • Three characteristics… • Mean, median, and mode are equal to one another • Perfectly symmetrical about the mean • Tails are asymptotic (get closer to horizontal axis but never touch)

4. The Normal Curve The normal or bell shaped curve

5. Hey, That’s Not Normal! • In general, many events occur in the middle of a distribution with a few on each end. How scores can be distributed

6. More Normal Curve 101 • For all normal distributions… • Almost 100% of scores will fit between –3 and +3 standard deviations from the mean • So…distributions can be compared • Between different points on the x-axis, a certain percentage of cases will occur

7. What’s Under the Curve? Distribution of cases under the normal curve

8. The z Score • A standard score that is the result of dividing the amount that a raw score differs from the mean of the distribution by the standard deviation. • What about those symbols?

9. ThezScore • Scores below the mean are negative (left of the mean), and those above are positive (right of the mean) • A z score is the number of standard deviations from the mean • z scores across different distributions are comparable

10. Using Excel to Compute z Scores

11. What z Scores Represent • The areas of the curve that are covered by different z scores also represent the probability of a certain score occurring. • So try this one… • In a distribution with a mean of 50 and a standard deviation of 10, what is the probability that one score will be 70 or above?

12. How to Figure Out the Probability

13. How to Figure Out the Probability

14. What z Scores Really Represent • Knowing the probability that a z score will occur can help you determine how extreme a z score you can expect before determining that a factor other than chance produced the outcome • Keep in mind…z scores are typically reserved for populations. 

15. Example • Normal Distribution • Mean = 0 • Standard Deviation = 1 • Answer the following: • What is the probability that a randomly selected value will be less than -0.1? • What is the probability that a randomly selected value will be greater than 0.6? • What is the probability that a randomly selected value will be between 0.6 and 0.9?

16. Question A Z-Score = (-0.1-0)/1 = -0.1 From chart on Slide 12, the area between the mean and the z-score is 3.98 Therefore, the probability is equal to 100-50-3.98 = 46.02% or 0.4602

17. Question B Z-Score = (0.6-0)/1 = 0.6 From chart on Slide 12, the area between the mean and the z-score is 22.24. Therefore, the probability is equal to 100-50-22.24 = 27.76% or 0.2776

18. Question C We have to compute two z-scores for this problem since we are finding the probability in between two values. We know that the probability of a value being above 0.6 is 27.76%. What is the probability of the value being above 0.9? Z-Score = (0.9-0)/1 = 0.9 Pick value from Table on Slide 13 which yields 31.59%. The odds of the value being above 0.9 is 100-50-31.59 = 18.41% or 0.1814 To figure the odds of being between: Subtract the two values. 27.76-18.41 = 9.35% or 0.0935 This is essentially the area shaded in the diagram

19. Acknowledgement The majority of the content of these slides were from the Sage Instructor Resources Website