Chapter 7: Normal Curves & Probability CRIM 483 Chapter 7: Normal Curves & Probability

Normal Curves & Probability The goal of statistics=estimate probability of an outcome—I.e., how likely is it that a particular outcome will occur Probability=the likelihood that an event will occur Provides the basis for determining the degree of confidence we have in stating that a particular outcome is “true” Basis of the normal curve and the foundation for inferential statistics One way to estimate the probability of an outcome is by using a normal curve=visual representation of a distribution of scores Provides the basis for understanding the probability associated with any possible outcome Also known as a bell curve

Normal Curve Characteristics Shape looks like a bell Mean, median, mode are equal to one another The mean represents the center of the distribution The shape is symmetrical 50% of the values lie above the mean and 50% of the values lie below this point Tails are asymptotic—tails come closer and closer to axis but never touch it

More Description The Normal Curve represents a distribution of events/scores Few events occur at the tails of the distribution—low probability that they will occur Most events occur within the middle—higher probability that they will occur Represents the basic nature of things—most distributions will naturally fall into this shape E.g., intelligence, height

Distribution Patterns In a any distribution of scores that follow a Normal Curve: Almost 100% of the scores will fit between -3 & +3 standard deviations from the mean REGARDLESS of the value of the mean and standard deviation

Distribution of Scores A certain percentage of cases will fall between different points on the axis Example: Mean=100; SD=10 % Within + Side - Side B/T Mean & 1 SD 34.13% 100-110 90-100 B/T Mean & 2 SD 13.59% 110-120 80-90 B/T Mean & 3 SD 2.15% 120-130 70-80 3 SD + .13% 130+ Below 70 50% * 2

Percentages & Probabilities The percentages or areas under the curve also represent probabilities of a certain score occurring Convert the percentage to a probability: 42%/100= .42 Convert a probability to a percentage: .42 * 100= 42%

Comparing Means Across Groups: The Z-Score Each group has a distribution—but in their original form, the groups are not comparable Each original score can be converted to a z-score, which is a standard score that can be compared across groups z=(x-mean)/s z=z-score x=score mean=mean of the distribution s=standard deviation of the distribution

Z-Score Characteristics Z-scores are distributed in a bell curve Mean, median, mode=0 SD=1 Scores below the meannegative z-score Scores above the meanpositive z-score Z-scores provide a way to compare group scores using a standard score Book Example: Page 127-128

Characteristics, Continued Given the standard distribution of scores within a normal curve, the following statements are true: 84% of the scores fall below z-score=1 16% of the scores fall above z-score=1 The more extreme the z-score, the farther it is from the mean

Comparing Distributions

Using z-Scores In distribution with mean of 100 and SD of 10 what is the probability that any one score will be 110 or greater? z-score(110)=1.00 Also know that 16% of the scores fall above this point 16%/100=.16 The probability that any one score will be 110 or greater=.16 Table of z-scores and related probabilities is provided for less straightforward z-score values (e.g., 1.28, .84, etc.) Using the z-score table—Table B1, Appendix B Provides a z-score for every possible point on the x-axis (Area between mean & z-score)+50=percent likelihood that the event will fall below z-score To find the area that falls above the score you must make the following computation: 100 – (the area provided in the chart+50%)

Applying z-Scores Finding the area between two z-scores E.g., the amount of area between a z score of 110 and 125 (mean=100; SD=10) This number corresponding to the area=the probability that a score will fall between these two scores of interest Find the area for each z-score and subtract it from one another

Comparing the Area B/T Scores (110-100)/10=10/10=1.00 (125-100)/10=25/10=2.50 Area between mean and 1.00=34.13% Area between mean and 2.50=49.38% Area between 1.00 and 2.50=15.25% Thus, the probability that a score will fall between 110 and 125 is .1525 The probability that a score will fall under 1.00=.50 + .3413=.8413

Using Probabilities Ultimately, the point of statistics is to estimate the probability of an outcome Can use z-scores to determine the probability of an event occurring AND we can determine if it is as likely, more likely, or less likely to occur than what would be expected by chance