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12.6 Normal Distributions Objectives • Use z-scores to find percentiles. • Thinking Skill: Explicitly assess information and draw conclusions Copyright © by Holt, Rinehart and Winston. All 12.6 Normal Distributions Glossary Terms normal curve normal distribution standard normal curve z-score Copyright © by Holt, Rinehart and Winston. All 12.6 Normal Distributions Rules and Properties Properties of Normal Distributions • Symmetric about the mean, x . • Total area under the curve is 1. • Mean, median, and mode are about equal. Copyright © by Holt, Rinehart and Winston. All Copyright © by Holt, Rinehart and Winston. All 12.6 Normal Distributions Rules and Properties Properties of Normal Distributions • About 68% of the area is within 1 standard deviation of the mean. Copyright © by Holt, Rinehart and Winston. All 12.6 Normal Distributions Rules and Properties Properties of Normal Distributions • About 95% of the area is within 2 standard deviations of the mean. Copyright © by Holt, Rinehart and Winston. All 12.6 Normal Distributions Rules and Properties Properties of Normal Distributions • About 99.8% of the area is within 3 standard deviations of the mean. Copyright © by Holt, Rinehart and Winston. All Translation of data values into standard scores • The z-score is a standard score. • z-score is the number of ______________ ____________ a score is from the __________ • Formula for z-score: Copyright © by Holt, Rinehart and Winston. All 12.6 Normal Distributions Rules and Properties z-Score x-x z= normal distribution mean: x any data value: x standard deviation: Copyright © by Holt, Rinehart and Winston. All • Questions on your homework? Copyright © by Holt, Rinehart and Winston. All Percentiles • The area under the entire curve is one or 100% of the scores • So area up to a score is the percentile for that score – the percent of scores lower than that score Copyright © by Holt, Rinehart and Winston. All Try this: • Standardized test scores are normally distributed with a mean of 100 and a standard deviation of 10. • What percent scored less than 95? Copyright © by Holt, Rinehart and Winston. All Indicate on the drawing what we are looking for. Find the z-score Can’t tell % using the Empirical rule. Copyright © by Holt, Rinehart and Winston. All This table gives the percents for any given z-score Copyright © by Holt, Rinehart and Winston. All • The z-score for a score of 95 is -.5 • The table shows that the percent of scores lower than a z-score of -.5 is 30.85% Copyright © by Holt, Rinehart and Winston. All Try some more: • What is the percent below 120? Copyright © by Holt, Rinehart and Winston. All • What is the percent higher than 112? (be careful!!) Copyright © by Holt, Rinehart and Winston. All • What is the percent scoring between 90 and 115? Copyright © by Holt, Rinehart and Winston. All