TOPICSLIDEWhat is the Standard Normal Distribution?2What do z-scores tell us?3The Empirical Rule10Steps for finding the area under the Normal Curve13• Tutorial: Using Excel 2007 to obtain z-scores• Tutorial: Using Excel 2007 to obtain the cumulativearea under the Normal CurveChapter 6The Standard Normal DistributionChapter 6➊ A distribution of scores called standard normal or z-scores➋ Standard normal or z-scores are used when the researcher wants to convert the original unit of measurement into a common unit of measurement (i.e., z-scores)• EXAMPLE: Suppose you are told someone’s test score is 278. Can you tell if this is a good or bad score? Why?The Standard Normal DistributionChapter 6➊ Z-scores are in standard deviation units• The value of a z-score tells you how many standard deviations that score is from its mean• EXAMPLE: Suppose a group of scores have a mean equal to 250 and a SD equal to 28. What score is one SD above the mean? Answer: 278. Therefore, the z-score for an original score of 278 is z = +1.00 • Remember – a z-score tells you how many SDs a score is from its meanThe Standard Normal DistributionChapter 6➊ EXAMPLE: The heights for a large sample of adults is normally distributed with a mean of 68” and a SD of 3”The Standard Normal DistributionHEIGHT IN INCHES68” 77”74”71”65”62”59”Chapter 6➊ The scores 65” and 71” are each one SD away from the mean, therefore:• The z-score for 65” is z = -1.00 and• The z-score for 71” is z = +1.00 The Standard Normal DistributionHEIGHT IN INCHES68” 77”74”71”65”62”59”Z = -1.00 Z = +1.00Chapter 6➊ The sign of a z-score tells you which side of the mean the score is on• +Positive z-scores are above the mean• -Negative z-scores are below the meanThe Standard Normal DistributionHEIGHT IN INCHES68” 77”74”71”65”62”59”Z = -1.00 Z = +1.00Chapter 6The Standard Normal DistributionHEIGHT IN INCHES68” 77”74”71”65”62”59”68%• We expect 68% of all scores to fall in the range z = -1.00 to z = +1.00• 68% of all scores are expected to fall within ±1.00 standard deviation of the meanChapter 6• We expect 95% of all scores to fall in the range z = -2.00 to z = +2.00• 95% of all scores are expected to fall within ±2.00 standard deviations of the meanThe Standard Normal DistributionHEIGHT IN INCHES68” 77”74”71”65”62”59”95%Chapter 6• We expect 99.7% of all scores to fall in the range z = -3.00 to z = +3.00• 99.7% of all scores are expected to fall within ±3.00 standard deviations of the meanThe Standard Normal DistributionHEIGHT IN INCHES68” 77”74”71”65”62”59”99.7%Chapter 6➊ We can also use z-scores to estimate the percentage of scores expected to occur in a given range➋ The Empirical Rule can help us understand how to estimate the percentage of scores expected in a given range:• 68% of all scores are expected to be in the range -1.00 to +1.00• 95% of all scores are expected to be in the range -2.00 to +2.00• 99.7% of all scores are expected to be in the range -3.00 to +3.00The Standard Normal DistributionChapter 6Area Under The Standard Normal Distribution0 +1.0099.7%68%95%+3.00+2.00-3.00 -1.00-2.00THE EMPIRICAL RULEChapter 6• What percentage of scores are expected to be less than or equal to z = 0?Area Under the Standard Normal DistributionZ0 +1.00 +3.00+2.00-3.00 -1.00-2.00Chapter 6➊ Sketch the normal distribution➋ Plot the approximate location of the z-score(s) in question➌ Shade-in the area in question➍ Use the Empirical Rule or Excel to obtain the estimated percentage of scores at or below the z-score(s)➎ When finding the percentage of scores between two z-scores, subtract the smaller area from the larger areaThe Standard Normal DistributionChapter 6• What percentage of scores are expected to be less than or equal to z = 0?• ANSWER  50% of all scores are expected to be equal to z = 0 or lessArea Under the Standard Normal DistributionZ0 +1.00 +3.00+2.00-3.00 -1.00-2.00Chapter 6• What percentage of scores are expected to be in the range of z = -2.00 to z = +1.00?Area Under the Standard Normal DistributionZ0 +1.00 +3.00+2.00-3.00 -1.00-2.00Chapter 6• What percentage of scores are expected to be in the range of z = -2.00 to z = +1.00?Area Under the Standard Normal DistributionZ0 +1.00 +3.00+2.00-3.00 -1.00-2.00Chapter 6• What percentage of scores are expected to be in the range of z = -3.00 to z = -1.00?Area Under the Standard Normal DistributionZ0 +1.00 +3.00+2.00-3.00 -1.00-2.00Chapter 6• What percentage of scores are expected to be greater than or equal to z = +1.00?Area Under the Standard Normal DistributionZ0 +1.00 +3.00+2.00-3.00 -1.00-2.00Chapter 6HEIGHT IN INCHES68” 77”74”71”65”62”59”• Using the height data, what percentage of the sample is expected to be in the range of 59” to 65”?Area Under the Standard Normal DistributionChapter 6HEIGHT IN INCHES68” 77”74”71”65”62”59”• Using the height data, what percentage of the sample is expected to be in the range of 59” to 65”?Area Under the Standard Normal DistributionChapter 6HEIGHT IN INCHES68” 77”74”71”65”62”59”• Using the height data, what percentage of the sample is expected to be 62” or taller?Area Under the Standard Normal DistributionChapter 6HEIGHT IN INCHES68” 77”74”71”65”62”59”• Using the height data, what percentage of the sample is expected to be 62” or taller?Area Under the Standard Normal DistributionEnd of Chapter 6 – Part