Chapter 6 The Standard Normal Distribution
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1 The Standard Normal Distribution TOPIC SLIDE What is the Standard Normal Distribution? 2 What do z-scores tell us? 3 The Empirical Rule 10 Steps for finding the area under the Normal Curve 13 Tutorial: Using Excel 2007 to obtain z-scores Tutorial: Using Excel 2007 to obtain the cumulative area under the Normal Curve
2 The Standard Normal Distribution ➊ A distribution of scores called standard normal or z- scores ➋ 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?
3 The Standard Normal Distribution ➊ 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 = Remember a z-score tells you how many SDs a score is from its mean
4 The Standard Normal Distribution ➊ EXAMPLE: The heights for a large sample of adults is normally distributed with a mean of 68 and a SD of 3 HEIGHT IN INCHES
5 The Standard Normal Distribution HEIGHT IN INCHES Z = Z = ➊ The scores 65 and 71 are each one SD away from the mean, therefore: The z-score for 65 is z = and The z-score for 71 is z =
6 The Standard Normal Distribution HEIGHT IN INCHES Z = Z = ➊ 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 mean
7 The Standard Normal Distribution HEIGHT IN INCHES 68% We expect 68% of all scores to fall in the range z = to z = % of all scores are expected to fall within ±1.00 standard deviation of the mean
8 The Standard Normal Distribution HEIGHT IN INCHES 95% We expect 95% of all scores to fall in the range z = to z = % of all scores are expected to fall within ±2.00 standard deviations of the mean
9 The Standard Normal Distribution HEIGHT IN INCHES % We expect 99.7% of all scores to fall in the range z = to z = % of all scores are expected to fall within ±3.00 standard deviations of the mean
10 The Standard Normal Distribution ➊ We can also use z-scores to estimate the percentage of scores expected to occur in a given range ➋ 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 to % of all scores are expected to be in the range to % of all scores are expected to be in the range to +3.00
11 Area Under The Standard Normal Distribution THE EMPIRICAL RULE 68% 95% 99.7%
12 Area Under the Standard Normal Distribution Z What percentage of scores are expected to be less than or equal to z = 0?
13 The Standard Normal Distribution ➊ Sketch the normal distribution ➋ Plot the approximate location of the z-score(s) in question ➌ Shade-in the area in question ➍ 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 area
14 Area Under the Standard Normal Distribution Z 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 less
15 Area Under the Standard Normal Distribution Z What percentage of scores are expected to be in the range of z = to z = +1.00?
16 Area Under the Standard Normal Distribution Z What percentage of scores are expected to be in the range of z = to z = +1.00?
17 Area Under the Standard Normal Distribution Z What percentage of scores are expected to be in the range of z = to z = -1.00?
18 Area Under the Standard Normal Distribution Z What percentage of scores are expected to be greater than or equal to z = +1.00?
19 Area Under the Standard Normal Distribution HEIGHT IN INCHES Using the height data, what percentage of the sample is expected to be in the range of 59 to 65?
20 Area Under the Standard Normal Distribution HEIGHT IN INCHES Using the height data, what percentage of the sample is expected to be in the range of 59 to 65?
21 Area Under the Standard Normal Distribution HEIGHT IN INCHES Using the height data, what percentage of the sample is expected to be 62 or taller?
22 Area Under the Standard Normal Distribution HEIGHT IN INCHES Using the height data, what percentage of the sample is expected to be 62 or taller?
23 End of Chapter 6 Part 1
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