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The Normal Distribution

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PSYC 6130, PROF. J. ELDER 2 is the mean is the standard deviation The height of a normal density curve at any point x is given by Normal Distribution

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PSYC 6130, PROF. J. ELDER 3 Normal Distribution A Normal Distribution has a symmetric, unimodal and bell- shaped density curve. The mean and standard deviation completely specify the curve. The mean, median, and mode are the same.

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PSYC 6130, PROF. J. ELDER 4 Probabilities and the Normal Distribution Shaded area = 0.683Shaded area = 0.954Shaded area = 0.997 and 99.7% chance of falling between and

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PSYC 6130, PROF. J. ELDER 5 Z-Scores The z-score is a normalized representation of a random variable with zero mean and unit variance. z-scores are dimensionless. The z-score tells you how many standard deviations a score lies from the mean.

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PSYC 6130, PROF. J. ELDER 6 The Standard Normal Table: (Appendix z) A table of areas (probabilities) under the standard normal density curve. The table entry for each value z is the area under the curve between the mean and z.

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PSYC 6130, PROF. J. ELDER 7

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8 Estimating percentiles using z-scores From a frequency table we can directly compute percentiles and percentile ranks. If we model the data as normal, we can also calculate percentiles and percentile ranks using z-scores.

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PSYC 6130, PROF. J. ELDER 9 Z-Scores

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PSYC 6130, PROF. J. ELDER 10 Z-Scores

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PSYC 6130, PROF. J. ELDER 11 Z-Scores

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PSYC 6130, PROF. J. ELDER 12 Z-Scores

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PSYC 6130, PROF. J. ELDER 13 Z-Scores

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PSYC 6130, PROF. J. ELDER 14 Sampling distribution of the mean

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PSYC 6130, PROF. J. ELDER 15 Sampling Distribution of the Mean

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PSYC 6130, PROF. J. ELDER 16 Properties of the Sampling Distribution of the Mean

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PSYC 6130, PROF. J. ELDER 17 Example Chest measurements of 5738 Scottish soldiers by Belgian scholar Lambert Quetelet (1796-1874) –First application of the Normal distribution to human data

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PSYC 6130, PROF. J. ELDER 18 The sample mean has a sampling distribution Sampling batches of Scottish soldiers and taking chest measurements. Pop mean = 39.8 in, Pop sd = 2.05 in

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PSYC 6130, PROF. J. ELDER 19 12 samples of size 24

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PSYC 6130, PROF. J. ELDER 20 Histograms from 100,000 samples

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PSYC 6130, PROF. J. ELDER 21 Example: Height of US Males, Sampling Distribution of the Mean 6065707580 0 0.5 1 1.5 Height of US Males (in) Probability p n=1 n=9 n=100

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PSYC 6130, PROF. J. ELDER 22 The Central Limit Theorem

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PSYC 6130, PROF. J. ELDER 23 Example: a uniform distribution n=1 n=2 n=10 n=100

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PSYC 6130, PROF. J. ELDER 24 Example: a chi-squared distribution n=1 n=2 n=10 n=100

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PSYC 6130, PROF. J. ELDER 25 Z-Scores for Groups

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PSYC 6130, PROF. J. ELDER 26 Example: IQ Tests

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PSYC 6130, PROF. J. ELDER 27 Underlying Assumptions Population is normally distributed Random sampling –Every sample of size n has the same probability of being selected. All individuals have the same probability of being selected. Selection of each individual is independent of the selection of all other individuals. Technically, sampling should be with replacement, but in Psychology, sampling is normally without replacement.

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End of Lecture 3 Sept 24, 2008

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