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Lecture 4

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Department
Psychology
Course
PSYC 2002
Professor
Steven Carroll
Semester
Winter

Description
Lecture 4: variability Consider the following distribution - What are the mean and median of each of these distributions? Sample 1 Sample 2 5 1 5 3 5 4 5 5 5 6 5 7 5 9 - So if the mean and the median are the same, can we say that there is no difference between them? Variability - The “spread” of scores in a distribution - The extent to which the scores in a distribution differ from one another Consider these data X 10 30 50 70 90 - What is the mean of this distribution? • Answer: 50 - Imagine these data represent scores on a test - If someone were to ask you “how did the class do on the test?” what would be your response? - But “the class average was 50” doesn’t tell the whole story - Only one person actually got a score of 50 on the test - Calculate the amount that each student’s score deviated from the class mean X X - µ 10 -40 30 -20 50 0 70 20 90 40 - NOW: on average, how much did people’s scores deviate from the mean? - In other words, what is the mean of that second column? X - µ -40 -20 0 20 40 - What is the formula for calculating a mean? X • µ = N - Except, in this case, we’re using ( X - µ ) instead of X Consider these data X - µ -40 -20 0 20 40 (X−µ) - µdiff= N - So what is the  ( X - µ )? -  ( X - µ ) = 0 - We now have no way to calculate the average amount that scores deviate from the mean! - The problem is this: the negative deviations are always going to cancel-out the positive deviations - What can we do to these values to get rid of the negatives without losing information about the nature of these deviations? Consider these data X - µ (X – µ)² -40 1600 -20 400 0 0 20 400 40 1600 - ( X - µ)² - By convention, we square the deviations Population variance ² X - µ (X – µ)² -40 1600 -20 400 0 0 20 400 40 1600 ❑ 2(X−µ)² - N - Another way to write  (X - µ)² is “SS”, which stands for “Sum of the Squared deviations” SS - ² = N - So: What is SS? • 1600 + 400 + 0 + 400 + 1600 = 4000 - What is 4000/ N ? • 4000 / 5 = 800 = ² But hang on a second... X X - m (X – m)2 10 -40 1600 30 -20 400 50 0 0 70 20 400 90 40 1600 - The original range of scores went from 10 to 90 - So what the hell does 800 mean? It’s too big a value to make sense out of SS -  = √ N - In this case, 800 = 28.28 - So the average amount that each of the scores in the distribution deviated from the mean “50” was 28.28 - 28.28 is the “standard deviation” of the population of scores An example µ = 100 -2 = 70 -1 = 850 = 100 1 = 115 2 = 130 - What is the standard deviation for the distribution of IQ scores? - This is a “probability distribution”: individual scores are more likely to fall closer to the mean, and less likely to fall close to the tails An example 13.59% 13.59% 34.13% 34.13% -2 -1 0 1 2 - What percentage of the population has an IQ that lies within 2 of the mean? - Answer: 95.44 Samples - What is the difference between a sample and a population? - Why do we use samples? Samples - The formulae for sample statistics differ from the formulae used to calculate population parameters SS SS -  = √ N ² = N SS SS - s = √n−1 s² = n−1 N – 1 ?! - What’s up with the denominat
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