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Lecture 3

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Department
Psychology
Course
PSYC 2002
Professor
Steven Carroll
Semester
Winter

Description
Lecture 3: tendency What is ‘central tendency’? - What does ‘central’mean? • In the middle - What does ‘tendency’mean? • An inclination. Away in which a person or thing is likely to behave What is ‘central tendency’? - Ameasure of central tendency is a single score that describes the behavior of a distribution of scores - For example: imagine I tell you that the class average on a test was 72% • Does that mean every single person in the class got a score of 72%? • Does it mean thatANYONE in the class got a score of 72%? • So what does it mean? So what is a good measure of central tendency? - Imagine I’m in the head of “Stats Co. Inc. Ltd.”, the world’s most influential and powerful stats company - I visit our Ottawa sales office, and ask how well the stats salespeople are doing - The manager tells me “weekly sales of stats are around 750$ per person” • Is this a good measure of the central tendency of sales? Stats Co. Inc. Ltd - There isn’t really enough information to tell - So I ask “what are the actual sales figures for the 5 sales people?” Salesperson Sales Mark $3350 Nathalie $100 Jessica $100 Trevor $100 Sharmili $100 What?! th - 19 century British Prime Minister Benjamin Disraeli is reputed to have said: “There are three kinds of lies: lies, damned lies, and statistics.” - We’ll see that the sales manager wasn’t really lying, but he wasn’t being particularly honest either The mean - This is more commonly known as the arithmetic average - The mean of a distribution of scores is the sum of the scores divided by the number of scores - The formula for the mean of a population is: - This is not a formula for a statistic! This is a formula for a parameter! • Statistics are used to make guesses about population. If you have all of the X’s in a population, you don’t need to guess. - Parameters are typically represented by Greek letters The mean - The formula for the mean of a sample is: - Some books and articles use M. Some use X - Note the difference in the denominator of the fraction • Why would they change it? Consider our population of scores - N=5 3350, 100, 100, 100, 100 - What is the mean of this population? X - µ= N 3350+100+100+100+100 ¿ - ¿ ¿ 3750 ¿ - ¿ = 750 ¿ So he wasn’t lying - The sales manager wasn’t technically lying: the average was 750$ - But what happened?! 4 of the 5 sales people only sold 100$ worth of stats? - How is it that the mean was so high? Aneat thing about the mean - N=5 3350, 100, 100, 100, 100 - µ= 750 - Subtract the mean from each one of the scores in the population of scores • These differences represent the amount that each score deviates from the mean - Add up these differences The sum of the deviations X X-µ 3350 2600 100 -650 100 -650 100 -650 100 -650 - ∑ X−µ=2600−650−650−650−650=0 The mean - Now: - Imagine the sales manager had either said “sales are around 800$ per person” or “sales are around 700$ per person” - Calculate the summed deviations of the scores from these two estimates The sum of the deviations X X-800 3350 2550 100 -700 100 -700 100 -700 100 -700 - ∑ X−800=−250 The sum of the deviations X X-800 3350 2650 100 -600 100 -600 100 -600 100 -600 - ∑ X−700=250 The mean brings balance! ∑ X−µ=0 - - The mean is the point at which there is as much negative deviation on one side as there is positive deviation on the other! - No other value will do this! Aweighted mean - This class has two sections: • n section= 146 n sectionV • = 290 - Pretend that, following Quiz 1, the sample averages are as follows: M sectionB • = 87.90 M • section= 83.43 - What is the class average? The wrong way - The wrong way to do it would be to add up the means and divide by 2: • (87.90 + 83.43) / 2 = 85.67 - This gives you a score that is right between the two averages - Why is this wrong? The right way X overall X 1X 2 - M overal= = noverall n1+n 2 12833+24194 37027 - 146+290 = 436 = 84.92 - Because the pajama group has more people, it exerts more influence over the class average. • As such, the “weighted mean” is lower Computing the mean of a frequency distribution X f 5 2 4 2 3 3 2 1 1 2 - This is one of the tables we saw during last week’s lectures X - M = n - What is  X? • 31 - What is n? • 10 X f Xf 5 2 10 4 2 8 3 3 9 2 1 2 1 2 2 -  X =  Xf = 31 - N = f = 10 - So what is the mean score of this frequency distribution? • 3.1 Other neat things about the mean - N = 5: 5, 4, 3, 2, 1 - µ = ? • µ= 3 - Add 10 to each of the scores and recalculate the mean Other neat things about the mean - N = 5: 15, 14, 13, 12, 11 - µ = ? • µ= 13 - What is the difference between the first mean and the second mean? • Difference of 10. If you add something to all of the individual scores, the mean will go up by that much as well Other neat things about the mean - N= 5: 5, 4, 3, 2, 1 - Multiply each of the scores by 10 and recalculate the mean - If you add something to all of the individual scores, the mean will go up
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