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 Xµ
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 X800
3350 2550
100 700
100 700
100 700
100 700
 ∑ X−800=−250
The sum of the deviations
X X800
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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