1. Reasoning in Psychology Using Statistics
2017

2. Descriptive statistics
Summaries or pictures of the distribution
Numeric descriptive statistics
Shape: modality, and skew (and kurtosis, not cover much)
Measures of Center: Mode, Median, Mean
Measures of Variability (Spread): Range, Inter-Quartile Range, Standard Deviation (& variance)
Descriptive statistics

3. Useful to summarize or describe distribution with single numerical value.
Value most representative of the entire distribution, that is, of all of the individuals
Central Tendency: 3 main measures
Mean (M)
Median (Mdn)
Mode
Note: “Average” may refer to each of these three measures, but it usually refers to Mean.
Measures of Center

4. The Mean Most commonly used measure of center Arithmetic average
Computing the mean
Divide by the total number in the population
Formula for population mean (a parameter):
Add up all of the X’s
Formula for sample mean (a statistic):
Divide by the total number in the sample M=
Note: Mean is mathematical result, not necessarily score on scale (e.g., average of 2.5 children)
The Mean

5. The Mean Conceptualizing the mean As the center of the distribution
As the representative score in the distribution
The Mean

6. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
The Mean

7. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
The Mean

8. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
The Mean

9. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
Balancing point
The Mean

10. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
Balancing point
1+10 = 11
Mean = 11/2 = 5.5
The Mean

11. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
Balancing points
The Mean

12. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
The Mean

13. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
The Mean

14. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
The Mean

15. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
The Mean

16. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
Balancing point
= 18
Mean = 18/3 = 5.5
The Mean

17. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
= 18
Mean = 18/3 =
6.0
The Mean

18. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
= 18
Mean = 18/3 = 6.0
The Mean

19. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
= 18
Mean = 18/3 = 6.0
The Mean

20. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
What happens if we add an observation to our distribution?
New Balancing point
= 18
Mean = 18/3 = 6.0
The Mean

21. The Mean Conceptualizing the mean As center of distribution
As the representative score in the distribution
To be fair, let’s give
everybody the
same amount.
Girl Scout bake sale for camping trip
$30 $6
$15
$12
$25
$18
$13
=119
119/7 = 17
The Mean

22. The Mean Conceptualizing the mean As center of distribution
As representative score in distribution
Girl Scout bake sale for camping trip
$17
$17
$17
$17
$17
$17
$17
=119
119/7 = 17
So everybody is represented by same score, the mean is the “standard”
=119
119/7 = 17
The Mean

23. A weighted mean Suppose that you combine 2 groups together.
How do you compute new group mean?
Average the 2 averages
But it only works this way when the two groups have exactly the same number of scores
A weighted mean

24. A weighted mean Suppose that you combine 2 groups together.
How do you compute new group mean?
$205!?
I only
have $191
Group 1
Group 2
New Group
A weighted mean

25. A weighted mean Suppose that you combine 2 groups together.
How do you compute new group mean?
Group 1
$12
$25
$30 $6
$18
$15
$13
Group 2
$25
$30
$17
New Group
=191
Mean = 191/10 = 19.1
$12
$30 $6
$13
$30
$18
$17
$25
$25
$15
A weighted mean

26. The mean is the representative score in
Suppose that you combine 2 groups together.
How do you compute new group mean?
The mean is the representative score in
the distribution
Group 1
$17
Group 2
$24
New Group
A weighted mean

27. Characteristics of a mean
Change/add/delete a given score, then the mean will change.
Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7= /7 = $16 (instead of $17) 17
Characteristics of a mean

28. Characteristics of a mean
Change/add/delete a given score, then the mean will change.
Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7= /7 = $16 (instead of $17) 17
Characteristics of a mean

29. Characteristics of a mean
Change/add/delete a given score, then the mean will change.
Suppose that one of the girl scouts discovered that she had really made $23 instead of $30. So now the total is 119-7= /7 = $16 (instead of $17) 16
Characteristics of a mean

30. Characteristics of a mean
Change/add/delete a given score, then the mean will change.
Add/subtract a constant to each score, then the mean will change by adding(subtracting) that constant.
Suppose that you want to factor out a $2 camping fee for each girl scout. Subtract 2 from each amount. Now the total is $105, so the mean is 105/7 = $15.
But notice you could have just subtracted $2 from the previous mean of $17 and arrived at the same answer.
Characteristics of a mean

31. Characteristics of a mean
Change/add/delete a given score, then the mean will change.
Add/subtract a constant to each score, then the mean will change by adding(subtracting) that constant.
Suppose that the troop sponsor agreed to match the money made by each girl scout (they give each girl scout an additional amount of money equal to however much each made on the sale). So now the total is $238, and the mean for each girl is 238/7 = $34
Which is 2 times the original mean
Multiply (or divide) each score by a constant, then the mean will change by being multiplied by that constant.
Characteristics of a mean

32. Median divides distribution in half: 50% of individuals in distribution have scores at or below the median.
Case1: Odd number of scores
Step1: put scores in order
$12
$25
$30 $6
$18
$15
$13
The median

33. Median divides distribution in half: 50% of individuals in distribution have scores at or below the median.
Case1: Odd number of scores
Step1: put scores in order
$15 $6
$25
$30
$12
$13
$18
Step2: find middle score
That’s the median, a score on scale
The median

34. Median divides distribution in half: 50% of individuals in distribution have scores at or below the median.
Case2: Even number of scores
Step1: put scores in order
$12
$25
$30
$18
$15
$13 $6
Step2: find middle 2 scores
Step3: find arithmetic average of 2 middle scores
That’s the median
Note: mathematical result not a score on scale
The median

35. The mode Mode: score or category with greatest frequency.
Pick variable in frequency table or graph with highest frequency (mode always a score on scale).
Mode = 5
Modes =
2, 8
T-shirt size
Mode =
Medium
The mode

36. Depends on a number of factors, like scale of measurement and shape.
The mean is the most preferred measure and it is closely related to measures of variability
However, there are times when the mean is not the appropriate measure.
Which center when?

37. Which center when? If data on nominal scale: Mode only
Unranked categories (e.g. eye color)
Not a numeric scale
Can not do arithmetic operations on values
Can not calculate cumulative percentages
Eye color
Green
Mode = Brown
Median =
Which center when?

38. Which center when? If data on ordinal scale: Median (plus Mode)
Not a numeric scale (e.g., T-shirt size)
Can not do arithmetic operations on values
Can calculate cumulative percentages on frequencies (median is score at 50th percentile)
Median of T-shirt size = Medium
Mode of T-shirt size = Medium
Which center when?

39. Which center when? Median If data on interval or ratio scale BUT:
Distributions open-ended
Response category like “5 or more”
Extreme values unknown, so can not calculate mean
Distributions skewed with long tails
Extreme values over influence mean
E.g., income sample of 50
47 middle income ($60,000-$100,000) and 3 millionaires or billionaires
Median = $80,000
Mean = $135,000 or $60,000,000
Median
(plus Mode)
Which center when?

40. If data on interval or ratio scale AND no exclusionary conditions: Mean (plus Median) (plus Mode)
Numeric scale
Can do arithmetic calculations on values
Have benefit of other statistics using the mean, such as standard deviation
Which center when?

41. In Summation Situation Most Representative Least Representative
Can’t Use
Nominal
Mode
Median/Mean
Ordinal
Median
Mean
Skewed interval or ratio
Open ended interval or ratio
Interval or Ratio
In Summation

42. Which center when? Impact of shape on center (interval or ratio scale)
mean
median
, 2 modes
mode
mean
median = = =
Positively skewed distribution
Negatively skewed distribution
median
mode
mode
median
mean
mean
>
>
<
<
Mean & median pulled toward tail
Which center when?

43. Chicago distributions
Mode , ,000
Median , ,600
Mean ? ,212
Chicago distributions
Check out your hometown:

44. Buyer beware: Know your distribution
The average price of houses in this neighborhood is …
Mode , ,000
Median , ,600
Mean ? ,212
selling
buying
When you say “average” are you talking about the median or the mean?
Buyer beware: Know your distribution

45. Wrap up Today’s lab Questions?
Compute mean, median, & mode both by hand & using SPSS
Questions?
Wrap up