1. Last class we discussed:
Ways of knowing reality (science is chief means for psychology)
Classification of Research Studies
(1) Design (experiment / correlational / descriptive)
(2) Setting (field / lab)
(3) Data-collection method (self-report / observation)

2. By the end of today’s class, you will be able to:
Define statistics and its two branches
Understand the four scales of measurement
Deal with measures of central tendancy
Calculate standard deviation
Discuss the properties of a distribution curve

3. Statistics
is the branch of mathematics that deals with the collection, organization, and analysis of numerical data.
Even if your data doesn’t begin as a set of numbers, you can quantify it (turn it into numbers) and use statistical analysis.

4. What are statistics?
Numerical representations of large amounts of data
Two Branches
Descriptive – tell us about the numbers
Inferential – tell us what the numbers mean
The work of the statistician is no longer confined to gathering and tabulating data (descriptive statistics), but is chiefly a process of interpreting the information (inferential statistics).

5. How do we obtain statistics?
Through Quantification (#)
All behaviour needs to be quantified to be analysed statistically
How do we quantify behaviour?
Though scales of measurement

6. What are Scales of Measurement?
Measurement is the assignment of numbers to objects or events in a systematic fashion.
Four levels of measurement are commonly distinguished:
Nominal
Ordinal
Interval
Ratio
“NOIR”

7. Nominal Based on Categories
No quantitative information is conveyed and no ordering of the items is implied.
Gender
Religion
Ethnicity

8. Nominal scale Classifies data according to a category only.
Measurement: Frequency distributions. How often do certain items or responses occur?

9. Ordinal
Ordered in the sense that higher numbers represent higher values.
However, the intervals between the numbers are not necessarily equal.
e.g., “Strongly Disagree, Disagree, Neither Agree nor Disagree, Agree, Strongly Agree”

10. Ordinal scale Classifies data according to rank.
The difference between mild and average hotness may not represent the same difference as the difference between a rating of hot and very hot.
There is no "true" zero point for ordinal scales since the zero point is chosen arbitrarily.

11. Interval
One unit on the scale represents the same magnitude on the trait or characteristic being measured across the whole range of the scale.
For example, if anxiety were measured on an interval scale, then a difference between a score of 10 and a score of 11 would represent the same difference in anxiety as would a difference between a score of 50 and a score of 51.

12. Interval scale Does not have true zero point (it is arbitrary).
On the anxiety scale: score of 30 NOT twice as anxious as score of 15. We know they are MORE anxious, but that’s about where it ends.
Similarly, a score of zero does not mean that they have zero anxiety.
True interval measurement is rare to nonexistent in the behavioral sciences. Most measurement scales in psychology are ordinal.

13. Ratio All of the previous + absolute zero
e.g. weight and height
This scale has an absolute zero.
Temperature of 300 Kelvin is twice as high as a temperature of 150 Kelvin.
Time can have an absolute zero (no time).

14. Ratio scale Similar to interval scales, but has a true zero point.
Racer C’s speed (below) was twice as fast as Racer A’s speed.

15. Levels of Measurement
Mutually Exclusive
Rank Order
Equal Intervals
Absolute Zero
Nominal 
Ordinal
Interval
Ratio
Note. Mutually Exclusive: Every point on every measurement scale is separate from every other point. You can’t be red and blue at the same time.

16. Concept Review Which rating scale is based on categories? Nominal
Which rating scale has an absolute zero?
Ratio
Most psychological scales, such as the CES-D, which measures depression, use which rating scale?
Ordinal
How do we interpret data?
Inferential Statistics

17. 2 main branches of these statistics: Inferential / Descriptive

18. Inferential Statistics
Draw a sample from larger population
The sample is smaller than the population and should be representative of the population.
Use inferential statistics to draw inferences about the population based on the sample.

19. Statistics prescribes ways to take measurements of smaller groups of people and infer the findings true for all people matching the defined criteria, such as every Canadian adult between ages 25 and 34. The smaller group is our sample, and we infer from the sample to the population.

20. Descriptive Statistics
Aims to describe sets of data values
Ex: a group of individual test scores.
Each data value is a single measurement of some attribute being observed.
The term data set refers to all data values considered in a set of statistical calculations.
Descriptive statistics summarize sets of information.

21. Descriptive Stats cont’d
Examples
Central Tendency (mean, median, mode)
Variability
Shape of Distributions
Relations (Correlation)
We’ll discuss each of these in turn

22. What are measures of central tendency?
Each is a score that represents the typical performance of the sample
Mode
Mean
Median

23. Mode
Answers the question, "Which number or object appears the most often in our data set?“
Can have more than one mode.
Example: Array of test scores 2,3,4,5,6,6,7,8,9,9  Mode is 6 & 9. We say it’s bimodal.
The mean is the sum of the individual scores divided by the total number of scores.

24. Looking ahead: median and mean are a little more slippery (but not much more).
It’s from the median and mean that we get the term “central tendency,” measures of the "center" (average) of the scores in a given data set.

25. Mean
sum of the individual scores divided by the total number of scores.
i.e. the average

26. Median …is the score that’s right in the middle of your data set
Median …is the score that’s right in the middle of your data set. 1/2 of the data set’s values less than or equal to the median; 1/2 of the numbers in the data set will have values equal to or greater than the median. To find the median of a finite list of numbers: Arrange all observations from lowest value to highest value and pick the middle one. If there are an even number of observations, you can take the mean of the two middle values.

27. Central Tendency Example
Hours watching The Bachelor per day for 10 people
Subject Hours
1. 2
2. 3
3. 2
4. 7
5. 3
6. 5
7. 2
8. 2
9. 1
Take a minute to calculate mean, median, and mode ?

28. Central Tendency Example
Hours watching The Bachelor per day for 10 people:
Subject Hours
1. 2
2. 3
3. 2
4. 7
5. 3
6. 5
7. 2
8. 2
9. 1
Mean = 3 (add all scores and divide by 10)
Median = 2.5
Mode = 2

29. Something important to keep in mind when writing papers:
A conclusion is simply the place where you got tired of thinking.

30. Variability

31. Consider Two Streets Conservative Street 45, 46, 47, 44, 43 Mean = ?
Wild Alley
25, 35, 45, 55, 65
Mean = ?

32. How are these streets different?
Consider Two Streets
Conservative Street
45, 46, 47, 44, 43
Mean = 45
Median =
Wild Alley
25, 35, 45, 55, 65
Mean = 45
Median =
How are these streets different?

33. Consider Two Streets Variability Conservative Street
45, 46, 47, 44, 43
Mean = 45
Median = 45
Wild Alley
25, 35, 45, 55, 65
Mean = 45
Median = 45
The means are the same. The medians are the same.
How are these streets different?
Variability

34. What is Variability? Extent of dispersion around the mean.
Are the scores all pretty close together or more spread out?

35. Variance
An index of variability (i.e. one way to express variability).

36. Variance Formula
- 1
(Take the difference between each score and the mean, squaring each of these deviations, and then calculating the mean of the squared deviations. Finally, you divide by the number of scores minus one.)

37. The problem with variance Has an ATTITUDE It is an index in the wrong units! Everything was SQUARED to get the variance. We need something else…

38. H m m m (ideas?)

39. Standard Deviation
Square root of the squared deviations about the mean.
i.e. square root of variance
(Wasn’t that disturbingly simple?)

40. Standard Deviation Formula
Definitional Formula
The standard deviation is equal to the square root of the sum of the squared deviations divided by the total number of scores
- 1

41. Practice Run: Calculate SD and Variance
Hours watching tv per day for 10 people
Subject Hours (Hours – mean) squared
1. 2
2. 3
3. 2
4. 7
5. 3
6. 5
7. 2
8. 2
9. 1
Calculate SD and variance for the data.

42. First step: Next: Then: raw score minus mean.
then square the deviations.
Sum these together.
Next:
Divide by n-1 to get variance.
Then:
Take the square root of this for SD.

43. Calculate SD and Variance
Hours watching tv per day for 10 people
Subject Hours (Hours – mean) squared – – –

44. Sum = 28
N – 1 = 9
28/9 = 3.11
Square Root of 3.11 = 1.76

45. When you collect data…
Assumption – normality. We assume that many things in life take the form of a “normal curve” when graphed, with most scores occurring close to the mean, and some scores more spread out.
Assumption sometimes violated
Example: Give a middle school-level math test to 6-year-olds and to 18-year-olds. Result will not be a normal curve; rather, you will see two clumps: high scores (18-yr-olds) and low scores (6-yr-olds).

46. Properties of a distribution curve
Skewness
Happens when the mean, median, and mode aren’t the same.
symmetry of a curve
determined by the tail in the curve

47. Positively skewed Tail on the right side of the distribution
Most scores cluster on the left side of the distribution
If this was a math test, most people had a low score and just a few performed very well.

48. Negatively skewed Tail on the left side of the distribution
Most scores cluster on the right side of the distribution
How did people perform on the math test this time?

49. Can’t use certain formula (t-tests, correlation)
Why is Skew a Problem?
Can’t use certain formula (t-tests, correlation)
Assumption of normality is important
Mathematically, we can manipulate the data to try to normalize distribution
Or, use different formula

50. Kurtosis Peakedness of the curve compared to the normal curve.
Are the scores grouped more toward the middle (the mean) or more toward the tails of the distribution

51. Leptokurtic Pointed shape Kurtosis >3
Most scores cluster at the middle of the distribution
Simultaneously “peaked” centre and “fat” tails

52. Platykurtic Flat shape Kurtosis <3
Many scores spread toward the tails of the distribution
Simultaneously less peaked and thinner tails

53. Mesokurtic
Normal curve distribution
Kurtosis = 0 (no skew)

54. Normal Distribution Bell shape Mean, median, and mode are equal
Located at centre of distribution

55. Area of the Normal Curve
34% is one SD above or below the mean
68% within one SD above and below
47.5% of scores is between 0 and 2 SD
95% within two SD above and below