1. Statistics Review
Levels of Measurement

2. Levels of Measurement Nominal scale
Nominal measurement consists of assigning items to groups or categories.
No quantitative information is conveyed and no ordering of the items is implied.
Nominal scales are therefore qualitative rather than quantitative.
Examples: Religious preference, race, and gender are all examples of nominal scales
Statistics: Sum, Frequency Distributions

3. Ordinal Scale
Measurements with ordinal scales are ordered: higher numbers represent higher values.
However, the intervals between the numbers are not necessarily equal.
There is no "true" zero point for ordinal scales since the zero point is chosen arbitrarily.
For example, on a five-point Likert scale, the difference between 2 and 3 may not represent the same difference as the difference between 4 and 5.
Also, lowest point was arbitrarily chosen to be 1. It could just as well have been 0 or -5.

4. Interval & Ratio Scales
On interval measurement scales, one unit on the scale represents the same magnitude on the trait or characteristic being measure across the whole range of the scale.
For example, on an interval/ratio scale of anxiety, a difference between 10 and 11 would represent the same difference in anxiety as between 50 and 51.

5. Statistics Review
Histograms

6. What can histograms tell you
A convenient way to summarize data (especially for larger datasets)
Shows the distribution of the variable in the population
Gives an approximate idea of the summary and spread of the variable

7. Distribution of No of Graphics on web pages (N=1873)
95.0
90.0
85.0
80.0
75.0
70.0
65.0
60.0
55.0
50.0
45.0
40.0
35.0
30.0
25.0
20.0
15.0
10.0
5.0
0.0
400
300
200
100
Mean = 17.93
Median = 16.00
Std. Dev = 17.92
N = 1873
Graphic Count

8. Statistics Review
Mean and Median

9. Mean and Median Mean is arithmetic average, median is 50% point
Mean is point where graph balances
Mean shifts around,
Median does not shift much, is more stable
Computing Median:
for odd numbered N
find middle number
For even numbered N
interpolate between middle 2,
e.g. if it is 7 and 9, then 8 is the median

10. Instability of mean Distribution of word count (N=1897)
WORDCNT2
4000.0
3600.0
3200.0
2800.0
2400.0
2000.0
1600.0
1200.0
800.0
400.0
0.0
800
600
400
200
Mean = 368.0
Median = 223
Minimum = 0
Maximum = 4132

11. Distribution of word count (N=1873), top 1% removed
2400.0
2200.0
2000.0
1800.0
1600.0
1400.0
1200.0
1000.0
800.0
600.0
400.0
200.0
0.0
500
400
300
200
100
Mean = 333.4
Median = 220
Minimum = 0
Maximum = 4132
WORDCNT2

12. Statistics Review
Standard Deviation

13. Standard Deviation: a measure of spread
The SD says how far away numbers
on a list are from their average.
Most entries on the list will be
somewhere around one SD away
from the average. Very few will be
more than two or three SD’s away.

14. Properties of the standard deviation
The standard deviation is in the same units as the mean
The standard deviation is inversely related to sample size (therefore as a measure of spread it is biased)
In normally distributed data 68% of the sample lies within 1 SD

15. Normal Probability Curve
Statistics Review
Normal Probability Curve

16. Properties of the Normal Probability Curve
The graph is symmetric about the mean (the part to the right is a mirror image of the part to the left)
The total area under the curve equals 100%
Curve is always above horizontal axis
Appears to stop after a certain point (the curve gets really low)

17. The graph is symmetric about the mean =
1 SD= 68%
2 SD = 95%
3 SD= 99.7%
The graph is symmetric about the mean =
The total area under the curve equals 100%
Mean to 1 SD = +- 68%
Mean to 2 SD = +- 95%
Mean to 3 SD = %
You can disregard rest of curve

18. It is a remarkable fact that many histograms in real life tend to follow the Normal Curve.
For such histograms, the mean and SD are good summary statistics.
The average pins down the center, while the SD gives the spread.
For histogram which do not follow the normal Curve, the mean and SD are not good summary statistics.
What when the histogram is not normal ...

19. Use inter quartile range
75th percentile - 25th percentile
Can be used when SD is too influenced by outliers
Note.
A percentile is a score below which a certain % of sample is

20. Statistics Review
Population and Sample

21. An investigator usually wants to generalize about a class of individuals/things (the population) For example: in forecasting the results of elections, population is all eligible voters

22. Usually there are some numerical facts about the population (parameters) which you want to estimate
You can do that by measuring the same aspect in the sample (statistic)
Depending on the accuracy of your measurement, and how representative your sample is, you can make inferences about the population

23. Scatter Plots and Correlations
Statistics Review
Scatter Plots and Correlations

24. Example Scatterplots y y x 4 High correlation Low correlation x x x x

25. What is a Correlation Coefficient
A measure of degree of relationship between two variables.
Sign refers to direction.
Based on covariance
Measure of degree to which large scores go with large scores, and small scores with small scores
Pearson’s correlation coefficient is most often used

26. Factors Affecting r Range restrictions Outliers Nonlinearity
e.g. anxiety and performance
Heterogeneous subsamples
Everyday examples

27. The effect of outliers on correlations
Dataset: 20 cases selected from darts and pros 80 60 40 20
-20
-40
Pros
DARTS
r = .80

28. Effect of linear transformations of data
no effect on Pearson's correlation coefficient.
Example: r between height and weight is the same regardless of whether height is measured in inches, feet, centimeters or even miles.
This is a very desirable property since choice of measurement scales that are linear transformations of each other is often arbitrary.

29. Non linear relationships13
Non linear relationships
Example: Anxiety and Performance
r = .07

30. The interpretation of a correlation coefficient
Ranges from –1 to 1
No correlation in the data means you will get a is 0 r or near it
Suffers from sampling error (like everything else!). So you need to estimate true population correlation from the sample correlation.

31. Statistics Review
Hypothesis Testing

32. Null and Alternative Hypothesis
Sampling error implies that sometimes the results we obtain will be due to chance (since not every sample will accurately resemble the population)
The null hypothesis expresses the idea that an observed difference is due to chance.
For example: There is no difference between the norms regarding the use of and voice mail

33. The alternative hypothesis
The alternative hypothesis (the experimental hypothesis) is often the one that you formulate:
For example: There is a correlation between people’s perception of a website’s reliability and the probability of their buying something on the site
Why bother to have a null hypothesis?
Can you reject the null hypothesis

34. One Tailed and Two Tailed tests
One tailed tests: Based on a uni-directional hypothesis
Hypothesis: Training will reduce number of problems users have with Powerpoint
Two tailed tests: Based on a bi-directional hypothesis
Hypothesis: Training will change the number of problems with PP

35. Implications of one and two tailed tests
Mean Usability Index
7.25
7.00
6.75
6.50
6.25
6.00
5.75
5.50
5.25
5.00
4.75
4.50
4.25
4.00
3.75
Sampling Distribution
Population for usability of Powerpoint
Frequency
1400
1200
1000
800
600
400
200
Std. Dev = .45
Mean = 5.65
N =
Identify region
Unidirectional hypothesis: .05 level
Bidirectional hypothesis: .05 level

36. Significance levels
PowerPoint example:
If we set significance level at .05 level,
5% of the time we will find a difference by chance
95% of the time the difference will be real
If we set significance level at .01 level
1% of the time we will find a difference by chance
99% of time difference will be real
What happens if we decrease our significance level from .01 to .05
Probability of finding differences that don’t exist goes up (criteria becomes more lenient)

37. Effect of decreasing significance level from .01 to .05
Probability of finding differences that don’t exist goes up
Also called Type I error (Alpha)
Effect of increasing significance from .01 to .001
Probability of not finding differences that exist goes up
Also called Type II error (Beta)

38. Significance levels for usability
For usability, if you are set out to find problems: setting lenient criteria might work better (you will identify more problems)

39. Degree of Freedom
The number of independent pieces of information remaining after estimating one or more parameters
Example: List= 1, 2, 3, 4 Average= 2.5
For average to remain the same three of the numbers can be anything you want, fourth is fixed
New List = 1, 5, 2.5, __ Average = 2.5

40. Comparing Means: t tests
Statistics Review
Comparing Means: t tests

41. Major Points T tests: are differences significant?
One sample t tests, comparing one mean to population
Within subjects test: Comparing mean in condition 1 to mean in condition 2
Between Subjects test: Comparing mean in condition 1 to mean in condition 2

42. One sample t test
Mean of population known, but standard deviation (SD) not known
Compute t statistic
Compare t to tabled values (for relevant degree of freedom) which show critical values of t

43. Factors Affecting t Difference between sample and population means
Magnitude of sample variance
Sample size

44. Factors Affecting Decision
Significance level
One-tailed versus two-tailed test

45. Within subjects/ Repeated Measures / Related Samples t test
Correlation between before and after scores
Causes a change in the statistic we can use
Advantages of within subject designs
Eliminate subject-to-subject variability
Control for extraneous variables
Need fewer subjects

46. Disadvantages of Within Subjects
Order effects
Carry-over effects
Subjects no longer naïve
Change may just be a function of time
Sometimes not logically possible

47. Between subjects t test
Distribution of differences between means
Heterogeneity of Variance
Nonnormality

48. Assumptions of Between Subjects t tests
Two major assumptions
Both groups are sampled from populations with the same variance
“homogeneity of variance”
Both groups are sampled from normal populations
Assumption of normality
Frequently violated with little harm.

49. Statistics Review
Analysis of Variance

50. Analysis of Variance
ANOVA is a technique for using differences between sample means to draw inferences about the presence or absence of differences between populations means.
Similar to t tests in two sample case
Can handle cases where there are more than two samples

51. Assumptions Observations normally distributed within each population
Population variances are equal
Homogeneity of variance or homoscedasticity
Observations are independent

52. Logic of the Analysis of Variance
Null hypothesis: Population means from different conditions are equal
Mean1 = Mean2 = Mean 3
Alternative hypothesis: H1
Not all population means equal.
Cont.

53. Lets visualize total amount of variance in an experiment
Total Variance = Mean Square Total
Between Group Differences
(Mean Square Group)
Error Variance
(Individual Differences + Random Variance)
Mean Square Error
F ratio is a proportion of the MS group/MS Error.
The larger the group differences, the bigger the F

54. When there are more than two groups
Significant F only shows that not all groups are equal
We want to know what groups are different.
Such procedures are designed to control familywise error rate.
Familywise error rate defined
Contrast with per comparison error rate

55. Multiple Comparisons
The more tests we run the more likely we are to make Type I error.
Good reason to hold down number of tests

56. How to make inferences What are significant effects in your results?
If one t test is significant, check the distribution, where does the difference lie: Is it in the mean, is it the SD, does one variable have much greater range than another.
Next conduct another independent analysis which can verify finding. For example: Check the