1. PSYC 6130C UNIVARIATE ANALYSIS
Prof. James Elder

2. Introduction

3. What is (are) statistics?
A branch of mathematics concerned with understanding and summarizing collections of numbers
A collection of numerical facts
Estimates of population parameters, derived from samples
PSYC 6130, PROF. J. ELDER

4. What is this course about?
Applied statistics
Emphasizes methods, not proofs
Descriptive statistics
Inferential statistics
PSYC 6130, PROF. J. ELDER

5. Fall Term
PSYC 6130, PROF. J. ELDER

6. Winter Term
PSYC 6130, PROF. J. ELDER

7. Some Background
(Howell Ch. 1)

8. Variables and Constants
Constants are properties that never change (e.g., the speed of light in a vacuum ~3x108m/s).
Most physiological and psychological parameters of interest vary considerably
Between individuals (e.g., intelligence quotient)
Within individuals (e.g., heart rate)
Any variable whose variation is somewhat unpredictable is called a random variable (rv).
PSYC 6130, PROF. J. ELDER

9. Scales of measurement
Nominal scale: values are categories, having no meaningful correspondence to numbers.
PSYC 6130, PROF. J. ELDER

10. Scales of measurement
Ordinal scale: ordering is meaningful, but exact numerical values (if they exist) are not.
PSYC 6130, PROF. J. ELDER

11. Scales of measurement
Interval scale: values are numerically meaningful, and interval between two values is meaningful.
Example: Celsius temperature scale. It takes the same amount of energy to raise the temperature of a gram of water from 20 °C to 21 °C as it does to raise it from 30 °C to 31 °C.
Ratio scale: ratio of two values is also meaningful.
Example: Kelvin temperature scale. A gram of H20 at 300 K has twice the energy of a gram of H20 at 150 K.
Ratio scales require a 0-point corresponding to a complete lack of the substance being measured.
Example: a gram of H20 at 0 K has no heat (particles are motionless).
PSYC 6130, PROF. J. ELDER

12. Continuous vs Discrete Variables
A continuous variable may assume any real value within some range
PSYC 6130, PROF. J. ELDER

13. Continuous vs Discrete Variables
A discrete variable may assume only a countable number of values: intermediate values are not meaningful.
PSYC 6130, PROF. J. ELDER

14. Independent vs Dependent Variables
Experiments involve independent and dependent variables.
The independent variable is controlled by the experimenter.
The dependent variable is measured.
We seek to detect and model effects of the independent variable on the dependent variable.
Example: In a visual search task, subjects are asked to find the odd-man-out in a display of discrete items (e.g., a horizontal bar amongst vertical bars).
The number of items in the display is an independent variable.
Reaction time is the main dependent variable.
Typically, we observe a roughly linear relationship between the number of items and the reaction time.
PSYC 6130, PROF. J. ELDER

15. Experimental vs Correlational Research
Experimental study:
Researcher controls the independent variable.
Seek to detect effects on the dependent variable.
Direction of causation may be inferred (but may be indirect).
Correlational study:
There are no independent or dependent variables.
No variables are under control of the researcher.
Seek to find statistical relationships (dependencies) between variables.
Direction of causation may not normally be inferred.
PSYC 6130, PROF. J. ELDER

16. Correlational Studies: Examples
PSYC 6130, PROF. J. ELDER

17. Populations vs Samples
In human science, we typically want to characterize and make inferences not about a particular person (e.g., Uncle Bob) but about all people, or all people with a certain property (e.g., all people suffering from a bipolar disorder).
These groups of interest are called populations.
Typically, these populations are too large and inaccessible to study.
Instead, we study a subset of the group, called a sample.
In order to make reliable inferences about the population, samples are ideally randomly selected.
The population properties of interest are called parameters.
The corresponding measurements made on our samples are called statistics. Statistics are approximations (estimates) of parameters.
PSYC 6130, PROF. J. ELDER

18. Different Types of Populations and Samples
Outside of human science, populations do not necessarily refer to humans
e.g. populations may be of bees, algae, quarks, stock prices, pork belly futures, ozone levels, etc…
In clinical and social psychology you will often be conducting large-n studies on human populations.
In cognitive psychology, you will often be doing small-n within-subject studies involving repeated trials on the same subject.
Here, you may think of the ‘population’ as being the infinite set of responses you would obtain were you able to continue the experiment indefinitely.
The sample is the set of responses you were able to collect in a finite number of trials (e.g., 5000) on the same subject.
PSYC 6130, PROF. J. ELDER

19. Summation Notation i Xi Yi 1 2 3 … N 4
PSYC 6130, PROF. J. ELDER

20. Some Summation Rules
PSYC 6130, PROF. J. ELDER

21. Summary What is (are) statistics Variables and constants
Scales of measurement
Continuous and discrete variables
Independent and dependent variables
Experimental and correlational research
Populations and samples
Summation Notation
PSYC 6130, PROF. J. ELDER

22. Descriptive Statistics (Howell, Ch 2)

23. Frequency Tables
1991 U.S. General Social Survey: Number of Brothers and Sisters
PSYC 6130, PROF. J. ELDER

24. Bar Graphs and Histograms
PSYC 6130, PROF. J. ELDER

25. Grouped Frequency Distributions
Statistics Canada 2001 Census
Age of Respondent
What are the apparent limits?
What are the real limits?
PSYC 6130, PROF. J. ELDER

26. Percentiles and Percentile Ranks
Percentile: The score at or below which a given % of scores lie.
Percentile Rank: The percentage of scores at or below a given score
PSYC 6130, PROF. J. ELDER

27. Linear Interpolation to Compute Percentile Ranks
Statistics Canada 2001 Census Age of Respondent
PSYC 6130, PROF. J. ELDER

28. Linear Interpolation to Compute Percentiles
Statistics Canada 2001 Census Age of Respondent
PSYC 6130, PROF. J. ELDER

29. Measures of Central Tendency
The mode – applies to ratio, interval, ordinal or nominal scales.
The median – applies to ratio, interval and ordinal scales
The mean – applies to ratio and interval scales
PSYC 6130, PROF. J. ELDER

30. The Mode Defined as the most frequent value (the peak)
Applies to ratio, interval, ordinal and nominal scales
Sensitive to sampling error (noise)
Distributions may be referred to as unimodal, bimodal or multimodal, depending upon the number of peaks
PSYC 6130, PROF. J. ELDER

31. The Median Defined as the 50th percentile
Applies to ratio, interval and ordinal scales
Can be used for open-ended distributions
PSYC 6130, PROF. J. ELDER

32. The Mean Applies only to ratio or interval scales
Sensitive to outliers
PSYC 6130, PROF. J. ELDER

33. Properties of the Mean 1. 2. 3.
PSYC 6130, PROF. J. ELDER

34. Properties of the Mean (Cntd…)
PSYC 6130, PROF. J. ELDER

35. Measures of Variability (Dispersion)
Range – applies to ratio, interval, ordinal scales
Semi-interquartile range – applies to ratio, interval, ordinal scales
Variance (standard deviation) – applies to ratio, interval scales
PSYC 6130, PROF. J. ELDER

36. Range Interval between lowest and highest values
Generally unreliable – changing one value (highest or lowest) can cause large change in range.
Range = 79 drinks
PSYC 6130, PROF. J. ELDER

37. Semi-Interquartile Range
The interquartile range is the interval between the first and third quartile, i.e. between the 25th and 75th percentile.
The semi-interquartile range is half the interquartile range.
Can be used with open-ended distributions
Unaffected by extreme scores
SIQ = 2.5 drinks
PSYC 6130, PROF. J. ELDER

38. Population Variance and Standard Deviation
PSYC 6130, PROF. J. ELDER

39. Sample Variance and Standard Deviation
PSYC 6130, PROF. J. ELDER

40. Degrees of Freedom
PSYC 6130, PROF. J. ELDER

41. Computational Formulas for Variance
PSYC 6130, PROF. J. ELDER

42. Properties of the Standard Deviation1.
PSYC 6130, PROF. J. ELDER

43. Properties of the Standard Deviation (cntd…)2.
PSYC 6130, PROF. J. ELDER

44. Standard Deviation Example
cf. SIQ = 2.5 drinks
range = 79 drinks
PSYC 6130, PROF. J. ELDER

45. Skew The mean and median are identical for symmetric distributions.
Skew tends to push the mean away from the median, toward the tail (but not always)
Median=3
Mean=6.7
PSYC 6130, PROF. J. ELDER

46. Skewness Properties of skewness
Positive for positive skew (tail to the right)
Negative for negative skew (tail to the left)
Dimensionless
Invariant to shifting or scaling data (adding or multiplying constants)
PSYC 6130, PROF. J. ELDER

47. Dealing with Outliers Trimming: Transforming
Throw out the top and bottom k% of values (k=5%, for example).
May be justified if there is evidence for confounding process interfering with the dependent variable being studied
Example: participant blinks during presentation of a visual stimulus
Example: participant misunderstands a question on a questionnaire.
Transforming
Scores are transformed by some function (e.g., log, square root)
Often done to reduce or eliminate skewness
PSYC 6130, PROF. J. ELDER

48. Log-Transforming Data
skewness=0.67
skewness=0.08
PSYC 6130, PROF. J. ELDER

49. End of Lecture 1
Sept 10, 2008

50. Kurtosis kurtosis>0: leptokurtic (Laplacian)
kurtosis=0: mesokurtic (Gaussian)
kurtosis<0: platykurtic
PSYC 6130, PROF. J. ELDER

51. Summary Measures of central tendency Measures of dispersion Skew
Kurtosis
PSYC 6130, PROF. J. ELDER