This document provides an overview of descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, measures of central tendency, and measures of variability. Inferential statistics allow generalization from samples to populations through hypothesis testing, which involves specifying a null hypothesis and alternative hypothesis. Statistical significance is determined by calculating a p-value and comparing it to the significance level alpha to either reject or fail to reject the null hypothesis, with Type I and Type II errors a possibility. Common inferential tests include t-tests, ANOVAs, and meta-analyses.

1. Fall 2013
Lecture 5: Chapter 5
Statistical Analysis of Data
…yes the “S” word

2. What is a Statistic????
Population
Sample
Sample
Sample
Sample
Parameter: value that describes a population
Statistic: a value that describes a sample
PSYCH  always using samples!!!

3. Descriptive & Inferential Statistics
Descriptive Statistics
• Organize
• Summarize
• Simplify
• Presentation of data
Inferential Statistics
• Generalize from
samples to pops
• Hypothesis testing
• Relationships
among variables
Describing data Make predictions

4. Descriptive Statistics
3 Types
1. Frequency Distributions 3. Summary Stats
2. Graphical Representations
# of Ss that fall
in a particular category
Describe data in just one
number
Graphs & Tables

5. 1. Frequency Distributions
# of Ss that fall
in a particular category
How many males and how many females are
in our class?
Frequency
(%)
? ?
?/tot x 100 ?/tot x 100
-----% ------%
total
scale of measurement?
nominal

6. 1. Frequency Distributions
# of Ss that fall
in a particular category
Categorize on the basis of more that one variable at same time
CROSS-TABULATION
Democrats
Republican
total
24 1 25
19 6 25
Total 43 7 50

7. 1. Frequency Distributions
How many brothers & sisters do you have?
# of bros & sis Frequency
7 ?
6 ?
5 ?
4 ?
3 ?
2 ?
1 ?
0 ?

8. 2. Graphical Representations
Graphs & Tables
Bar graph (ratio data - quantitative)

9. Histogram of the categorical variables
2. Graphical Representations

10. Polygon - Line Graph
2. Graphical Representations

11. 2. Graphical Representations
Graphs & Tables
How many brothers & sisters do you have?
Lets plot class data: HISTOGRAM
# of bros & sis Frequency
7 ?
6 ?
5 ?
4 ?
3 ?
2 ?
1 ?
0 ?

12. Altman, D. G et al. BMJ 1995;310:298
Central Limit Theorem: the larger the sample size, the closer a distribution
will approximate the normal distribution or
A distribution of scores taken at random from any distribution will tend to
form a normal curve
jagged
smooth

13. 2.5% 2.5%
5% region of rejection of null hypothesis
Non directional
Two Tail
body temperature, shoe sizes, diameters of trees,
Wt, height etc…
IQ
68%
95%
13.5%
13.5%
Normal Distribution:
half the scores above
mean…half below
(symmetrical)

15. Summary Statistics
describe data in just 2 numbers
Measures of central tendency
• typical average score
Measures of variability
• typical average variation

16. Measures of Central Tendency
• Quantitative data:
– Mode – the most frequently occurring
observation
– Median – the middle value in the data (50 50 )
– Mean – arithmetic average
• Qualitative data:
– Mode – always appropriate
– Mean – never appropriate

17. Mean
• The most common and most
useful average
• Mean = sum of all observations
number of all observations
• Observations can be added in
any order.
• Sample vs population
• Sample mean = X
• Population mean =m
• Summation sign =
• Sample size = n
• Population size = N
Notation


18. Special Property of the Mean
Balance Point
• The sum of all observations expressed as
positive and negative deviations from the
mean always equals zero!!!!
– The mean is the single point of equilibrium
(balance) in a data set
• The mean is affected by all values in the data
set
– If you change a single value, the mean changes.

19. The mean is the single point of equilibrium (balance) in a data set
SEE FOR YOURSELF!!! Lets do the Math

20. Summary Statistics
describe data in just 2 numbers
Measures of central tendency
• typical average score
Measures of variability
• typical average variation
1. range: distance from the
lowest to the highest (use 2
data points)
2. Variance: (use all data points)
3. Standard Deviation
4. Standard Error of the Mean

21. Descriptive & Inferential Statistics
Descriptive Statistics
• Organize
• Summarize
• Simplify
• Presentation of data
Inferential Statistics
• Generalize from
samples to pops
• Hypothesis testing
• Relationships
among variables
Describing data Make predictions

22. Measures of Variability
2. Variance: (use all data points):
average of the distance that each score is from
the mean (Squared deviation from the mean)
Notation for variance
s2
3. Standard Deviation= SD= s2
4. Standard Error of the mean = SEM = SD/ n

23. Inferential Statistics
Population
Sample
Draw inferences about the
larger group
Sample
Sample
Sample

24. Sampling Error: variability among
samples due to chance vs population
Or true differences? Are just due to
sampling error?
Probability…..
Error…misleading…not a mistake

25. Probability
• Numerical indication of how likely it is that a
given event will occur (General
Definition)“hum…what’s the probability it will rain?”
• Statistical probability:
the odds that what we observed in the sample did
not occur because of error (random and/or
systematic)“hum…what’s the probability that my results
are not just due to chance”
• In other words, the probability associated with
a statistic is the level of confidence we have that
the sample group that we measured actually
represents the total population

26. data
Are our inferences valid?…Best we can do is to calculate probability
about inferences

27. Inferential Statistics: uses sample data
to evaluate the credibility of a hypothesis
about a population
NULL Hypothesis:
NULL (nullus - latin): “not any”  no
differences between means
H0 : m1 = m2
“H- Naught”
Always testing the null hypothesis

28. Inferential statistics: uses sample data to
evaluate the credibility of a hypothesis
about a population
Hypothesis: Scientific or alternative
hypothesis
Predicts that there are differences
between the groups
H1 : m1 = m2

29. Hypothesis
A statement about what findings are expected
null hypothesis
"the two groups will not differ“
alternative hypothesis
"group A will do better than group B"
"group A and B will not perform the same"

30. Inferential Statistics
When making comparisons
btw 2 sample means there are 2
possibilities
Null hypothesis is true
Null hypothesis is false
Not reject the Null Hypothesis
Reject the Null hypothesis

31. Possible Outcomes in
Hypothesis Testing (Decision)
Null is True Null is False
Accept
Reject
Correct
Decision
Correct
Decision
Error
Error
Type I Error
Type II Error
Type I Error: Rejecting a True Hypothesis
Type II Error: Accepting a False Hypothesis

32. Hypothesis Testing - Decision
Decision Right or Wrong?
But we can know the probability of being right
or wrong
Can specify and control the probability of
making TYPE I of TYPE II Error
Try to keep it small…

33. ALPHA
the probability of making a type I error  depends on the
criterion you use to accept or reject the null hypothesis =
significance level (smaller you make alpha, the less likely
you are to commit error) 0.05 (5 chances in 100 that the
difference observed was really due to sampling error – 5%
of the time a type I error will occur)
Possible Outcomes in
Hypothesis Testing
Null is True Null is False
Accept
Reject
Correct
Decision
Correct
Decision
Error
Error
Type I Error
Type II Error
Alpha (a)
Difference observed is really
just sampling error
The prob. of type one error

34. When we do statistical analysis… if alpha
(p value- significance level) greater than 0.05
WE ACCEPT THE NULL HYPOTHESIS
is equal to or less that 0.05 we
REJECT THE NULL (difference btw means)

35. 2.5% 2.5%
5% region of rejection of null hypothesis
Non directional
Two Tail

36. 5%
5% region of rejection of null hypothesis
Directional
One Tail

37. BETA
Probability of making type II error  occurs when we fail
to reject the Null when we should have
Possible Outcomes in
Hypothesis Testing
Null is True Null is False
Accept
Reject
Correct
Decision
Correct
Decision
Error
Error
Type I Error
Type II Error
Beta (b)
Difference observed is real
Failed to reject the Null
POWER: ability to reduce type II error

38. POWER: ability to reduce type II error
(1-Beta) – Power Analysis
The power to find an effect if an effect is present
1. Increase our n
2. Decrease variability
3. More precise measurements
Effect Size: measure of the size of the difference
between means attributed to the treatment

39. Inferential statistics
Significance testing:
Practical vs statistical significance

40. Inferential statistics
Used for Testing for Mean Differences
T-test: when experiments include only 2 groups
a. Independent
b. Correlated
i. Within-subjects
ii. Matched
Based on the t statistic (critical values) based on
df & alpha level

41. Inferential statistics
Used for Testing for Mean Differences
Analysis of Variance (ANOVA): used when
comparing more than 2 groups
1. Between Subjects
2. Within Subjects – repeated measures
Based on the f statistic (critical values) based on
df & alpha level
More than one IV = factorial (iv=factors)
Only one IV=one-way anova

42. Inferential statistics
Meta-Analysis:
Allows for statistical averaging of results
From independent studies of the same
phenomenon