Basic Statistics

Michael Hylin

Scientific Method

Start w/ a question Gather information and

resources (observe) Form hypothesis Perform experiment and

collect data Analyze data Interpret data & draw

conclusions form new hypotheses

Retest (frequently done by other scientists) i.e. replicate & extend

Example

Pavlov noticed that when the dogs saw the lab tech they salivated the same as when they saw meat powder (Observation)

Predicted that other stimuli could elicit this response when paired w/ meat powder (Hypothesis)

Example

Pavlov found that when a bell was paired w/ the presence of meat powder an association occurred (Experimentation)

Concluded that pairing of US w/ CS could lead to CR (Interpretation)

Research since Pavlov has demonstrated the mechanism of how CC works (e.g. Aplysia)

Basics of Experimental Design

Types of Variables Types of Comparisons Types of Groups

Types of Variables

Independent Variable Manipulated by the experimenter May have several

Dependent Variable Dependent upon the IV The data

IV → DV

Types of Comparisons

Between-subjects Comparing one group to another

Within-subjects Comparing a subject’s results at one

point to another point Usually referred to as repeated-measures

Types of Groups

Experimental Group Receives experimental manipulation

Control Group “controls” for the effect of manipulation

Example

A researcher has a new drug (M100) that improves semantic memory in normal individuals.

The researcher decides to test M100’s effectiveness by giving the drug to participants and testing their ability to memorize a list of words. Other participants are given a sugar pill and told to memorize the list as well.

Example

What is the IV? the DV? Additional IVs & DVs

What was the control? What type of comparison was being

done? Could it be different?

What about statistics?

Why do we need statistics? Cannot rely solely upon anecdotal

evidence Make sense of raw data Describe behavioral outcomes Test hypotheses

Measures of Central Tendency

Mode Frequency, most common ‘score’

Median Point at or below 50% of scores fall when the

data is arranged in numerical order Used typically w/ non-normal distributions

Mean (Often expressed ) Sum of the scores divided by the number of

scores

X

Mean

n

XX

nXXXX .....21

Example

Data for number of words recalled

8, 14, 17, 10, 8 Mode = 8 Median = 10 (8 , 8, 10, 14, 17) Mean = 8+14+17+10+8 = 11.4

5

Measures of Variability

Range Difference between highest and lowest

scores Variance (s2) Standard Deviation (s) Standard Error of the Mean (S.E.M.)

Variance

Equation for Variance

)1(

)( 22

n

XXs

22

2

2

12 .....)( XXXXXXXX n

Where:

Variance

Another Equation for Variance

1

2

2

2

nn

XX

s

222

21

2 ..... nXXXX Where:

2

21

2.....

nXXXX

&

Standard Deviation

Equation for Standard Deviation

)1(

)( 2

n

XXs

Or

1

2

2

nn

XX

s

Example

Data for number of words recalled

8, 14, 17, 10, 8 Range = 17 – 8 = 9 Variance = 15.8 Standard Deviation = 3.97

Example

11.4 8 8 - 11.4 = -3.4 11.56

14 14 - 11.4 = 2.6 6.76

17 17 - 11.4 = 5.6 31.36

10 10 - 11.4 = -1.4 1.96

8 8 - 11.4 = -3.4 11.56

63.2

X X XX 2XX

2

XX

4

2.632 s

8.152 s

4

2.63s

97.3s

Variance

Standard Deviation

Mean & Standard Deviation

Null Hypothesis

Start w/ a research hypothesis “Manipulation” has an effect e.g. Students given study techniques

have a higher GPA Set up the null hypothesis

“Manipulation” has NO effect e.g. Students w/ techniques are no diff.

than those w/o techniques

Null Hypothesis

Does the manipulation have an effect

Use a critical value to test our hypothesis Usually 0.05

controltreatH 0

controltreatH 1

Hypothesis Testing

Type II Error

p = β

Correct decision

p = 1 - αAccept H0

Correct decision

p = 1 – β = Power Type I Error

p = α

Reject H0

H0 FalseH0 TrueDecision

True State of the World

Hypothesis Testing

Not truly ‘proving’ our hypothesis In reality we are setting up a situation

where there is no relationship between the variables and then testing whether or not we can reject this (null hypothesis)

Independent T-Test

Test whether our samples come from the same population or different populations

1X

2X

Equation for Independent T-Test

2121

2

2

222

1

2

121

21

112 nnnn

n

XX

n

XX

XXt

2.803.26

2.402.95

2.102.98

3.103.16

2.543.41

GPAGPAGroup 2 (no techniques)Group 1 (study techniques)

76.151 X 94.122 X

80.4921 X 07.342

2 X51 n 52 n

15.31 X 58.22 X

51

51

255

594.12

07.34576.15

80.49

58.215.322

t

2.02.08

544.167

07.345

38.24880.49

57.0

t

4.08

49.3307.3468.4980.49

57.0

t

4.08

58.012.0

57.0

t

4.0870.0

57.0

t

035.0

57.0t

187.0

57.0t

4.00875.0

57.0

t

04.3t

Since our observed t = 3.04 which is greater than 2.306 we can reject the null hypothesis

Therefore the probability of the difference we observed occurring when the null hypothesis is true is less than 0.05 (5%)

As a result our effect is likely due to the training

Degrees of Freedom

6, 8, 10 Mean = 8

If we change two numbers the other is determine if we want to keep Mean = 8 67 & 1013 then the final number is 4

4202420241

8

3

208

3

208

3

137

YYY

YYY

IV with more than two levels

Sometimes we want to compare more that just two groups

Cannot just due multiple t-tests Increase alpha

Simple analysis of variance 1-way ANOVA

Multiple IVs

Factoral ANOVA Allow for comparison of more than one IV IVs can be between or within If both its called mixed ANOVA (repeated

measures) Interaction of IVs E.g. 2x2 ANOVA

IV1 Study group (no study vs. study) IV2 Time at testing (pre. vs. post.)

Example

0

1

2

3

4

Pre Post

GP

A Study

No Study

ANOVA Table

  Sum of Squares df Mean

Square F Sig.

Test 0.5445 1 0.5445 36.3 0.00

Test * Group 0.8405 1 0.8405 56.03333 0.00

Error(Test) 0.12 8 0.015  

Group 0.7605 1 0.7605 5.827586 0.04

Error 1.044 8 0.1305    

Example

0

1

2

3

4

Pre Post

GP

A Study

No Study

ANOVA Table

  Sum of Squares df Mean

Square F Sig.

Test 0.002 1 0.002 0.148148 0.710342

Test * Group 0 1 0 0 1

Error(Test) 0.108 8 0.0135  

Group 0.002 1 0.002 0.017241 0.898775

Error 0.928 8 0.116    

F-Score Equation

error

group

MS

MSF

What about further group comparisons

Significant main effects with more than 2 levels Post hoc comparisons

Significant interactions Simple effects