psych 3400 statistics for the behavioral sciences cuny brooklyn college, department of psychology

PSYCH 3400Statistical Methods

CUNY Brooklyn College, Department of Psychology

Alla [email protected]

mailto:[email protected]

Approach of the Course• In this class you will learn both the theory and

practice of statistics.

• Homework is practice for the exams• Essay type answers• Statistical calculations by hand• SPSS analysis

Lab Format

• Announcements (make sure you are on time

• Demonstration of new computer techniques required for that week’s homework

• Period of questions and answers

• Opportunity for you to work with SPSS whenyour TA is present

You should think of the lab section as training, you will complete most of the homework on your own time.

http://psychfiles.net

• Contact info• Syllabus/ Semester Schedule• Lecture Slides• Homework Assignments/Problem Sets

Definition of a Statistic

OUR WORKING DEFINITION:A number that organizes, summarizes or makes understandable a collection of data.

THE FORMAL DEFINITION:A number calculated on sample data that quantifies a characteristic of the sample.

“In our calculations, we noted large differences in pupil size between males and females. The male group had pupil diameters (mm) of 3.2, 4.1, 4.6, 7.2, 4.1, 5.3, 8.1, 6.3, 4.8, 4.6, 4.8, while females had the following pupil diameters: 4.6, 7.1, 4.7, 3.7, 8.0, 4.8, 6.2, 4.5, 4.9, 7.1, 6.8. Obviously, there is a noticeable difference.”

vs.

“In our calculations, we noted large differences in pupil size between males and females. The male group had an average pupil diameter of 4.9, while females had an average pupil diameter of 6.1. Obviously, there is a noticeable difference.”

Which of these makes more sense?

Hours worked

Pay

Hours worked

Pay

Hours workedPa

y

We can also use statistics to describe relationships that we can depict graphically, such as in these

SCATTERPLOTS.

How do we acquire knowledge?

AuthorityIntuition

Scientific Method

Rationality

WHY do I have to learn Statistics?

Some VERY important definitions:• Experimental vs. Observational Methods• Population – the complete set of individuals, objects, or

scores that the investigator is interested in studying.• Sample – a subset of the population.• Variable – any property or characteristic of some event,

object, or person that may have different values at different times depending on the conditions– Independent: the variable that is systematically manipulated by the

investigator– Dependent: the variable that is measured to determine the effect of

the independent variable• Data - the measurements made on the subjects of an

experiment• Statistic – a number calculated on sample data that

quantifies a characteristic of the sample. (Note: Parameter).– Descriptive vs. inferential statistics

The Concept of a Variable

Textile Workers

45

50

55

60

65

70

75

80 100 120 140 160

Weight (lbs)

Hie

ght (

inch

es)

Height (y-axis)Weight (x-axis)

Any measurable property of a person, event or object that may take on different values at different times or under different conditions.

Compare with aCONSTANT like p

Continuous and Discrete Variables

1 2 3 4 5 6

2.51/2

2.1251/8

2.251/4

Discrete Variable

2 3

Continuous Variable

Can dividein halfinfinitely

Scales of Measurement

Nominal Names or categories

Order: a sense of greateror lesser but not by how much

Ordinal

Ordinal and how much greater& lesser: each interval is equal

Interval

Interval scale with an absolute zero - ratios of scores have meaning.

Ratio

Summarizing Samples with Math and Graphs

=S Gi NominalOrdinalIntervalRatio

Class Heights (Raw Scores)

0

5

10

15

54 55 56 57 58 59 60 61 62 63 64

Height (inches)

Freq

uenc

y (n

umbe

r of

indi

vidu

als)

Significant Figures and RoundingIt does not make sense to carry our calculations beyond the real limits of the variables we measure.Ex: On a thermometer the smallest unit is half of a degree.

By convention, in this class we will round all numbers to the hundredths place (two places after the decimal).

5.624 5.62 when the 3rd decimal place is ≤4.1.287 1.29 when the 3rd decimal place is ≥5.

Mathematical Notation

This is probably new to you.S

It means “summation”

Mathematical Notation: Summation Calculation

Student Grade ID (X) 1 93 2 75 3 88 4 77 5 65 6 55 7 97

Average of the variable X:

S X1n ( )

SX =

= (1/7) 550= 78.57

93 + 75 +88 + 77 + 65 + 55 + 97

SX = 550

Order of Operations

Order of operations:Parentheses, Exponents,Summation, Multiplication/Division, Addition/Subtraction

Read them like Englishsentences or lists of things to do in order

Important Example

x: { 1, 2, 3}

S x2 (S x )2

“Sum of the squared x’s” “Square of the summed x’s”

x123

x2

(1)2=1(2)2=4(3)2=9

14

x123

6 62 = 36

Here is a set of 15 height measurements (in inches).{ 55, 56, 56, 58, 60, 61, 57, 57, 59, 60, 60, 61, 54, 57, 57}

How can data be described? Summarized?

Value Frequency54 155 156 257 458 159 160 361 2

Frequency Table

Frequency Histogram

HEIGHT

61.060.059.058.057.056.055.054.0

HEIGHT

Freq

uenc

y

5

4

3

2

1

0

Std. Dev = 2.20

Mean = 57.9

N = 15.00

How can data be described? Summarized?How to create a detailed frequency table:

Set of scores: x: {2, 1, 5, 0, 2, 1, 2, 0, 1, 1, 3, 1, 2, 1, 1, 0, 0, 2, 3 , 1}Example: How many siblings do you have?

Value012345



Value012345

Frequency



Value012345

Frequency4



Value012345

Frequency485201

Total 20



Value012345

Frequency485201

Total 20

Percent



Value012345

Frequency485201

Percent

Total 20

= (4/20) x 100= .20 x 100= 20

2020



Value012345

Frequency485201

Percent

Total 20

2040251005

CumulativeFrequency41217191920

CumulativePercent2060859595100

20


Set of scores: x: {100, 23, 65, 98, 84, 72, 50, 49, 52, 99, 83, 79, 89, 9056, 63, 72, 92, 83, 100}

What if our range is very large?- We use class intervals instead of single values - Rule for # of intervals for use in this class: 10- To determine the width that each interval should be given the range

of data we have, use the following formula:

= (Highest score – Lowest score)/10= (100 – 23)/10= 77/10= 7.7 round this to the next whole number, 8.

Example: TEST GRADES!!?


Set of scores: x: {100, 23, 65, 98, 84, 72, 50, 49, 52, 99, 83, 79, 89, 9056, 63, 72, 92, 83, 100}


Intervals23-3031-3839-4647-5455-6263-7071-7879-8687-9495-102


Set of scores: x: {100, 23, 65, 98, 84, 72, 50, 49, 52, 99, 83, 79, 89, 9056, 63, 72, 92, 83, 100}


Intervals23-3031-3839-4647-5455-6263-7071-7879-8687-9495-102

Frequency1003122434


Set of scores: x: {100, 23, 65, 98, 84, 72, 50, 49, 52, 99, 83, 79, 89, 9056, 63, 72, 92, 83, 100}


Intervals23-3031-3839-4647-5455-6263-7071-7879-8687-9495-102

Frequency1003122434

Percent5001551010201520

CumulativeFrequency1114579131620

CumulativePercent555202535456580100

Choice of Interval is Important

HEIGHT

64.059.555.050.546.0

HEIGHT

Fre

quen

cy

30

20

10

0

43-48 49-54 55-60 61-66 67-72

HEIGHT

65.062.560.057.555.052.550.047.545.0

HEIGHT

Freq

uenc

y

20

10

0

45-47 48-50 51-53 54-56 57-59 60-62 63-65 66-68 69-71

Frequency Polygons

HEIGHT

61.0060.0059.0058.0057.0056.0055.0054.00

Cou

nt5.0

4.0

3.0

2.0

1.0

0.0

HEIGHT

61.060.059.058.057.056.055.054.0

HEIGHTFr

eque

ncy

5

4

3

2

1

0

Std. Dev = 2.20

Mean = 57.9

N = 15.00

HEIGHT

61.0060.0059.0058.0057.0056.0055.0054.00

Cou

nt

5.0

4.0

3.0

2.0

1.0

0.0

By Comparison…

HEIGHT

61.060.059.058.057.056.055.054.0

HEIGHTFr

eque

ncy

5

4

3

2

1

0

Std. Dev = 2.20

Mean = 57.9

N = 15.00

HEIGHT

61.0060.0059.0058.0057.0056.0055.0054.00

Cou

nt

5.0

4.0

3.0

2.0

1.0

0.0

By Comparison…These are commonly referred

to as DISTRIBUTIONS

Common Shapes of Frequency Distributions

HEIGHT

60.059.058.057.056.055.054.0

HEIGHT

Freq

uenc

y

7

6

5

4

3

2

1

0

HEIGHT

60.059.058.057.056.055.054.0

HEIGHT

Freq

uenc

y

7

6

5

4

3

2

1

0

HEIGHT

60.059.058.057.056.055.054.0

HEIGHT

Freq

uenc

y

7

6

5

4

3

2

1

0

SymmetricalBell-shaped

PositivelySkewed

NegativelySkewed


Multimodal Distributions

HEIGHT

60.059.058.057.056.055.054.0

HEIGHT

Freq

uenc

y

8

6

4

2

0

HEIGHT

62.061.060.059.058.057.056.055.054.0

HEIGHT

Fre

quen

cy

14

12

10

8

6

4

2

0

When describing a distribution, always specify:-Is it unimodal, bimodal, multimodal?- Is it symmetrical?- Is it skewed, positive or negative?

Psych Stats 3400 First Exam GradesN=66 students

0

2

4

6

8

10

12

14

16

20-28 29-36 37-44 45-52 53-60 61-68 69-76 77-84 85-92 93-100

Grade

Freq

uenc

yA real example…

IT’S THE HUMAN HISTOGRAM!

Is this a histogram?