Describing Data

Lesson 3

Psychology & Statistics Goals of Psychology

Describe, predict, influence behavior & cognitive processes

Role of statistics Descriptive statistics

Describe, organize & summarize data Efficient communication

Inferential statistics Draw conclusions about data Aid decision making ~

Organizing Data

Describing distribution of variables enumeration: list raw data

Frequency distributions organize tables or graphs highlight important characteristics

range, most frequent value ~

Distributions as Tables f = Frequency

# of times a value of variable occurs f = n calculate proportions & percentages

Tabular frequency distributions ordered list of all values of variable &

their frequencies logical order (usually descending) ~

TabularFrequency Distribution

X f19 118 216 315 314 513 212 611 710 3 9 6 8 5 7 3 6 2 5 2

50

# of presentations to be able to recall 100%

8 9 7 816 710111614

121312131214 8 91512

18141412 81111 9 918

1511 7 9 5 6 8101111

101416 61115 91912 5

Enumeration

Grouped Frequency Distribution

Group by class intervals report f for intervals Lose information: exact values

General rules each interval same width consecutive & do not overlap ~

GroupedFrequency DistributionX f19-20 117-18 215-16 613-14 711-12 13 9-10 9 7- 8 8 5- 6 4

50

TabularFrequency Distribution

X f19 118 216 315 314 513 212 611 710 3 9 6 8 5 7 3 6 2 5 2

50

Distributions as graphs

Summarizes data focus on clear communication

Bar Graphs nominal or ordinal data

Histograms & Frequency Polygons Interval/ratio data

continuous & discrete variables ~

Bar Graphs

f

Political affiliation

2

6

10

14

18

Rep Dem Ind

OrdinalNominal

f

Exam Grades

2

6

10

14

18

A B C D F

Histograms X-axis

Class intervals of variables

Y-axis Frequencies

vertical bars ~

f

# of presentations

2

6

10

14

18

5 7 9 11 13 15 17 19 215-6 7-8 9-10 11-12 13-14 15-16 17-18 19-20

Frequency polygons

Frequency represented as points Contains same info as histogram ~

f

# of presentations

2

6

10

14

18

5 7 9 11 13 15 17 19 21# of presentations

f

Relative Frequency

Distributions: 3 useful features

Summarizes important characteristics of data

1. What is shape of the distribution?

2. Where is middle of distribution?

3. How wide is distribution?

Shapes of distributions

Unimodal distribution single value is most

frequent

Bimodal (or multimodal ) 2 most frequently

occurring values May indicate relevant

subgroups ~

X

f

X

f

Symmetry of distributions Symmetric

if right side mirror-image of left

Skewed - asymmetric a few extreme values Positively skewed:

right tail longer Negatively skewed:

left tail longer ~

X

f

0 +2 +4-2-4

X

f

0 +2 +4-2-4

f

The Normal Distribution

Bell-shaped 3 characteristics

Unimodal symmetric asymptotic

Many naturally-occurring variables approximately normally distributed Makes statistics useful ~

f

Central Tendency Describes most typical values

Depends on level of measurement Mode (all levels)

Most frequently occurring value Median (only ordinal & interval/ratio)

value where ½ observations above & ½ below

Mean (only interval/ratio) Arithmetic average ~

f

Political affiliation

2

6

10

14

18

Rep Dem Ind

f

# of presentations

2

6

10

14

18

5 7 9 11 13 15 17 19 21

f

# of presentations

2

6

10

14

18

5 7 9 11 13 15 17 19 21

f

exam grades

2

6

10

14

18

A B C D F

Mode Most frequently occurring value ~

Median Midpoint of a data set

values ½ smaller, ½ larger ~

10 20 30 40 5060 70 80 90 10 20 30 40 50 60 70 80 90

Finding the Median

1. List all values from largest smallest

if f=3, then list 3 times

2. Odd # entries median = middle value

3. Even # entries = half way b/n middle 2 values ~

Mean Summarizes quantitative data

May not be actual value in data set Introduces error Most commonly used

Computing the mean

Sum of all observations

Number of observationsMean =

Statistical Notation

N

X Formula for mean:

Σ: summate add all that follows

X: observation value of an observation

N: number of observations Or data points ~

Populations & Samples Population: all individuals of interest

Depends on goal of researchers Parameter: value describing population

all observations used in calculation an exact value – no error

Sample: a portion of group of interest represents the whole population

Statistic: value describing sampleEstimate of parameterError introduced ~

Populations & Samples: Notation

Different symbols Often different formulas for

calculation Population: Greek letters

Population mean = μ Sample: Roman letters

Sample mean = APA style: M ~

X

Formulas for Mean

Population mean Parameter

Sample mean Statistic Estimate / error Sometimes n used for

sample ~

N

X

N

XX