edpsy 511-001 chp. 2: measurement and statistical notation

EDPSY 511-001

Chp. 2: Measurement and Statistical Notation

Populations vs. Samples

• Population– The complete set of individuals

• Characteristics are called parameters

• Sample– A subset of the population

• Characteristics are called statistics.

– In most cases we cannot study all the members of a population

Descriptive vs. Inferential

• Descriptive statistics– Summarize/organize a group of numbers from

a research study

• Inferential statistics– Draw conclusions/make inferences that go

beyond the numbers from a research study– Determine if a causal relationship exists

between the IV and DV

Common Research Designs

• Correlational– Do two qualities “go together”.

• Comparing intact groups– a.k.a. causal-comparative and ex post facto designs.

• Quasi-experiments– Researcher manipulates IV

• True experiments– Must have random assignment.

• Why?

– Researcher manipulates IV

Variables

• Variables– Characteristics that takes on different values

• Achievement• Age• Condition

– Independent variable (IV)• Manipulated or Experimental

– Condition• Subject

– Personality– Gender

– Dependent variable (DV)• The outcome of interest

– Achievement– Drop-out status

Measurement

• Is the assignment of numerals to objects.• Nominal

– Examples: Gender, party affiliation, and place of birth

• Ordinal– Examples: SES, Student rank, and Place in race

• Interval– Examples: Test scores, personality and attitude scales.

• Ratio– Examples: Weight, length, reaction time, and number of

responses

Categorical, Continuous and Discontinuous

• Categorical (nominal)– Gender, party affiliation, etc.

• Discontinuous– No intermediate values

• Children, deaths, accidents, etc.

• Continuous– Variable may assume an value

• Age, weight, blood sugar, etc.

Values

• Exhaustive– Must be able to assign a value to all objects.

• Mutually Exclusive– Each object can only be assigned one of a set

of values.

• A variable with only one value is not a variable.– It is a constant.

Chapter 2: Statistical Notation• Nouns, Adjectives, Verbs and

Adverbs.– Say what?

• Here’s what you need to know– X

• Xi = a specific observation– N

• # of observations– ∑

• Sigma– Means to sum

– Work from left to right• Perform operations in

parentheses first• Exponentiation and square

roots• Perform summing operations• Simplify numerator and divisor• Multiplication and division• Addition and subtraction

N

iiX

1

• Pop Quiz (non graded)– In groups of three or four

• Perform the indicated operations.

• What was that?

)1(

)( 22

NN

XXN

Chapter 3

Exploratory Data Analysis

Exploratory Data Analysis

• A set of tools to help us exam data– Visually representing data makes it easy to

see patterns.• 49, 10, 8, 26, 16, 18, 47, 41, 45, 36, 12, 42, 46, 6,

4, 23, 2, 43, 35, 32

– Can you see a pattern in the above data?• Imagine if the data set was larger.

– 100 cases– 1000 cases

Three goals

• Central tendency– What is the most common score?– What number best represents the data?

• Dispersion– What is the spread of the scores?

• What is the shape of the distribution?

Frequency Tables

• Let say a teacher gives her students a spelling test and wants to understand the distribution of the resultant scores.– 5, 4, 6, 3, 5, 7, 2, 4, 3, 4

Value F Cumulative F % Cum%

7 1 1 10% 10%

6 1 2 10% 20%

5 2 4 20% 40%

4 3 7 30% 70%

3 2 9 20% 90%

2 1 10 10% 100%

N=10

As groups

• Create a frequency table using the following values.– 20, 19, 17, 16, 15, 14, 12, 11, 10, 9

Banded Intervals

• A.k.a. Grouped frequency tables

• With the previous data the frequency table did not help.– Why?

• Solution: Create intervals

• Try building a table using the following intervals<=13, 14 – 18, 19+

Stem-and-leaf plots

• Babe Ruth– Hit the following number of Home Runs from 1920 –

1934.• 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22

– As a group let’ build a stem and leaf plot

– With two classes’ spelling scores on a 50 item test.

• Class 1: 49, 46, 42, 38, 34, 33, 32, 30, 29, 25 • Class 2: 39, 38, 38, 36, 36, 31, 29, 29, 28, 19

– As a group let’ build a stem and leaf plot

Landmarks in the data

• Quartiles– We’re often interested in the 25th, 50th and 75th

percentiles.• 39, 38, 38, 36, 36, 31, 29, 29, 28, 19

– Steps• First, order the scores from least to greatest.• Second, Add 1 to the sample size.

– Why?• Third, Multiply sample size by percentile to find location.

– Q1 = (10 + 1) * .25– Q2 = (10 + 1) * .50– Q3 = (10 + 1) * .75

» If the value obtained is a fraction take the average of the two adjacent X values.

Box-and-Whiskers Plots (a.k.a., Boxplots)

Shapes of Distributions

• Normal distribution

• Positive Skew– Or right skewed

• Negative Skew– Or left skewed

How is this variable distributed?

87654321

score

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Frequency

Mean = 4.3Std. Dev. = 1.494N = 10

How is this variable distributed?

7.006.005.004.003.002.001.000.00

right

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Frequency

Mean = 2.80Std. Dev. = 1.75119N = 10

How is this variable distributed?

8.007.006.005.004.003.002.00

left

3.0

2.5

2.0

1.5

1.0

0.5

0.0

Frequency

Mean = 5.40Std. Dev. = 1.42984N = 10

Descriptive Statistics

Statistics vs. Parameters

• A parameter is a characteristic of a population.– It is a numerical or graphic way to summarize data

obtained from the population

• A statistic is a characteristic of a sample.– It is a numerical or graphic way to summarize data

obtained from a sample

Types of Numerical Data

• There are two fundamental types of numerical data:

1) Categorical data: obtained by determining the frequency of occurrences in each of several categories

2) Quantitative data: obtained by determining placement on a scale that indicates amount or degree

Measures of Central Tendency

Central Tendency

Average (Mean) Median Mode

1

1

n

ii

N

ii

XX

n

X

N

Mean (Arithmetic Mean)

• Mean (arithmetic mean) of data values– Sample mean

– Population mean

1 1 2

n

ii n

XX X X

Xn n

1 1 2

N

ii N

XX X X

N N

Sample Size

Population Size

Mean

• The most common measure of central tendency

• Affected by extreme values (outliers)

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Mean = 5 Mean = 6

Median

• Robust measure of central tendency• Not affected by extreme values

• In an Ordered array, median is the “middle” number– If n or N is odd, median is the middle number– If n or N is even, median is the average of the two

middle numbers

0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14

Median = 5 Median = 5

Mode• A measure of central tendency• Value that occurs most often• Not affected by extreme values• Used for either numerical or categorical data• There may may be no mode• There may be several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 9

0 1 2 3 4 5 6

No Mode