Statistics Review

PSY379

Basic concepts

• Measurement scales

• Populations vs.samples

• Continuous vs.discrete variable

• Independent vs.dependent variable

• Descriptive vs.inferential stats

Common analyses

• Independent T-test

• Paired T-test

• One-way ANOVA

• Two-way ANOVA

• Regression

Measurement Scales

Before we can examine a variable statistically, we

must first observe, or measure, the variable.

Measurement is the assignment of numbers to

observations based on certain rules.

Measurement attempts to transform attributes into

numbers.

How much is high vs. low stress? How much fast vs.

slow learning of a maze? How much is good vs. bad

memory?

Measurement Scales

• Non-metric (or qualitative)

Nominal scale (Categories):

Numbers indicate difference in kind; no order info

(e.g., ethnicity, gender, id#s)

Say that ‘men’ is assigned ‘0’ and ‘women’ is

assigned ‘1’; doesn’t mean 1 is better than 0

Ordinal scale (Orders):

Numbers represent rank orderings; distances are

not equal

(e.g., grades, rank orderings on a survey)

Measurement Scales

• Metric (or quantitative)

Interval scale:

Equal intervals, “arbitrary” zero

Ratios have no meaning

(e.g., temperature in degrees F;

50 - 30 F = 120 - 100 F; 60 F 2 X 30 F)

Ratio scale:

Equal intervals, absolute zero

Equal ratios are equivalent

(e.g., weight, height)

Populations vs. Samples

• Population: all members of a specific group.

parametric: a measure (e.g., mean and variance)

computed for the population

• Sample: a finite subset of a predefined population.

statistic: a measure (e.g., mean and variance)

computed for the sample

Continuous vs. Discrete

• Discrete variable:

one in which a measure can take on distinct values but not

intermediate values (e.g., number of children -- it is either

1 or 2, but not 1.2). The most common form of discrete

variable is based on counting.

• Continuous variable:

approximations of the exact value; it is not possible to

obtain the exact measure on a continuous variable,

because there are always infinitely smaller gradation of

measure (e.g., height – we can say someone is 72 inches

tall, but this is really approximating between 71.5 and

72.5 inches).

Independent vs. Dependent

• Independent variable:

one manipulated by the experimenter, or the

observed variable thought to cause or predict the

dependent variable. In the relation Y = f(X), X is

independent variable because the value of X does not

depend on the value of Y.

• Dependent variable:

one thought to result from the independent variable.

In the relation Y = f(X), Y is the dependent variable

because the value it takes on depends on the value of

X

Descriptive Statistics

- Descriptive Statistics (a.k.a. Summary Statistics)

- Primarily concerned with the summary and description

of a collection of data

- Serves to reduce a large and unmanageable set of data to

a smaller and more interpretable set of information

Descriptive Statistics

Frequency distribution & histogram

• a function that summarizes the membership of

individual observation to measurement classifications.

• Can be constructed regardless of whether the scale is

nominal, ordinal, interval or ratio, as long as each and

every observation goes into one and only one class.

Descriptive Statistics

One of the goals in stats is to compare distributions of data,

one data distribution with another data distribution.

This would be easier if each data distribution can be

summarized into one or two numbers.

Central Tendency & Variability

-- what is the descriptive central number and how much do

individual scores vary from the number?

Descriptive Statistics

Measures of Central Tendency

Mean: typical/average score, sensitive to extreme

scores

Median: middlemost score; useful for skewed

distribution

Mode: most “common or frequent” score

Descriptive Statistics

IQ scores

Fre

qu

en

cy

11410810296908478

100

80

60

40

20

0

Descriptive Statistics

Measures of Variability

Variance (dispersion or spread): degree of spread in

X (variable)

Standard deviation (SD): a measure of variability in

the original metric units of X (variable); the square

root of the variance

Variance

S2= (xi – x )2

n-1

x

Variance

S2= (xi – x )2

n-1

x

IQ score

Fre

qu

en

cy

15213311495765738

70

60

50

40

30

20

10

0

IQ scores

Fre

qu

en

cy

156.0136.5117.097.578.058.539.0

100

80

60

40

20

0

IQ score

Fre

qu

en

cy

16014012010080604020

100

80

60

40

20

0

IQ score

Fre

qu

en

cy

16014012010080604020

80

70

60

50

40

30

20

10

0

Other Measures

Skewness is a measure of symmetry, or more precisely,

the lack of symmetry.

Kurtosis is a measure of whether the data are peaked or

flat relative to a normal distribution. That is, data sets with

high kurtosis tend to have a distinct peak near the

mean, decline rather rapidly, and have heavy tails. Data

sets with low kurtosis tend to have a flat top near the mean

rather than a sharp peak.

Pop Quiz!

Variance is the average of the squared differences between

data points and the mean. Then why are the differences

squared?

Standard deviation is the square root of variance. Then

why is the variance square rooted?

Inferential Statistics

- A formalized method for solving a class of problems

relating to the inference of properties to a large set of

data from examination of a small set of data

- Goal is to predict or to estimate characteristics of a

population based on information obtained from a sample

drawn from that population

Inferential Statistics

We want to know about these: We have this to work with:

RandomSelection

Inference

Parameter Statistic

PopulationSample

(Population mean) (Sample mean)

Normal distribution

67% of data

within 1 SD of

mean

95% of data

within 2 SD of

mean

Poisson distribution

mean

Mostly, nothing happens (lots of zeros)

Basic concepts

• Measurement scales

• Populations vs.samples

• Continuous vs.discrete variable

• Independent vs.dependent variable

• Descriptive vs.inferential stats

Common analyses

• Independent T-test

• Paired T-test

• One-way ANOVA

• Two-way ANOVA

• Regression

Hypothesis testing

1. Assume ‘null hypothesis (H0)’ (e.g., the two sets of

samples come from the same population)

2. Construct ‘alternative hypothesis (H1)’ (e.g., the two

sets of samples do not come from the same

population)

3. Calculate ‘test statistic’

4. Decide on rejection region for null hypothesis (e.g.,

95% confidence in rejecting null hypothesis)

• Null (H0): no effect of our experimental

treatment, “status quo”

• Alternative (H1): there is an effect

Hypotheses

T-tests

• One sample t-test – compare a group to a known

value

(e.g., comparing the IQ of a specific group to the

known average of 100)

• Paired samples t-test – compare one group at two

points in time (e.g., comparing pretest and posttest

scores)

• Independent samples t-test – compare two groups to

each other

Paired t-test

More examples…

• Before-and-after observations on the same subjects

(e.g. students’ diagnostic test results before and after

a particular module or course)

• A comparison of two different methods of

measurement or two different treatments where the

measurements or treatments are applied to the same

subjects (e.g. blood pressure measurements)

Paired t-test

1. Calculate the difference between the two

observations on each pair, making sure you

distinguish between positive and negative

differences.

2. Calculate statistics (mean, SD etc.) for these

difference scores.

3. Calculate the t-statistic (T). Under the null

hypothesis, this statistic follows a t-distribution

with n 1 degrees of freedom (n = sample size).

4. Use tables of the t-distribution to compare your value

for T to the tn distribution.

Paired t-test

-316195

+224224

+117163

+425212

+422181

differencePost-testPre-testStudent

Example: Suppose a sample of n students were given a

diagnostic test before studying a particular subject and then

again after completing it.

Question: Do two samples come from

different populations?

Independent t-test

A B

DATA

NO YESH0

Depends on whether the difference betweensamples is much greater than difference withinsample.

Independent t-test

A B

A B

Between >> Within…

Degrees of freedom (df)

df = (number of independent observations) – (number of

restraints)

or

df = (number of independent observations) – (number of

population estimates)

df = (a) (n - 1)

a = number of different groups; n = number of

observations (i.e., sample size)

How many degrees of freedom when sample

sizes are different?

Independent t-test

(n1-1) + (n2-1)

T-tables

.05.10.20df (two-

tailed)

2.7762.1321.5334

1.9601.6451.282infinity

3.1822.3531.6383

4.3032.9201.8862

12.7066.3143.0781

.025.05.10df (one-

tailed)

Two samples, each n=3, with t-statistic of 2.50:

significantly different?

T-tables

.05.10.20df (two-

tailed)

2.7762.1321.5334

1.9601.6451.282infinity

3.1822.3531.6383

4.3032.9201.8862

12.7066.3143.0781

.025.05.10df (one-

tailed)

Two samples, each n=3, with t-statistic of 2.50:

significantly different? No!

General form of the t-test; can have more

than 2 samples

One-way (factor) ANOVA

H0:

All samples the same…

H1:

At least one sample

different

General form of the t-test; can have more

than 2 samples

One-way (factor) ANOVA

A B C

AB C

A BC

A BC

DATAH0 H1

Just like t-test, compares differences between

samples to differences within samples

One-way (factor) ANOVA

A B C

Difference between means

Standard error within sample

MS between groups

MS within group

T-test statistic (t)

ANOVA statistic (F)

ANOVA table

SSEdf (E)

SSX

df (X)

MS

MSXMSE

F

Look

up !

p

SSTdf (T)Total

SSEdf (E)Error

(within groups)

SSXdf (X)Treatment

(between groups)

SSdf

}

}

alpha = 0.05, F2,12 = 3.89

2.92

34.5

MS

11.8

F

?

p

10414Total

3512Error

(within groups)

692Treatment

(between groups)

SSdf

Suppose there are 3 groups of treatment (i.e., one factor with

three levels), and there are 5 observations per group.

Just like one-way ANOVA, except subdivides thetreatment SS into:

• Treatment 1

• Treatment 2

• Interaction between 1 & 2

Two-way ANOVA

Suppose there are two groups of treatment 1 and twogroups of treatment 2, and there are 10 observations ineach group:

• Treatment 1 (2 levels, so df = 1)

• Treatment 2 (2 levels, so df = 1)

• Treatment 1 x Treatment 2 interaction (1df x 1df = 1df)

• Error?

Two-way ANOVA

df = k(n-1) = 4 (10-1) = 36

MS(T1)

MSE

MS(T1)SS(T1)1Treatment 1

MS(T2)

MSE

MS(T2)SS(T2)1Treatment 2

MSE

MS(T1XT2)

MS

MS(Int)

MSE

F

SST39Total

SSE36Error

(within groups)

SS(T1XT2)1Treatment 1 x

Treatment 2

SSdfv

Combination of treatments gives additive effect

Interactions

Additive effect:

T1 level 1 T1 level2

T2 level2

T2 level2

Combination of treatments gives non-additive

effect

Interactions

Anything not parallel!

How to report

Independent t-test:

(Example)

There was no overall difference in performance on control RAT

items between younger and older adults, Ms = 0.39 and 0.32,

respectively, t(18) = 1.34, p > .05.

How to report

ANOVA (or F-test):

(Example)Reading time (in seconds) on the control story was compared to the mean

reading time for the four stories with distraction using a 2 (Age: young and

old) X 2 (Story Type: without and with distraction) ANOVA with age as a

between-subject variable and story type as a within-subject variable. Older

adults were slower overall than younger adults, M = 66.45 and 51.33,

respectively, F (1, 18) = 18.94, p < .01, the stories with distraction took

longer to read than the stories without distraction, M = 79.83 and 37.95,

respectively, F (1, 18) = 202.44, p < .01, and, in replication of the earlier

work, the slowdown between the stories with and without distraction was

greater for older than for younger adults, F (1, 18) = 7.43, p < .05.