Statistics

• Summary numbers, or indices, that result from an analysis of data (numbers) – pg. 2

• All the procedures and tools used to organize and interpret facts, events, and observations that can be expressed numerically – pg. 2

Imaginary World

Snap

Crackle

Imaginary World #2

Snap Crackle

Pop

if orand

then

Imaginary World #3

Variables

• n. an element, feature, or factor that is liable to vary or change

• adj. not consistent or having a fixed pattern; liable to change

The Goal of Statistics(and all of science)

• Figure out how things vary together.

• If you succeed, you can answer the questions:– How, why, when, where, what

Q: Why is little Sally afraid of dogs?A: What are the variables, and how do they vary together?

(the presence or absence of dogs, Sally’s emotional state, Sally’s neural structure as it relates to these other events, Sally’s history and how it relates to the above elements, etc.)

Q: Why does my roommate drink my milk and leave the empty jug in the refrigerator?

Making Sense of the Variability

Break it down:• Independent Variables: the variables we are

manipulating (or using for categorization) – E.g., shock intensity

• Dependent Variables: the variables we are measuring– E.g., amplitude of scream

IV

DV

Unaccounted For Variables

• Variables that we do not know the status of

• Because we didn’t bother to include them– E.g., eye color

• Because we couldn’t include them (even if we wanted to)– E.g., exact electro-chemical state of the nervous

system before, during, and after the shock

Example 1

• Octavian is interested in the effect of breathing exercises on time spent on task.– What is the independent variable?• Breathing exercises• Time on task

– What is the dependent variable?• Breathing exercises• Time on task

Example 2

• Adelpha wants to know if pronunciation can be improved if one practices speaking with marbles in ones mouth.– What is the independent variable?• Clarity of pronunciation• Practice with marbles

– What is the dependent variable?• Clarity of pronunciation• Practice with marbles

Unaccounted For Variables: What To Do?

• The influence of unaccounted for variables can be controlled for by designing your study in such a way that unaccounted for variables are not likely to influence one group or condition more than any other group or condition.

• How is this accomplished?

Design

1. Random Selection– Random = each member of the population has an

equal chance of being selected.– If you stop here, you can’t make causal statements.

2. Random Assignment– Random = each selected participant has an equal

chance of being exposed to each condition.– This is the only way you can make causal

statements.– This is an experiment.

Example

• Romulus selects the first 50 students that walk into the Bernhard Center to participate in a study on WMU students. The students are asked if they are smokers or nonsmokers and then they fill out a short survey that is supposed to measure their general level of anxiety.

• Do we have random selection , random assignment , neither , or both ?

What’s wrong with it?

1. No random selection: not all students go to the Bernhard Center on a regular basis, and many might only go late in the afternoon. (selection bias)

2. No random assignment: students were already in the category of interest, smoker or nonsmoker, so the IV was not manipulated. (not an experiment)

So what?

1. No random selection: means that you cannot say anything about the target population (WMU students).

2. No random assignment: means that you cannot make causal statements about the IV on the DV (e.g., you can’t say that smoking caused students to have a higher level of anxiety).

3. This experiment is a big failure, no matter how good the statistical analysis is.

What’s the Lesson?

Design Trumps Analysis

• Whenever possible, a study should include both random selection and random assignment.

• If you can’t randomly assign, your study might still be useful, but you can’t make causal statements.

Food for Thought

Statistics and Variability

Variability ≠ RandomnessRandomness is an Illusion

Complexity

• Variability is brought about by COMPLEXITY, not randomness.

• Human behavior is very complex (i.e., there are many relevant variables), but because it is orderly, we can predict and influence behavior.

• Statistics makes this prediction and influence much easier.

Population vs. Sample

Population

Every person or object in a group

(past, present, future, often infinite)

Sample

Part of the population

Summary Measure = Statistic_

Ex: X = sample means2 = sample variances = sample standard

deviation

Summary Measure = Parameter

Ex: μ = pop. meanσ2 = pop. varianceσ = pop. standard

deviationNotice: Greek symbols are used for parameters, Roman symbols for statistics

Random• simple• stratified

Biased

Why Sample?

Population

Sample

SamplingInference

We want to make inferences about the population.

Inference

Definition: a conclusion reached on the basis of evidence [in our case, sample data] and reasoning [in our case, statistical analysis]

Why not take data onthe population?

• The population might be huge.– If the population is U.S. adults over age 65 you would need to

take data on about 40 million people (also see below).• Part of the population might be inaccessible.

– A study on infants born this year should apply to infants born next year.

– A study on heroin addicts should apply to people who were addicted to heroin but died of an overdose.

• But if you can get population data, you should.– If I am interested in how many hours students in my class spend

studying for quizzes, I can (and should) take population data.

Inferential vs. Descriptive Statistics

Inferential Statistics: using statistical methods to make inferences about the population given sample dataEx: How likely is it that my depression intervention results (from my sample) will generalize to the population?

Descriptive Statistics: using statistical methods to describe a set of dataEx: What is the mean depression score on the Hamilton Rating Scale for clients I am working with?

INFER

DESCRIBE

Sampling Error

Sampling error: the error caused by observing a sample instead of the population.– Represented by the formula:

• statistic – parameter = sampling error• Ex:

– The ONLY way to avoid sampling error entirely is to take data on the entire population

– What can we do to reduce sample error?1. Random Sampling 2. Large Samples 3. Multiple Samples

4. Stratified Random Sampling

1. Decide which characteristic of the population you will use for stratification. (Ex: sex)

2. Randomly sample from each strata.3. The number drawn from each strata should be

proportional to the number in each strata in the population.

Ex: Total pig population (16 pigs): You think the color of the pig might be an important variable, but you can only afford to do the study with half of the pigs. How many black and how many pink pigs would you include in your study?

That’s right! Six pink pigs and two black pigs.

Try This

Male: 3,500,000

Female: 2,000,000U. A. E.

Population:People age 16-65living in the U. A. E.

What should yourStratified samplelook like if yourtotal sample size is100?

Here’s how I solved it:

I wrote the number of females as a fraction of the total population, and put my unknown over my sample size on the other side.

So, your sample should look like this:

Male: 64

Female: 36Then I solved for X.

With and Without Replacement

• Sampling with replacement means that after each person/object is selected, it is returned to the population so that it could be selected again.– This is done so that the probability of an individual being

selected remains constant.• Sampling without replacement means that after each

person/object is selected, it is not returned to the population and, therefore, cannot be selected again.– In most behavioral studies, this is the only way to sample.

Why is that?

Measurement

Definition: the assignment of numbers or labels to objects or events

Labels? How is that measurement?

1. Nominal Scales

• Naming of objects or events

• Naming ≠ measurement

• Naming allows us to categorize and deal with individuals or groups separately

• Labeling is required to get frequency data, which is type of measurement

Nominal Scales

• If we took the last seven digits of each of your phone numbers and averaged them, what could we learn?

• NOTHING. Be careful, just because it’s a number doesn’t mean it’s not a name.

• What if we wrote down the area code of each student in class and found the number of times each area code occurs (frequency)?

• This would work! Let’s try it.

2. Ordinal Scales

• Naming objects or events and putting them in order

• Ranking or rank ordering

• Example: advisory system

Ordinal Scales

1234

???

Ordinal scales give us limited data.

Interval Scales

• Naming of objects or events and putting them in order using units with equal intervals

• The distance between 1 and 2 should be exactly the same as the distance between 2 and 3 or 3 and 4.

• Common example: temperature in degrees Fahrenheit or Celcius

Interval Scales

• In psychology: whether or not Likert scales are interval or ordinal scales is an issue of debate.

• Why do you think this might be?

Note: We will treat them as interval scales in this course, but only for convenience.

Ratio Scales

• Naming of objects or events and putting them in order using units with equal intervals on a scale with a true zero

• Ask yourself, does zero really mean nothing?

• Common example: temperature in degrees Kelvin

Homework

• Prepare for Quiz #2 over Chapters 1 & 2.

• Read Chapter 3 and be prepared to answer basic questions.

• Complete Homework assignment #1 and prepare to hand it in at the beginning of the next class.

• Check WebCT and/or the course website for the homework assignment and other study materials.