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descriptive stats

Used to describe characteristics

(central tendency & dispersion) of population

(central tendency & dispersion) of population

inferential stats

Use sample data to...

(1) test hypothesis about single variable

or association between two variables

AND

(2) Determine whether we can generalize

results back to population

(1) test hypothesis about single variable

or association between two variables

AND

(2) Determine whether we can generalize

results back to population

population

set of elements researcher

wants to understand (ie. adolescents in U.S.)

wants to understand (ie. adolescents in U.S.)

sample

subset of population that is actually

studied

studied

We use probability theory to...

Identify type(s) of samples that would most

likely be drawn from given population

distribution

Make theoretical connections between given

population & corresponding sample

Make inferences (draw conclusions) from

sample back to given population

likely be drawn from given population

distribution

Make theoretical connections between given

population & corresponding sample

Make inferences (draw conclusions) from

sample back to given population

behavioral stats graph

population--probability theory-->sample,sample (inf. stats)--generalization-->pop.

probability

defined as

proportion, fraction, or percent of all possible

outcomes

proportion, fraction, or percent of all possible

outcomes

Random Sampling

To make inferences about population from

sample using probability theory, select

elements by this

Each element in population has an EQUAL

and KNOWN chance of being selected

Probabilities must stay constant from one

selection to next (replacement)

sample using probability theory, select

elements by this

Each element in population has an EQUAL

and KNOWN chance of being selected

Probabilities must stay constant from one

selection to next (replacement)

Empirical vs. Theoretical Probability

Distributions

Distributions

Until now, we have been working with

EMPIRICAL distributions - based on real data

THEORETICAL distributions have many of

same properties as EMPIRICAL distributions

But constructed using probability theory

rather than actual observations

EMPIRICAL distributions - based on real data

THEORETICAL distributions have many of

same properties as EMPIRICAL distributions

But constructed using probability theory

rather than actual observations

The Normal Distribution

Many real-life phenomena approximate

NORMAL distribution (ie. height, athletic

ability, social attitudes, etc)

THEORETICAL because it is based upon

what distribution of scores on variable from

infinitely large population would look like

Use what we know about this THEORETICAL

distribution to make inferences about

population from sample data

NORMAL distribution (ie. height, athletic

ability, social attitudes, etc)

THEORETICAL because it is based upon

what distribution of scores on variable from

infinitely large population would look like

Use what we know about this THEORETICAL

distribution to make inferences about

population from sample data

Properties of Normal Distribution

Bell-shaped

Perfectly symmetrical

Highest frequency in middle of curve

Frequencies tapering off as move toward

either extreme

Perfectly symmetrical

Highest frequency in middle of curve

Frequencies tapering off as move toward

either extreme

Always True for Normal Distribution

Symmetric (not skewed)

Mean, median, mode are equivalent & fall in

center of curve

Specific proportion of cases (area under

the normal curve) fall within 1 SD of mean,

2 SD of mean, 3 SD of mean

Mean, median, mode are equivalent & fall in

center of curve

Specific proportion of cases (area under

the normal curve) fall within 1 SD of mean,

2 SD of mean, 3 SD of mean

In theoretical normal distribution...

68.26% of cases fall within 1 standard

deviation of mean

95.44% of cases fall within 2 standard

deviations of mean

99.74% of cases fall within 3 standard

deviations of mean

68-95-99 RULE

deviation of mean

95.44% of cases fall within 2 standard

deviations of mean

99.74% of cases fall within 3 standard

deviations of mean

68-95-99 RULE

Why are properties of normal

distribution important?

distribution important?

Many naturally occurring (empirical)

distributions follow normal curve

PROBABILITY of random observation having

value between x1 and x2 equals

PROPORTION of cases between x1 and x2

Normal distribution is guaranteed in certain

circumstances

distributions follow normal curve

PROBABILITY of random observation having

value between x1 and x2 equals

PROPORTION of cases between x1 and x2

Normal distribution is guaranteed in certain

circumstances

Due to these properties we can...

Calculate how many standard deviations

from sample mean given observation is

AND

Make accurate predictions about probability

of variable taking on a specific value or an

event occurring

from sample mean given observation is

AND

Make accurate predictions about probability

of variable taking on a specific value or an

event occurring

What is a z-score good for?

Just standardized score that indicates the distance, in

SD units, of particular value from mean

In other words, z equals number of standard

deviations observation is from mean

When observation is standardized & variable is

normally distributed...

We can use properties of normal curve to PREDICT

probability of random observation taking on any

range of values

SD units, of particular value from mean

In other words, z equals number of standard

deviations observation is from mean

When observation is standardized & variable is

normally distributed...

We can use properties of normal curve to PREDICT

probability of random observation taking on any

range of values

Steps to Determine Probability

Step #1: Translate raw scores to z-scores (standardize)/Step #2: Interpret z-scores & visualize area

under curve/Step #3: Determine area under normal curve that falls

between values of interest/Step#4:Look up z-score

under curve/Step #3: Determine area under normal curve that falls

between values of interest/Step#4:Look up z-score

What does this have to do with

inferential stats?

inferential stats?

Take sample data & ask how extreme a zscore

to expect before proclaiming that

observed value is not due to chance, but to

some other factor

This is especially important when we are

using sample statistics to estimate population

parameters

Because sampling procedures involve certain

amount of error

to expect before proclaiming that

observed value is not due to chance, but to

some other factor

This is especially important when we are

using sample statistics to estimate population

parameters

Because sampling procedures involve certain

amount of error