The Normal Curve

Theoretical Symmetrical Known Areas For Each

Standard Deviation or Z-score

FOR EACH SIDE: 34.13% of scores in

distribution are b/t the mean and 1 s from the mean

13.59% of scores are between 1 and 2 s’s from the mean

2.28% of scores are > 2 s’s from the mean

Z SCORE FORMULA Z = Xi – X S Xi = 120; X = 100; s=10

Z= 120 – 100 = +2.00

10

The point is to convert your particular metric (e.g., height, IQ scores) into the metric of the normal curve (Z-scores). If all of your values were converted to Z-scores, the distribution will have a mean of zero and a standard deviation of one.

Normal Curve Probability Important Link

Normal curve is limited as most real world data is not “normally distributed”

More important use has to do with probability theory and drawing samples from a population

Probability Basics What is the probability of picking a red marble out

of a bowl with 2 red and 8 green?

There are 2 outcomes that

are red

THERE ARE 10 POSSIBLE

OUTCOMES

p(red) = 2 divided by 10p(red) = .20

Frequencies and Probability The probability of picking a color relates to the

frequency of each color in the bowl 8 green marbles, 2 red marbles, 10 total p(Green) = .8 p(Red) = .2

Frequencies & Probability What is the probability of randomly selecting an

individual who is extremely liberal from this sample?

p(extremely liberal) = 32 = .024 (or 2.4%) 1,319

THINK OF SELF AS LIBERAL OR CONSERVATIVE

32 2.3 2.4 2.4171 12.3 13.0 15.4186 13.4 14.1 29.5486 35.0 36.8 66.3

205 14.8 15.5 81.9

198 14.3 15.0 96.9

41 3.0 3.1 100.0

1319 95.1 100.062 4.5

6 .468 4.9

1387 100.0

1 EXTREMELY LIBERAL2 LIBERAL3 SLIGHTLY LIBERAL4 MODERATE5 SLGHTLYCONSERVATIVE6 CONSERVATIVE7 EXTRMLYCONSERVATIVETotal

Valid

8 DK9 NATotal

Missing

Total

Frequency Percent Valid PercentCumulative

Percent

PROBABILITY & THE NORMAL DISTRIBUTION

We can use the normal curve to estimate the probability of randomly selecting a case between 2 scores

Probability distribution: Theoretical distribution

of all events in a population of events, with the relative frequency of each event

Normal Curve, Mean = .5, SD = .7

3.072.211.36.50-.36-1.21-2.07

1.2

1.0

.8

.6

.4

.2

0.0

PROBABILITY & THE NORMAL DISTRIBUTION

The probability of a particular outcome is the proportion of times that outcome would occur in a long run of repeated observations. 68% of cases fall within +/- 1

standard deviation of the mean in the normal curve

The odds (probability) over the long run of obtaining an outcome within a standard deviation of the mean is 68%

Normal Curve, Mean = .5, SD = .7

3.072.211.36.50-.36-1.21-2.07

1.2

1.0

.8

.6

.4

.2

0.0

Probability & the Normal Distribution

Suppose the mean score on a test is 80, with a standard deviation of 7. If we randomly sample one score from the population, what is the probability that it will be as high or higher than 89?

Z for 89 = 89-80/7 = 9/7 or 1.29 Area in tail for z of 1.29 = 0.0985 P(X > 89) = .0985 or 9.85%

ALL WE ARE DOING IS THINKING ABOUT “AREA UNDER CURVE” A BIT DIFFERENTLY (SAME MATH)

Probability & the Normal Distribution

Bottom line:Normal distribution can also be thought of as

probability distributionProbabilities always range from 0 – 1

0 = never happens 1 = always happens In between = happens some percent of the time

This is where our interest lies

Inferential Statistics Inferential statistics are used to generalize from

a sample to a population We seek knowledge about a whole class of similar

individuals, objects or events (called a POPULATION)

We observe some of these (called a SAMPLE) We extend (generalize) our findings to the entire

class

WHY SAMPLE? Why sample?

It’s often not possible to collect info. on all individuals you wish to study

Even if possible, it might not be feasible (e.g., because of time, $, size of group)

WHY USE PROBABILITY SAMPLING?

Representative sample One that, in the aggregate, closely approximates the

population from which it is drawn

PROBABILITY SAMPLING Samples selected in accord with probability theory,

typically involving some random selection mechanism If everyone in the population has an equal chance of being

selected, it is likely that those who are selected will be representative of the whole group EPSEM – Equal Probability of SElection Method

PARAMETER & STATISTIC Population

the total membership of a defined class of people, objects, or events

Parameter the summary description of a given variable in a

population Statistic

the summary description of a variable in a sample (used to estimate a population parameter)

INFERENTIAL STATISTICS Samples are only estimates of the population

Sample statistics will be slightly off from the true values of its population’s parameters Sampling error:

The difference between a sample statistic and a population parameter

μ = 4.5 (N=50)

x=7x=0 x=3x=1 x=5x=8 x=5 x=3

x=8 x=7x=4 x=6

x=2 x=8 x=4 x=5 x=9 x=4

x=5 x=9x=3 x=0x=6 x=5

x=1 x=7 x=3x=4 x=5x=6

EXAMPLE OF HOW SAMPLE STATISTICSVARY FROM A POPULATION PARAMETER

X=4.0

X=5.5

X=4.3

X=5.3 X=4.7

CHILDREN’S AGE IN YEARS

By Contrast: Nonprobability Sampling Nonprobability sampling may be more appropriate

and practical than probability sampling: When it is not feasible to include many cases in the

sample (e.g., because of cost) In the early stages of investigating a problem (i.e.,

when conducting an exploratory study)

It is the only viable means of case selection: If the population itself contains few cases If an adequate sampling frame doesn’t exist

Nonprobability Sampling: 2 Examples

1. CONVENIENCE SAMPLING When the researcher simply selects a requisite

number of cases that are conveniently available

2. SNOWBALL SAMPLING Researcher asks interviewed subjects to suggest

additional people for interviewing

Probability vs. Nonprobability Sampling:Research Situations

For the following research situations, decide whether a probability or nonprobability sample would be more appropriate:

1. You plan to conduct research delving into the motivations of serial killers.

2. You want to estimate the level of support among adult Duluthians for an increase in city taxes to fund more snow plows.

3. You want to learn the prevalence of alcoholism among the homeless in Duluth.

(Back to Probability Sampling…)The “Catch-22” of Inferential Stats:

When we collect a sample, we know nothing about the population’s distribution of scores We can calculate the mean (X) & standard deviation

(s) of our sample, but (population mean) and (population standard deviation) are unknown

The shape of the population distribution (normal?) is also unknown Exceptions: IQ, height

PROBABILITY SAMPLING 2 Advantages of probability sampling:

1. Probability samples are typically more representative than other types of samples

2. Allow us to apply probability theory This permits us to estimate the accuracy or

representativeness of the sample

SAMPLING DISTRIBUTION From repeated random sampling, a

mathematical description of all possible sampling event outcomes (and the probability of each one)

Permits us to make the link between sample and population… & answer the question: “What is the

probability that sample statistic is due to chance?”

Based on probability theory

μ = 4.5 (N=50)

x=7x=0 x=3x=1 x=5x=8 x=5 x=3

x=8 x=7x=4 x=6

x=2 x=8 x=4 x=5 x=9 x=4

x=5 x=9x=3 x=0x=6 x=5

x=1 x=7 x=3x=4 x=5x=6

Imagine if we did this an infinite amount of times…

X=4.0

X=5.5

X=4.3

X=5.3 X=4.7

CHILDREN’S AGE IN YEARS

What would happen…(Probability Theory) If we kept repeating the samples from the

previous slide millions of times? What would be our most common sample mean?

The population mean What would the distribution shape be?

Normal This is the idea of a sampling distribution

Sampling distribution of means

Relationship between Sample, Sampling Distribution & Population

POPULATION

SAMPLING DISTRIBUTION

(Distribution of sample outcomes)

SAMPLE

• Empirical (exists in reality) but unknown

• Nonempirical (theoretical or hypothetical)

Laws of probability allow us to describe its characteristics(shape, central tendency,dispersion)

• Empirical & known (distribution shape, mean, standard deviation)

TERMINOLOGY FOR INFERENTIAL STATS Population

the universe of students at the local college Sample

200 students (a subset of the student body) Parameter

25% of students (p=.25) reported being Catholic; unknown, but inferred from sample statistic

Statistic Empirical & known: proportion of sample that is Catholic is

50/200 = p=.25 Random Sampling (a.k.a. “Probability”)

Ensures EPSEM & allows for use of sampling distribution to estimate pop. parameter (infer from sample to pop.)

Representative EPSEM gives best chance that the sample statistic will

accurately estimate the pop. parameter