4 measures of central tendency

8/6/2019 4 Measures of Central Tendency

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Measures of CentralMeasures of Central

TendencyTendency

to be or not to be

Normal


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TOPICS

Normal Distributions

Skewness & Kurtosis

Normal Curves and Probability Z- scores

Confidence Intervals

Hypothesis Testing

The t-distribution


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Is this normal ?

VAR00001

500.0400.0300.0200.0100.0

3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0

St

.

= 160.68

M

an = 178.3

= 6.00

Statistics

VAR00001

6

0

178.3333

2.242

.845

5.2191.741

Vali

Missin

M

an

Sk

n ss

St

. Error o Sk

n ss

rtosisSt

. Error o

rtosis

VAR00001

1 16.7 16.7 16.7

2 33.3 33.3 50.0

2 33.3 33.3 83.3

1 16.7 16.7 100.06 100.0 100.0

70.00

100.00

150.00

500.00

Total

Vali

Fr

q

ncy Percent Vali

Percent

Cumulati

e

Percent


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Normal DistributionsNormal Distributions

Are your curves normal?

Why do we care about normal curves?

What do normal curves tell us?

Answer:

The curves tell us something about the distribution

of the population

The curves allow us to make statistical inferencesregarding the probability of some outcomes

within some margin of error


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The normal distributionThe normal distribution

A distribution is easily

depicted in a graph

where the height of the

line determined by thefrequency of cases for

the values beneath it

Most cases cluster

near the middle of adistribution if close to

normal


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The Normal CurveThe Normal Curve

Bell-shaped distribution or curve

Perfectly symmetrical about the mean

Mean median mode

Tails are asymptotic: closer and closer to

horizontal axis but never reach it


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Skewness and Sample DistributionsNot all curves are normal, even if still bell-shaped


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Skewness

Formula for skewness

Sy

medianmeanSkewness

)(3 !


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Kurtosis (Its not a disease)Kurtosis (Its not a disease)

Beyond skewness, kurtosis tells us whenour distribution may have high or lowvariance, even if normal

The kurtosis value for a normal distributionwill equal Anything above this is apeaked value (low variance) and anything

below is platykurtic (high variance)


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Back to normal distributions

The power of normal distributions, or those

close to it, is that we can predict where

cases will fall within a distribution

probabilistically

For example, what are the odds, given the

population parameter of human height, that

someone will grow to more than eight feet?

Answer, likely less than a probability


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Sample DistributionSample Distribution W

hat doesA

ndre theGiant do to the sampledistribution?

What is the probabilityof finding someonelike Andre in thepopulation?

Are you ready formore inferential

statistics?

Answer: Oh boy, yes!!


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Normal Curves and probabilityNormal Curves and probability

We have answered the question of whatAndre and the Sumo wrestler would do tothe distribution

But what about the probability of findingsomeone the same height as Andre in thepopulation?

What is the probability of finding someonethe same height as Dr Pea or DrBoehmer?


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More on normal curves and

probability

Andre would be hereDr Boehmer would be here


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ZZ--Scores (no sleeping!!)Scores (no sleeping!!)

We can standardize the central tendency

away from the mean across different

samples with z-scores

The basic unit of the z-score is the standard

deviation

s

XXz i

)( !


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We can use the z-score to score each

observation as a distance from the

mean.

How far is a given observation from the

mean when its z-score = 2?

Answer: standard deviations

Approximately what percentage of cases

is a given case higher than if its z-score

= 2?

Answer: %


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Random Sampling Error

Ever hear a poll report a margin of error? What

is that?

andom Sampling Error standard deviation/ square

root of the sample sizeOr

NW As the variance of the

population increases, sodoes the chance that a

sample could not reflect the

population parameters


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Standard Error

We often refer to both the random samplingerror with both the chance to err when

sampling but also the error of a specific

sample statistic, the mean We typicallyuse the term Standard Error

Asample statistic standard error is thedifference between the mean of a sample

and the mean of the population from which

it is drawn


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Standard Error

Example: What if most humans werepounds and only million globally were

pounds?

The random sampling error would be low

since the chance of collecting a sample

consisting heavily of those heavier humans

would be unlikely There would not be

much error in general from sampling

because of the low variance


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Standard Error

Example continued Now, when we take asample, each sample has a mean If apopulation has low variance, so should thesamples We should see this reflected in

low standard error in the mean of thesample, the sample statistic

Of course, higher variance in thepopulation also causes higher error in

samples taken from it


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Some more notation

Distributions Mean Standard Dev.

Sample of

observed data sPopulation

epeated

Sampling

X

NW

Error in a Samples mean is the Standard Error

Random Sampling Error

ns


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Central Limit TheoremCentral Limit Theorem

Remember that if we took an infinite number

of samples from a population, the means

of these samples would be normally

distributed

Hence, the larger the sample relative to the

population, the more likely the sample

mean will capture the population mean


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Confidence IntervalsConfidence Intervals

We can actually use the information wehave about a standard deviation from themean and calculate the range of values forwhich a sample would have if they were tofall close to the mean of the population

This range is based on the probability that

the sample mean falls close to thepopulation mean with a probability of ,or % error


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How Confident Are You?How Confident Are You?

Are you % sure? Social scientists use a % as a threshold

to test whether or not the results are

product of chance That is, we take out of chances to be

wrong

What do you MEAN?

We build a % confidence interval to makesure that the mean will be within thatrange


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Confidence Interval (CI)Confidence Interval (CI)

s yZY WE 2/

Y mean

Z Z score related with a % CI

standard error

rorstandarder*)2(96.1 or sa ple eans


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Building a CIBuilding a CI

N

y

Y

W

W!

Assume the following

40015

100

!

!

!

N

y

y

W

Q

750.400

15!!

yW


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CI

53.98

47.101

)750.0)(96.1(100

!

!

s

Lower

Upper

Why do we use ?


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Calculating a % CICalculating a % CI

Lets look at the class population

distribution of height

Is it a normal or s

kew distribution?

Lets build a % CI around the mean

height of the class


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Why do we care about CI?Why do we care about CI?

We use CI interval for hypothesis testing

For instance, we want to know if there is

an income difference betweenE

lP

asoand Boston

We want to know whether or not taking

class at Kaplan makes a difference in our

GRE scores


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Mean Difference testing

Income levels

El Paso BostonLas Cruces

Mean U

SA


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4 measures of central tendency

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