Deviations from NormalitySkewness and Kurtosis

Why do we care if the distribution is not normal?

• It helps you understand how a characteristic exhibits itself in your sample or in the population.

• It impacts what descriptive statistics you might use.

• It impacts the inferential statistics you might use.

SkewnessThe majority of scores do not fall in the middle of

the distribution.

The distribution is asymmetrical

You label the kind of skew according to the longer tail of the distribution.

Normal versus Skewed Distributionsen.wikipedia.org/wiki/Image:Standard_deviation_diagram.png

Freq

uenc

y

Negative skewPositive skew

Long tail is on the negative endLong tail is on

the positive end

Mean

Median

Mode

Normal Distribution

What Positive Skew MeansIf this represented the results of a quiz, the majority of the participants scored very low— almost no one scored in the highest range.

12

34

5

0 10 20 30 40 50 60 70 80 90 100

This test must have been very

easy or difficult?

Median

Mode

Mean

What Negative Skew MeansIf this represented the results of a quiz, the majority of the participants scored high— almost no one scored in the lowest range.

This test must have been very

easy or difficult?

0 10 20 30 40 50 60 70 80 90 100

12

34

5

Mean

Median

Mode

Normal vs. Skewed Distributions

•

Income Distributionshttp://www.city-data.com/city/Chicago-Illinois.html

Kurtosis

leptokurticplatykurtic

KurtosisLeaping

leptokurticdistribution

0 10 20 30 40 50 60 70 80 90 100

Platykurtic

0 10 20 30 40 50 60 70 80 90 100

Platykurtic like a

platypus distribution

leptokurtic

Normal (mesokurtic)

platykurtic

Sprinthall - Quick Kurtosis RuleWhen you have a distribution and know its standard deviation and range, you can estimate its kurtosis.

Fact: For a normal distribution, the standard deviation is about 1/6 of the range.

If the standard deviation is more than 1/6 of the range, then a distribution is platykurtic.

If the standard deviation is less than 1/6 of the range, then a distribution is leptokurtic.

How you determine this:Range = Standard Value

6Compare actual sd to SV

If sd > SV, platykurtic

If sd < SV, leptokurtic

Dr. Bellini’s MCC Research

47/6 = SV

SV = 7.83

8.98 > 7.83

Platykurtic

Dr. Bellini’s MCC Research

19/6 = SV

SV = 3.17

3.52 > 3.17

Platykurtic

Skew

$20,000 $40,000 $60,000

Beginning Salary

25

50

75

100

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nt

$25,000 $50,000 $75,000 $100,000 $125,000

Current Salary

0

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50

75

100

Cou

nt

C:\Program Files\SPSS\University of Florida graduate salaries.sav

10000 20000 30000 40000 50000 60000

Starting Salary

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nt

Statistics

Starting Salary1100

26064.2026000.00

200006967.982

58300

NMeanMedianModeStd. DeviationRange

Measurement Scales

Measurement

• Assigning numbers to observations following a set of rules.

How are numbers assigned to observations?

What scale is used?1. Nominal2. Ordinal3. Interval4. Ratio

Nominal Data• Using numbers to label categories, but the

numbers have no inherent numerical qualities• Male = 1 Female = 2

• social security number • jersey numbers• race/ethnicity

Other examples of nominal scaling• Whether a participant does or does not have a

driver’s license (0,1)

• Whether the participant belongs to the experimental group or the control group (0,1)

• The school the participant attends (1,2,3,4,5)

Uses of nominal data• Generally, the most you can do with nominal data

is count it.

Categorical or Continuous?• Is a variable that uses a nominal scale of

measurement categorical or continuous?

Ordinal Scaling• The assigned number provides information about

the rank of an observation.

• Ordinal scales put observations in order.

• Rating scales are often considered ordinal in how they measure characteristics.

Example

Strongly Agree Neither agree Disagree Strongly agree nor disagree disagree

Statistical 1 2 3 4 5Thinking is my favorite Xclass

Hospital pain scales (1-10)

Ordinal Scales have two basic rules:1. Equality/non-equality rule

2. Greater-than-or-less-than rule

Ranking of Tennis Players http://sports.espn.go.com/sports/tennis/rankings?sport=WOMRANK

Uses of ordinal dataYou can express if something is greater than or less than (but you can’t express how much greater than or less than).

< > • Strongly agreeing is more than simply agreeing• Being tenth in the class is lower than being ninth

How much lower?

IssuesThere are limitation of what you can do statistically

with ordinal data

Degree of violation Consensus

NOIR Interval ScalingSatisfies all the requirements of an ordinal scale

(there is a high to low structure to the scale) and

• The intervals between the points on the scale become meaningful because the distance between successive points on an interval scale are equal.

Examples• Degrees Fahrenheit or Celsius• Calendar years

Interval scales do not have a meaningful zero point. They may contain a zero point, but it is arbitrary.

NOIR Ratio• Satisfies all the requirements of an interval scale• There is a real and meaningful zero point on a

ratio scale• Weight, height, heart rate, breaths per minute,

degrees Kelvin, annual income, miles per hour, pulse, etc.

HINT: If your scale has negative numbers, like with temperature, then it is interval but it probably isn’t ratio.

Our Survey

R N O N N R N O O R

Measures of Central Tendency and Dispersion

Describing data

Describing Data vs. Describing People

Measures of Central Tendency

•Mean (M or X)•Median (Md)•Mode (Mo)

Issues with Describing Data1 2 4 61 2 4 61 2 4 6 n = 341 2 4 6 Mean = 3.15 ???1 2 42 3 42 3 4 What does that mean?2 3 42 3 52 4 5

The MeanMost common synonym is the “average”

But what does the mean “mean”?

Definitions: • “arithmetic average”• a descriptor of the center of the data, when data

are distributed “normally”

When a mean value is most useful• To simply summarize a data set that is normally

distributed • To summarize data from a sample that can be used

to estimate information about an entire population

When a mean can be less useful as s descriptor

• To summarize data that is skewed (especially when it used as the only descriptor)

• To summarize data where there is an outlier

• To summarize data measured using a nominal scale

Mean and Skew- Find the mean annual income

$ 7,200$ 9,011 $ 20,074$ 24,999$ 36,567$ 32,145$ 54,158$567,987

$94,018How well does this number represent a measure of the center of our data set?

Mediana.k.a. Middle score in a ranked set of scores.It divides the distribution of scores into equal halves.

When there is an odd number of scores:1 2 2 3 5 5 7 8 10 15 16 16 21

When there is an even number of scores:4 5 7 15 16 19 31 32

Average = (15+16)/2 = 15.5 Median Score

Mean and Skew- Find the mean annual income

$28,572$7,200

$9,011

$20,074

$24,999

$32,145

$36,567

$54,158

$567,987

How well does this number represent a measure of the center of our data set?

10000.00 20000.00 30000.00 40000.00 50000.00

VAR00002

0.0

0.5

1.0

1.5

2.0C

ount

ModeMost frequently occurring score is a group of scores.

Exam scores:

78 85 92 55 87 85 98 84 71

88 85 78 65 99 100 85 62 100

Measures of central tendencyGuide to which measures of central tendency are appropriate to use

with each scale of measurement:Mean Median Mode

Nominal XOrdinal (X)? X XInterval X X XRatio X X X

Measures of central tendencyGuide to which measures of central tendency are appropriate to use

with each scale of measurement:Mean Median Mode

Nominal XOrdinal (X)? X X

Interval X X XRatio X X X

Mean, Median and/or Mode

Class Survey Age VariableDescribe this dataset in terms of its

Mean

Median

Mode

Range

Descriptive Statistics

14 37 36.79 10.71414

ageValid N (listwise)

N Range Mean Std. Deviation

30 40 50

age

0

2

4

6

Cou

nt

Class Survey Commute Variable

Describe this dataset in terms of its

Mean

Median

Mode

Range

What about this data set makes it difficult to describe using measures of central tendency?

10.0 20.0 30.0 40.0 50.0 60.0

travel

2

4

6

8

Cou

ntDescriptive Statistics

14 64.5 10.914 16.312414

travelValid N (listwise)

N Range Mean Std. Deviation

Commute

Giving more information, such as how skewed this variable is, would be helpful in the description.