Basic Statistics

Measures of Variability

Measures of Variability

The Range Deviation Score The Standard Deviation The Variance

STRUCTURE OF STATISTICS

STATISTICS

DESCRIPTIVE

INFERENTIAL

TABULAR

GRAPHICAL

NUMERICAL

CONFIDENCEINTERVALS

TESTS OF HYPOTHESIS

Continuing with numerical approaches.

NUMERICAL

STRUCTURE OF STATISTICSNUMERICAL DESCRIPTIVE MEASURES

DESCRIPTIVE

TABULAR

GRAPHICAL

NUMERICAL

CENTRALTENDENCY

VARIABILITY

SYMMETRY

STRUCTURE OF STATISTICSNUMERICAL DESCRIPTIVE MEASURES

NUMERICAL

CENTRALTENDENCY

VARIABILITY

SYMMETRY

RANGE

VARIANCE

STANDARDDEVIATION

You are an elementary school teacher who has been assigned a class of fifth graders whose mean IQ is 115. Because children with IQ of 115 can handle more complex, abstract material, you plan many sophisticated projects for the year.

Do you think your project will succeed ?

115100 130 145857055

General population

85%

We need the variability of IQs in the class!

Having graduated from college, you are considering two offers of employment. One in sales and the other in management. The pay is about

same for both. After checking out the statistics for salespersons and managers at the library, you find that those who have been working for 5

years in each type of job also have similar averages.

Can you conclude that the pay for two occupations is equal?

Is the average salary enough? We need the variability!

management Sales

Much moreMuch less

$20,000

Group of scores

Single scoreCentral Tendency measures

IQ of 100 students

Mean IQ=118

Central Tendency Measures

???

Measures of Central Tendency do not tell you the differences that exist among the scores

More homogeneous

Central Tendency

1 23

4

Same Mean---Different Variability So What?

60

How many are out here?

1. The Range = The difference between the largest (Xmax) and the smallest (Xmin).

21

25, 21, 22, 23, 28, 26, 24, 29

24 25 26 28 29

Range = 29 –21 = 8

22 23

A large range means there is a lot of variability in data.

10, 28, 26, 27, 29

10 26 28 29

Range = 29 –10 = 19

27

?

Drawbacks of The Range

Range = 29 –26=3

The Range depends on only the two extreme scores

Because the range is determined by just two scores in the group, it ignores the spread of all scores except the largest and smallest.

One aberrant score or outlier can be greatly increase the range

Range and Extreme Observations

R

R R

Range and Measurement Scales

1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3,

Country Code SES F Age

3-1=2 3-1=2 3-1=2 3-1=21=American 2=Asian 3=Mexican

1=Upper 2=Middle 3=Lower

Before you determine the Range, all scores must be arranged in order

3. The Variance

3566

53

88

83

35 42

6341

72

81

9549

4177

5762

7849

27

35

44

4941

81

49

53

27

3553

78

49

66

66

88?

Differences among Scores

Differen

ces amo

ng

Sco

res

Total Variability = Sum of Individual Variability

How can you determine the variability of each individual in group?

72

67

7055

2222

31212

The amount of Individual difference entirely depends on comparison criteria.

Can you figure out total amount of differences among scores ?

Can you figure out how much each score is different from other scores ?

48

Reference score? Mean Score

You need a Common Criteria for computing Total Variability

46

47

53

50

52 45

51

49

47

46

49

51

52

50

53

48

Reference score?49

You need a Common Criteria for computing Total Variability

45

-2

+4

-1-3

0

+2

-4+3

+1

Deviation Scores

A Deviation score tells you that a particular score deviate, or differs from the mean

DEVIATION SCORE = (Xi - Mean)

A score a great distance from the mean will have large deviation score.

MeanA B C D E F

1 2

3

Total amount of variability?!

Sum of all distance values!

Sum of Deviation Scores

No way!

conceptually

mathematically

The idea makes sense…but

If you compute the sum of the deviation scores, the sum of the deviation scores equals zero!

Sum of Deviation scores =(-4) + (-3) + (-2) + 0 + (1) + (2) + (3) + (4) = 0

The Sum of Absolute Deviation Scores

Sum of absolute deviation scores ( 4 + 3 + 2 + 1 + 0 +1 + 2 + 3 + 4) = 20

The sum of absolute deviations is rarely used as a measure of variability because the process

of taking absolute values does not provide meaningful information for inferential statistics.

Sum of Squares of deviation scores

“SS”

Conceptually

And

Mathematically

Sum of Squares of Deviation Scores, SS

Instead of working with the absolute values of deviation scores, it is preferable to (1) square each deviation score and (2) sum them to obtain a quantity know as the Sum of Squares.

SS=(-4)+(-3)+(-2)+(-1)+0+(1)+(2)+(3)+(4) =16+9+4+1+0+1+2+9+16 =60

2 2 2 2 2 2 2 2 2

SS=i

Group of scores“A”

Group of scores“B”

SS(A)=30 SS(B)=40

Can you say that the variability of the data in Group B is greater than the data

in Group A?

So !

3, 4 3, 4 3, 4

What happens to SS when we look at some data?

Group A Group B

Mean = 3.5 Mean = 3.5

SS = (3 - 3.5) + (4 - 3.5) +

(3 - 3.5) + (4 - 3.5)

=1.00

2 2 2 2

2 2

SS = (3 - 3.5) + (4 - 3.5)

=.50

i=1

N

i

SS tends to increase as number of data(N) increase.

SS is not appropriate for comparing variability among groups having unequal sample size.

How can you overcome the limitation of SS

Mean

If SS is divided by N

The resulting value will beMean of the Deviation Scores (Mean

Square)

VARIANCE

N

SS

N

)X(XS

2i2

3, 4 3, 4 3, 4

Group A Group B

Mean = 3.5 Mean = 3.5

V = (3 - 3.5) + (4 - 3.5)

=.50/2 = .25

V = (3 - 3.5) + (4 - 3.5) +

(3 - 3.5) + (4 - 3.5)

= 1.00/4 = 2.5

2 2 2 2

2 2

Variance

22

( )XN

Population Variance

22

1s X Xn

( )

Sample Variance

N

)X(X 2i

POPULATION VARIANCE

Sigma Square Population size

Population meanIndividual value

22

( )XN

SAMPLE VARIANCE SAMPLE VARIANCE

22

1s X Xn

( )

Sample variance

The sample variance (S2) is used to estimate the population variance (2)

Individual value Sample Mean

Sample size-1Degree of freedom

Why n-1 instead n ?2

2

1s X Xn

( )

Sampling n

Sampling error

=

?<100<?

100

population

sample

The value of the squared deviations is less from X than from any other score .

Hence, in a sample, the value of (X-X) n would be less than n.

>n n

Ideally, a sample variance would be based on (x - )2. This is impossible since is not known if one has only a sample of n cases. is substituted by .X

>

Ideal sample variance

One could correct for this bias by dividing by a factor somewhat less than n

n-1

sample n=5

If we know that the mean is equal to 5, and the first 4

scores add to 18, then the last score MUST equal 7.

n-1 are free to change

Degree of freedom

7

5

?5

n

XX

We know that ? must equal 25.

4. Standard Deviation SD

Positive square root of the variance

Population Sample

The Standard Deviation and the Mean with Normal Distribution

Normal Distribution

-1-2-3 +2 +3+1

Relationship between and

Normal Distribution

Relationship between and S

-1S -2S -3S +1S +2S +3S

68%

95%

99.9%

EMPIRICAL RULE

• For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within 1 standard deviation of the mean; approximately 98% within 2 standard deviations of the mean; and approximately 99.9% within 3 standard deviations of the mean.

You can approximately reproduce your data!

If a set of data has a Mean=50

and SD=10,then…

50403020 60 70 80

68%

95%

99%