Measure of Variability (Dispersion, Spread) 1.Range 2.Inter-Quartile Range 3.Variance, standard...

Measure of Variability (Dispersion, Spread)

1. Range

2. Inter-Quartile Range

3. Variance, standard deviation

4. Pseudo-standard deviation

Measure of Central Location

1. Mean

2. Median

1. Range

R = Range = max - min

2. Inter-Quartile Range (IQR)

Inter-Quartile Range = IQR = Q3 - Q1

Example

The data Verbal IQ on n = 23 students arranged in increasing order is:

80 82 84 86 86 89 90 94 94 95 95 96 99 99 102 102 104 105 105 109 111 118 119

Q2 = 96Q1 = 89 Q3 = 105min = 80 max = 119

Range and IQR

Range = max – min = 119 – 80 = 39

Inter-Quartile Range

= IQR = Q3 - Q1 = 105 – 89 = 16

3. Sample Variance

Let x1, x2, x3, … xn denote a set of n numbers.

Recall the mean of the n numbers is defined as:

n

xxxxx

n

xx nn

n

ii

13211

The numbers

are called deviations from the the mean

xxd 11

xxd 22

xxd 33

xxd nn

The sum

is called the sum of squares of deviations from the the mean.

Writing it out in full:

or

n

ii

n

ii xxd

1

2

1

2

223

22

21 ndddd

222

21 xxxxxx n

The Sample Variance

Is defined as the quantity:

and is denoted by the symbol

111

2

1

2

n

xx

n

dn

ii

n

ii

2s

The Sample Standard Deviation s

Definition: The Sample Standard Deviation is defined by:

Hence the Sample Standard Deviation, s, is the square root of the sample variance.

111

2

1

2

n

xx

n

ds

n

ii

n

ii

Example

Let x1, x2, x3, x4, x5 denote a set of 5 denote the set of numbers in the following table.

i 1 2 3 4 5

xi 10 15 21 7 13

Then

= x1 + x2 + x3 + x4 + x5

= 10 + 15 + 21 + 7 + 13

= 66

and

5

1iix

n

xxxxx

n

xx nn

n

ii

13211

2.135

66

The deviations from the mean d1, d2, d3, d4, d5 are given in the following table.

i 1 2 3 4 5

x i 10 15 21 7 13-3.2 1.8 7.8 -6.2 -0.2

10.24 3.24 60.84 38.44 0.04i id x x

22i id x x

The sum

and

n

ii

n

ii xxd

1

2

1

2

22222 2.02.68.78.12.3

80.112

04.044.3884.6024.324.10

2.28

4

8.112

11

2

2

n

xxs

n

ii

Also the standard deviation is:

31.52.28

4

8.112

11

2

2

n

xxss

n

ii

Interpretations of s

• In Normal distributions– Approximately 2/3 of the observations will lie

within one standard deviation of the mean– Approximately 95% of the observations lie

within two standard deviations of the mean– In a histogram of the Normal distribution, the

standard deviation is approximately the distance from the mode to the inflection point

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

s

Inflection point

Mode

s

2/3

s

Example

A researcher collected data on 1500 males aged 60-65.

The variable measured was cholesterol and blood pressure.

– The mean blood pressure was 155 with a standard deviation of 12.

– The mean cholesterol level was 230 with a standard deviation of 15

– In both cases the data was normally distributed

Interpretation of these numbers

• Blood pressure levels vary about the value 155 in males aged 60-65.

• Cholesterol levels vary about the value 230 in males aged 60-65.

• 2/3 of males aged 60-65 have blood pressure within 12 of 155. i.e. between 155-12 =143 and 155+12 = 167.

• 2/3 of males aged 60-65 have Cholesterol within 15 of 230. i.e. between 230-15 =215 and 230+15 = 245.

• 95% of males aged 60-65 have blood pressure within 2(12) = 24 of 155. Ii.e. between 155-24 =131 and 155+24 = 179.

• 95% of males aged 60-65 have Cholesterol within 2(15) = 30 of 230. i.e. between 230-30 =200 and 230+30 = 260.

A Computing formula for:

Sum of squares of deviations from the the mean :

The difficulty with this formula is that will have many decimals.

The result will be that each term in the above sum will also have many decimals.

n

ii xx

1

2

x

The sum of squares of deviations from the the mean can also be computed using the following identity:

n

x

xxx

n

iin

ii

n

ii

2

1

1

2

1

2

To use this identity we need to compute:

and 211

n

n

ii xxxx

222

21

1

2n

n

ii xxxx

Then:

n

x

xxx

n

iin

ii

n

ii

2

1

1

2

1

2

11 and

2

1

1

2

1

2

2

nn

x

x

n

xxs

n

iin

ii

n

ii

11

and

2

1

1

2

1

2

nn

x

x

n

xxs

n

iin

ii

n

ii

Example

The data Verbal IQ on n = 23 students arranged in increasing order is:

80 82 84 86 86 89 90 94

94 95 95 96 99 99 102 102

104 105 105 109 111 118 119

= 80 + 82 + 84 + 86 + 86 + 89

+ 90 + 94 + 94 + 95 + 95 + 96 + 99 + 99 + 102 + 102 + 104

+ 105 + 105 + 109 + 111 + 118 + 119 = 2244

= 802 + 822 + 842 + 862 + 862 + 892

+ 902 + 942 + 942 + 952 + 952 + 962 + 992 + 992 + 1022 + 1022 + 1042

+ 1052 + 1052 + 1092 + 1112

+ 1182 + 1192 = 221494

n

iix

1

n

iix

1

2

Then:

n

x

xxx

n

iin

ii

n

ii

2

1

1

2

1

2

652.2557

23

2244221494

2

You will obtain exactly the same answer if you use the left hand side of the equation

11 and

2

1

1

2

1

2

2

nn

x

x

n

xxs

n

iin

ii

n

ii

26.116

22

652.2557

2223

2244221494

2

11 Also

2

1

1

2

1

2

nn

x

x

n

xxs

n

iin

ii

n

ii

26.116

22

652.2557

2223

2244221494

2

782.10

A quick (rough) calculation of s

The reason for this is that approximately all (95%) of the observations are between

and

Thus

4

Ranges

sx 2.2sx

sx 2max .2min and sx .22minmax and sxsxRange

s4

4

Range Hence s

Example

Verbal IQ on n = 23 students min = 80 and max = 119

This compares with the exact value of s which is 10.782.The rough method is useful for checking your calculation of s.

75.94

39

4

80-119s

The Pseudo Standard Deviation (PSD)

Definition: The Pseudo Standard Deviation (PSD) is defined by:

35.1

Range ileInterQuart

35.1

IQRPSD

Properties

• For Normal distributions the magnitude of the pseudo standard deviation (PSD) and the standard deviation (s) will be approximately the same value

• For leptokurtic distributions the standard deviation (s) will be larger than the pseudo standard deviation (PSD)

• For platykurtic distributions the standard deviation (s) will be smaller than the pseudo standard deviation (PSD)

Example

Verbal IQ on n = 23 students Inter-Quartile Range

= IQR = Q3 - Q1 = 105 – 89 = 16

Pseudo standard deviation

This compares with the standard deviation

85.1135.1

16

35.1

IQRPSD

782.10s

• An outlier is a “wild” observation in the data

• Outliers occur because– of errors (typographical and computational)– Extreme cases in the population

• We will now consider the drawing of box-plots where outliers are identified

Box-whisker Plots showing outliers

• An outlier is a “wild” observation in the data

• Outliers occur because– of errors (typographical and computational)– Extreme cases in the population

• We will now consider the drawing of box-plots where outliers are identified

To Draw a Box Plot we need to:

• Compute the Hinge (Median, Q2) and the Mid-hinges (first & third quartiles – Q1 and Q3 )

• To identify outliers we will compute the inner and outer fences

The fences are like the fences at a prison. We expect the entire population to be within both sets of fences.

If a member of the population is between the inner and outer fences it is a mild outlier.

If a member of the population is outside of the outer fences it is an extreme outlier.

Lower outer fence

F1 = Q1 - (3)IQR

Upper outer fence

F2 = Q3 + (3)IQR

Lower inner fence

f1 = Q1 - (1.5)IQR

Upper inner fence

f2 = Q3 + (1.5)IQR

• Observations that are between the lower and upper fences are considered to be non-outliers.

• Observations that are outside the inner fences but not outside the outer fences are considered to be mild outliers.

• Observations that are outside outer fences are considered to be extreme outliers.

• mild outliers are plotted individually in a box-plot using the symbol

• extreme outliers are plotted individually in a box-plot using the symbol

• non-outliers are represented with the box and whiskers with– Max = largest observation within the fences– Min = smallest observation within the fences

Inner fencesOuter fence

Mild outliers

Extreme outlierBox-Whisker plot representing the data that are not outliers

Example

Data collected on n = 109 countries in 1995.

Data collected on k = 25 variables.

The variables

1. Population Size (in 1000s)

2. Density = Number of people/Sq kilometer

3. Urban = percentage of population living in cities

4. Religion

5. lifeexpf = Average female life expectancy

6. lifeexpm = Average male life expectancy

7. literacy = % of population who read

8. pop_inc = % increase in popn size (1995)

9. babymort = Infant motality (deaths per 1000)

10. gdp_cap = Gross domestic product/capita

11. Region = Region or economic group

12. calories = Daily calorie intake.

13. aids = Number of aids cases

14. birth_rt = Birth rate per 1000 people

15. death_rt = death rate per 1000 people

16. aids_rt = Number of aids cases/100000 people

17. log_gdp = log10(gdp_cap)

18. log_aidsr = log10(aids_rt)

19. b_to_d =birth to death ratio

20. fertility = average number of children in family

21. log_pop = log10(population)

22. cropgrow = ??

23. lit_male = % of males who can read

24. lit_fema = % of females who can read

25. Climate = predominant climate

The data file as it appears in SPSS

Consider the data on infant mortality

Stem-Leaf diagram stem = 10s, leaf = unit digit

0 4455555666666666777778888899 1 0122223467799 2 0001123555577788 3 45567999 4 135679 5 011222347 6 03678 7 4556679 8 5 9 4 10 1569 11 0022378 12 46 13 7 14 15 16 8

median = Q2 = 27

Quartiles

Lower quartile = Q1 = the median of lower half

Upper quartile = Q3 = the median of upper half

Summary Statistics

1 3

12 12 66 6712, 66.5

2 2Q Q

Interquartile range (IQR)

IQR = Q1 - Q3 = 66.5 – 12 = 54.5

lower = Q1 - 3(IQR) = 12 – 3(54.5) = - 151.5

The Outer Fences

No observations are outside of the outer fences

lower = Q1 – 1.5(IQR) = 12 – 1.5(54.5) = - 69.75

The Inner Fences

upper = Q3 = 1.5(IQR) = 66.5 – 1.5(54.5) = 148.25

upper = Q3 = 3(IQR) = 66.5 – 3(54.5) = 230.0

Only one observation (168 – Afghanistan) is outside of the inner fences – (mild outlier)

Box-Whisker Plot of Infant Mortality

0

0 50 100 150 200

Infant Mortality

Example 2

In this example we are looking at the weight gains (grams) for rats under six diets differing in level of protein (High or Low) and source of protein (Beef, Cereal, or Pork).

– Ten test animals for each diet

TableGains in weight (grams) for rats under six diets

differing in level of protein (High or Low)and source of protein (Beef, Cereal, or Pork)

Level High Protein Low protein

Source Beef Cereal Pork Beef Cereal Pork

Diet 1 2 3 4 5 6

73 98 94 90 107 49

102 74 79 76 95 82

118 56 96 90 97 73

104 111 98 64 80 86

81 95 102 86 98 81

107 88 102 51 74 97

100 82 108 72 74 106

87 77 91 90 67 70

117 86 120 95 89 61

111 92 105 78 58 82

Median 103.0 87.0 100.0 82.0 84.5 81.5

Mean 100.0 85.9 99.5 79.2 83.9 78.7

IQR 24.0 18.0 11.0 18.0 23.0 16.0

PSD 17.78 13.33 8.15 13.33 17.04 11.05

Variance 229.11 225.66 119.17 192.84 246.77 273.79

Std. Dev. 15.14 15.02 10.92 13.89 15.71 16.55

Non-Outlier MaxNon-Outlier Min

Median; 75%25%

Box Plots: Weight Gains for Six Diets

Diet

We

igh

t G

ain

40

50

60

70

80

90

100

110

120

130

1 2 3 4 5 6

High Protein Low Protein

Beef Beef Cereal Cereal Pork Pork

Conclusions

• Weight gain is higher for the high protein meat diets

• Increasing the level of protein - increases weight gain but only if source of protein is a meat source

Measures of Shape

Measures of Shape• Skewness

• Kurtosis

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 250

0.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 250

-3 -2 -1 0 1 2 3

0

-3 -2 -1 0 1 2 3

Positively skewed

Negatively skewed

Symmetric

PlatykurticLeptokurticNormal

(mesokurtic)

• Measure of Skewness – based on the sum of cubes

• Measure of Kurtosis – based on the sum of 4th powers

n

ii xx

1

3

n

ii xx

1

4

The Measure of Skewness

3

11 3

22

1

n

ii

n

ii

n x x

g

x x

The Measure of Kurtosis

4

12

2

1

3

n

ii

n

ii

x xg

n x x

The 3 is subtracted so that g2 is zero for the normal distribution

Interpretations of Measures of Shape

• Skewness

• Kurtosis

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25

0

-3 -2 -1 0 1 2 3

0

-3 -2 -1 0 1 2 3

g1 > 0 g1 = 0 g1 < 0

g2 < 0 g2 = 0 g2 > 0

Descriptive techniques for Multivariate data

In most research situations data is collected on more than one variable (usually many variables)

Graphical Techniques

• The scatter plot

• The two dimensional Histogram

The Scatter Plot

For two variables X and Y we will have a measurements for each variable on each case:

xi, yi

xi = the value of X for case i

and

yi = the value of Y for case i.

To Construct a scatter plot we plot the points:

(xi, yi)

for each case on the X-Y plane.

(xi, yi)

xi

yi

Data Set #3

The following table gives data on Verbal IQ, Math IQ,Initial Reading Acheivement Score, and Final Reading Acheivement Score

for 23 students who have recently completed a reading improvement program

Initial FinalVerbal Math Reading Reading

Student IQ IQ Acheivement Acheivement

1 86 94 1.1 1.72 104 103 1.5 1.73 86 92 1.5 1.94 105 100 2.0 2.05 118 115 1.9 3.56 96 102 1.4 2.47 90 87 1.5 1.88 95 100 1.4 2.09 105 96 1.7 1.7

10 84 80 1.6 1.711 94 87 1.6 1.712 119 116 1.7 3.113 82 91 1.2 1.814 80 93 1.0 1.715 109 124 1.8 2.516 111 119 1.4 3.017 89 94 1.6 1.818 99 117 1.6 2.619 94 93 1.4 1.420 99 110 1.4 2.021 95 97 1.5 1.322 102 104 1.7 3.123 102 93 1.6 1.9

Scatter Plot

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Verbal IQ

Mat

h I

Q

Scatter Plot

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Verbal IQ

Mat

h I

Q

(84,80)

Scatter Plot

60

70

80

90

100

110

120

130

60 70 80 90 100 110 120 130

Verbal IQ

Mat

h I

Q

Some Scatter Patterns

-100

-50

0

50

100

150

200

250

40 60 80 100 120 140

• Circular

• No relationship between X and Y

• Unable to predict Y from X

0

20

40

60

80

100

120

140

160

40 60 80 100 120 140

• Ellipsoidal

• Positive relationship between X and Y

• Increases in X correspond to increases in Y (but not always)

• Major axis of the ellipse has positive slope

0

20

40

60

80

100

120

140

160

40 60 80 100 120 140

Example

Verbal IQ, MathIQ

Scatter Plot

60

70

80

90

100

110

120

130

60 70 80 90 100 110 120 130

Verbal IQ

Mat

h I

Q

Some More Patterns

0

20

40

60

80

100

120

140

40 60 80 100 120 140

• Ellipsoidal (thinner ellipse)

• Stronger positive relationship between X and Y

• Increases in X correspond to increases in Y (more freqequently)

• Major axis of the ellipse has positive slope

• Minor axis of the ellipse much smaller

0

20

40

60

80

100

120

140

40 60 80 100 120 140

• Increased strength in the positive relationship between X and Y

• Increases in X correspond to increases in Y (almost always)

• Minor axis of the ellipse extremely small in relationship to the Major axis of the ellipse.

0

20

40

60

80

100

120

140

40 60 80 100 120 140

• Perfect positive relationship between X and Y

• Y perfectly predictable from X

• Data falls exactly along a straight line with positive slope

0

20

40

60

80

100

120

140

40 60 80 100 120 140

• Ellipsoidal

• Negative relationship between X and Y

• Increases in X correspond to decreases in Y (but not always)

• Major axis of the ellipse has negative slope slope

0

20

40

60

80

100

120

140

40 60 80 100 120 140

• The strength of the relationship can increase until changes in Y can be perfectly predicted from X

0

20

40

60

80

100

120

140

40 60 80 100 120 140

Some Non-Linear Patterns

0

200

400

600

800

1000

1200

-20 -10 0 10 20 30 40 50

• In a Linear pattern Y increase with respect to X at a constant rate

• In a Non-linear pattern the rate that Y increases with respect to X is variable

Growth Patterns

-20

0

20

40

60

80

100

120

0 10 20 30 40 50

-150

-100

-50

0

50

100

150

0 10 20 30 40 50

-20

0

20

40

60

80

100

120

0 10 20 30 40 50

• Growth patterns frequently follow a sigmoid curve

• Growth at the start is slow

• It then speeds up

• Slows down again as it reaches it limiting size

0

20

40

60

80

100

120

0 10 20 30 40 50

Measures of strength of a relationship (Correlation)

• Pearson’s correlation coefficient (r)

• Spearman’s rank correlation coefficient (rho, )

Assume that we have collected data on two variables X and Y. Let

(x1, y1) (x2, y2) (x3, y3) … (xn, yn)

denote the pairs of measurements on the on two variables X and Y for n cases in a sample (or population)

From this data we can compute summary statistics for each variable.

The means

and

n

xx

n

ii

1

n

yy

n

ii

1

The standard deviations

and

11

2

n

xxs

n

ii

x

11

2

n

yys

n

ii

y

These statistics:

• give information for each variable separately

but

• give no information about the relationship between the two variables

x yxs ys

Consider the statistics:

n

iixx xxS

1

2

n

iiyy yyS

1

2

n

iiixy yyxxS

1

The first two statistics:

• are used to measure variability in each variable

• they are used to compute the sample standard deviations

n

iixx xxS

1

2

n

iiyy yyS

1

2and

1

n

Ss xx

x 1

n

Ss yy

y

The third statistic:

• is used to measure correlation• If two variables are positively related the sign of

will agree with the sign of

n

iiixy yyxxS

1

xxi

yyi

•When is positive will be positive.

•When xi is above its mean, yi will be above its

mean

•When is negative will be negative.

•When xi is below its mean, yi will be below its

mean

The product will be positive for most cases.

xxi yyi

xxi yyi

yyxx ii

This implies that the statistic

• will be positive

• Most of the terms in this sum will be positive

n

iiixy yyxxS

1

On the other hand

• If two variables are negatively related the sign of

will be opposite in sign to

xxi

yyi

•When is positive will be negative.

•When xi is above its mean, yi will be below its

mean

•When is negative will be positive.

•When xi is below its mean, yi will be above its

mean

The product will be negative for most cases.

xxi yyi

xxi yyi

yyxx ii

Again implies that the statistic

• will be negative

• Most of the terms in this sum will be negative

n

iiixy yyxxS

1

Pearsons correlation coefficient is defined as below:

n

ii

n

ii

n

iii

yyxx

xy

yyxx

yyxx

SS

Sr

1

2

1

2

1

The denominator:

is always positive

n

ii

n

ii yyxx

1

2

1

2

The numerator:

• is positive if there is a positive relationship between X ad Y and

• negative if there is a negative relationship between X ad Y.

• This property carries over to Pearson’s correlation coefficient r

n

iii yyxx

1

Properties of Pearson’s correlation coefficient r

1. The value of r is always between –1 and +1.2. If the relationship between X and Y is positive, then

r will be positive.3. If the relationship between X and Y is negative,

then r will be negative.4. If there is no relationship between X and Y, then r

will be zero.

5. The value of r will be +1 if the points, (xi, yi) lie on a straight line with positive slope.

6. The value of r will be -1 if the points, (xi, yi) lie on a straight line with negative slope.

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r =1

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = 0.95

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = 0.7

0

20

40

60

80

100

120

140

160

40 60 80 100 120 140

r = 0.4

-100

-50

0

50

100

150

200

250

40 60 80 100 120 140

r = 0

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = -0.4

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = -0.7

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = -0.8

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = -0.95

0

20

40

60

80

100

120

140

40 60 80 100 120 140

r = -1

Computing formulae for the statistics:

n

iixx xxS

1

2

n

iiyy yyS

1

2

n

iiixy yyxxS

1

n

x

xxxS

n

iin

ii

n

iixx

2

1

1

2

1

2

n

yx

yx

n

ii

n

iin

iii

11

1

n

y

yyyS

n

iin

ii

n

iiyy

2

1

1

2

1

2

n

iiixy yyxxS

1

To compute

first compute

Then

xxS yyS xyS

n

iixC

1

2

n

iii yxE

1

n

iiyD

1

2

n

iiyB

1

n

iixA

1

n

ACSxx

2

n

BDS yy

2

n

BAESxy

Example

Verbal IQ, MathIQ

Data Set #3

The following table gives data on Verbal IQ, Math IQ,Initial Reading Acheivement Score, and Final Reading Acheivement Score

for 23 students who have recently completed a reading improvement program

Initial FinalVerbal Math Reading Reading

Student IQ IQ Acheivement Acheivement

1 86 94 1.1 1.72 104 103 1.5 1.73 86 92 1.5 1.94 105 100 2.0 2.05 118 115 1.9 3.56 96 102 1.4 2.47 90 87 1.5 1.88 95 100 1.4 2.09 105 96 1.7 1.7

10 84 80 1.6 1.711 94 87 1.6 1.712 119 116 1.7 3.113 82 91 1.2 1.814 80 93 1.0 1.715 109 124 1.8 2.516 111 119 1.4 3.017 89 94 1.6 1.818 99 117 1.6 2.619 94 93 1.4 1.420 99 110 1.4 2.021 95 97 1.5 1.322 102 104 1.7 3.123 102 93 1.6 1.9

Scatter Plot

60

70

80

90

100

110

120

130

60 70 80 90 100 110 120 130

Verbal IQ

Mat

h I

Q

Now

Hence

2214941

2

n

iix 227199

1

n

iii yx234363

1

2

n

iiy

23071

n

iiy2244

1

n

iix

652.255723

2244221494

2

xxS

87.296023

2307234363

2

yyS

043.2116

23

23072244227199 xyS

Thus Pearsons correlation coefficient is:

yyxx

xy

SS

Sr

769.087.2960652.2557

043.2116

Measure of Variability (Dispersion, Spread) 1.Range 2.Inter-Quartile Range 3.Variance, standard...

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Transcript of Measure of Variability (Dispersion, Spread) 1.Range 2.Inter-Quartile Range 3.Variance, standard...