numerical measures

Numerical Measures

• Measures of Central Tendency (Location)

• Measures of Non Central Location

• Measure of Variability (Dispersion, Spread)

• Measures of Shape

Measures of Central Tendency (Location)

• Mean

• Median

• Mode

0 5 10 15 20 25

Central Location

Measures of Non-central Location

• Quartiles, Mid-Hinges

• Percentiles

0 5 10 15 20 25

Non - Central Location

Measure of Variability (Dispersion, Spread)

• Variance, standard deviation

• Range

• Inter-Quartile Range

0 5 10 15 20 25

Variability

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

-3 -2 -1 0 1 2 3

Summation Notation

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

Then the symbol

denotes the sum of these n numbers

x1 + x2 + x3 + …+ xn

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 the symbol

denotes the sum of these 5 numbers

x1 + x2 + x3 + x4 + x5

= 10 + 15 + 21 + 7 + 13

Meaning of parts of summation notation

i in expression

Quantity changing in each term of the sum

Starting value for i

Final value for i

each term of the sum

Example

Again 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 the symbol

denotes the sum of these 3 numbers

= 153 + 213 + 73

= 3375 + 9261 + 343

= 12979

32 xxx

Measures of Central Location (Mean)

Then the mean of the n numbers is defined as:

Example

Again 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 the mean of the 5 numbers is:

5554321

1 xxxxxx

137211510

Interpretation of the Mean

Then the mean, , is the centre of gravity of those the n numbers.

That is if we drew a horizontal line and placed a weight of one at each value of xi , then the balancing point of that system of mass is at the point .

x1 x2x3 x4xn

107 15 2113

In the Example

100 20

The mean, , is also approximately the center of gravity of a histogram

60 - 70 70 - 80 80 - 90 90 - 100 100 - 110 110 - 120 120 - 130 130 - 140 140 - 150

Measures of Central Location (Median)

The Median

Then the median of the n numbers is defined as the number that splits the numbers into two equal parts.

To evaluate the median we arrange the numbers in increasing order.

If the number of observations is odd there will be one observation in the middle.

This number is the median.

If the number of observations is even there will be two middle observations.

The median is the average of these two observations

Example

Again let x1, x2, x3, 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

The numbers arranged in order are:

7 10 13 15 21

Unique “Middle” observation – the median

Example 2

Let x1, x2, x3 , x4, x5 , x6 denote the 6 denote numbers:

23 41 12 19 64 8

Arranged in increasing order these observations would be:

8 12 19 23 41 64

Two “Middle” observations

Median

= average of two “middle” observations =

Example

The data on N = 23 students

Variables

• Verbal IQ

• Math IQ

• Initial Reading Achievement Score

• Final Reading Achievement Score

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

Total 2244 2307 35.1 48.3

IQ IQ Acheivement AcheivementMeans 97.57 100.30 1.526 2.100

Computing the Median

Stem leaf Diagrams

Median = middle observation =12th observation

Summary

IQ IQ Acheivement AcheivementMeans 97.57 100.30 1.526 2.100Median 96 97 1.5 1.9

Some Comments

• The mean is the centre of gravity of a set of observations. The balancing point.

• The median splits the obsevations equally in two parts of approximately 50%

• The median splits the area under a histogram in two parts of 50%

• The mean is the balancing point of a histogram

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

xmedian

0 5 10 15 20 25

• For symmetric distributions the mean and the median will be approximately the same value

50% 50%

xMedian &

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

xmedian

• For Positively skewed distributions the mean exceeds the median

• For Negatively skewed distributions the median exceeds the mean

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

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

• The mean is altered to a significant degree by the presence of outliers

• Outliers have little effect on the value of the median

• This is a reason for using the median in place of the mean as a measure of central location

• Alternatively the mean is the best measure of central location when the data is Normally distributed (Bell-shaped)

Review

Summarizing Data

Graphical Methods

8 0 2 4 6 6 9

9 0 4 4 5 5 6 9 9

10 2 2 4 5 5 9

11 1 8 9

70 to 80 80 to 90 90 to100

100 to110

110 to120

120 to130

Histogram

Stem-Leaf Diagram

Verbal IQ Math IQ70 to 80 1 180 to 90 6 290 to 100 7 11

100 to 110 6 4110 to 120 3 4120 to 130 0 1

Grouped Freq Table

Numerical Measures

• Measures of Central Tendency (Location)

• Measures of Non Central Location

• Measure of Variability (Dispersion, Spread)

• Measures of Shape

The objective is to reduce the data to a small number of values that completely describe the data and certain aspects of the data.

Measures of Central Location (Mean)

Then the mean of the n numbers is defined as:

Interpretation of the Mean

Then the mean, , is the centre of gravity of those the n numbers.

That is if we drew a horizontal line and placed a weight of one at each value of xi , then the balancing point of that system of mass is at the point .

x1 x2x3 x4xn

The mean, , is also approximately the center of gravity of a histogram

60 - 70 70 - 80 80 - 90 90 - 100 100 - 110 110 - 120 120 - 130 130 - 140 140 - 150

The Median

Then the median of the n numbers is defined as the number that splits the numbers into two equal parts.

To evaluate the median we arrange the numbers in increasing order.

If the number of observations is odd there will be one observation in the middle.

This number is the median.

If the number of observations is even there will be two middle observations.

The median is the average of these two observations

Measures of Non-Central Location

• Percentiles

• Quartiles (Hinges, Mid-hinges)

DefinitionThe P×100 Percentile is a point , xP ,

underneath a distribution that has a fixed proportion P of the population (or sample) below that value

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

P×100 %

Definition (Quartiles)The first Quartile , Q1 ,is the 25 Percentile , x0.25

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

The second Quartile , Q2 ,is the 50th

Percentile , x0.50

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

• The second Quartile , Q2 , is also the

median and the 50th percentile

The third Quartile , Q3 ,is the 75th Percentile , x0.75

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

The Quartiles – Q1, Q2, Q3

divide the population into 4 equal parts of 25%.

00.020.040.060.080.1

0.120.140.16

0 5 10 15 20 25

25 % 25 %

Q1 Q2 Q3

Computing Percentiles and Quartiles

• There are several methods used to compute percentiles and quartiles. Different computer packages will use different methods

• Sometimes for small samples these methods will agree (but not always)

• For large samples the methods will agree within a certain level of accuracy

Computing Percentiles and Quartiles – Method 1• The first step is to order the observations in

increasing order.

• We then compute the position, k, of the P×100 Percentile.

k = P × (n+1)

Where n = the number of observations

ExampleThe data on n = 23 students

Variables

• Verbal IQ

• Math IQ

• Initial Reading Achievement Score

• Final Reading Achievement Score

We want to compute the 75th percentile and

the 90th percentile

The position, k, of the 75th Percentile.

k = P × (n+1) = .75 × (23+1) = 18

The position, k, of the 90th Percentile.

k = P × (n+1) = .90 × (23+1) = 21.6

When the position k is an integer the percentile is the kth observation (in order of magnitude) in the data set.

For example the 75th percentile is the 18th (in size) observation

When the position k is an not an integer but an integer(m) + a fraction(f).

i.e. k = m + f

then the percentile is

xP = (1-f) × (mth observation in size)

+ f × (m+1st observation in size)

In the example the position of the 90th percentile is:

k = 21.6

x.90 = 0.4(21st observation in size)

+ 0.6(22nd observation in size)

When the position k is an not an integer but an integer(m) + a fraction(f).i.e. k = m + fthen the percentile is

xP = (1-f) × (mth observation in size)+ f × (m+1st observation in size)

xp = (1- f) ( mth obs) + f [(m+1)st obs]

(m+1)st obsmth obs

obs obs 1

obs obs 1obs 1

obs obs 1

obs thst

thstth

mfmfthst

obs obs 1

When the position k is an not an integer but an integer(m) + a fraction(f).

i.e. k = m + f

xp = (1- f) ( mth obs) + f [(m+1)st obs]

(m+1)st obsmth obs

mxthst

obs obs 1

Thus the position of xp is 100f% through the interval between the mth observation and the (m +1)st observation

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

x0.75 = 75th percentile = 18th observation in size =105

(position k = 18)

x0.90 = 90th percentile

= 0.4(21st observation in size)

+ 0.6(22nd observation in size)

= 0.4(111)+ 0.6(118) = 115.2

(position k = 21.6)

An Alternative method for computing Quartiles – Method 2• Sometimes this method will result in the

same values for the quartiles.

• Sometimes this method will result in the different values for the quartiles.

• For large samples the two methods will result in approximately the same answer.

The first step in Method 2 is to arrange the numbers in increasing order.

From the arranged numbers we compute the median.

This is also called the Hinge

ExampleConsider the 5 numbers:

10 15 21 7 13Arranged in increasing order:

7 10 13 15 21

The median (or Hinge) splits the observations in half

Median (Hinge)

The lower mid-hinge (the first quartile) is the “median” of the lower half of the observations (excluding the median).

The upper mid-hinge (the third quartile) is the “median” of the upper half of the observations (excluding the median).

Consider the five number in increasing order:

7 10 13 15 21

Median (Hinge)

Lower Half

Upper Half

Upper Mid-Hinge

(First Quartile)

(7+10)/2 =8.5

Upper Mid-Hinge

(Third Quartile)

(15+21)/2 = 18

Computing the median and the quartile using the first method:Position of the median: k = 0.5(5+1) = 3

Position of the first Quartile: k = 0.25(5+1) = 1.5

Position of the third Quartile: k = 0.75(5+1) = 4.5

7 10 13 15 21

Q2 = 13Q1 = 8. 5 Q3 = 18

• Both methods result in the same value

• This is not always true.

Example

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

Median (Hinge)

Lower Mid-Hinge

(First Quartile)

Upper Mid-Hinge

(Third Quartile)

Computing the median and the quartile using the first method:Position of the median: k = 0.5(23+1) = 12

Position of the first Quartile: k = 0.25(23+1) = 6

Position of the third Quartile: k = 0.75(23+1) = 18

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 = 105

• Many programs compute percentiles, quartiles etc.

• Each may use different methods.

• It is important to know which method is being used.

• The different methods result in answers that are close when the sample size is large.

Box-PlotsBox-Whisker Plots

• A graphical method of displaying data

• An alternative to the histogram and stem-leaf diagram

To Draw a Box Plot

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

• We also compute the largest and smallest of the observations – the max and the min

• The five number summary

min, Q1, Q2, Q3, max

Example

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

The Box Plot is then drawn

• Drawing above an axis a “box” from Q1 to Q3.

• Drawing vertical line in the box at the median, Q2

• Drawing whiskers at the lower and upper ends of the box going down to the min and up to max.

BoxLower Whisker

Upper Whisker

Q2Q1Q3min max

Example

min = 80

Q1 = 89

Q2 = 96

Q3 = 105

max = 119

This is sometimes called the five-number summary

70 80 90 100 110 120 130

Box Plot of Verbal IQ

Box Plot can also be drawn vertically

Box-Whisker plots(Verbal IQ, Math IQ)

Box-Whisker plots(Initial RA, Final RA )

Summary Information contained in the box plot

Middle 50% of population

25% 25% 25% 25%

Advance Box Plots

• 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 )

• The difference Q3– Q1 is called the inter-quartile range (denoted by IQR)

• 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.

Inner fences

Lower inner fence

f1 = Q1 - (1.5)IQR

Upper inner fence

f2 = Q3 + (1.5)IQR

Outer fences

Lower outer fence

F1 = Q1 - (3)IQR

Upper outer fence

F2 = Q3 + (3)IQR

• Observations that are between the lower and upper inner 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

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 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

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

Next topic:Numerical Measures of Variability

numerical measures

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