1. Numerical Descriptive Techniques
Chapter 4
Numerical Descriptive Techniques

2. 4.2 Measures of Central Location
Usually, we focus our attention on two types of measures when describing population characteristics:
Central location (e.g. average)
Variability or spread
The measure of central location reflects the locations of all the actual data points.

3. 4.2 Measures of Central Location
The measure of central location reflects the locations of all the actual data points.
How?
With two data points,
the central location
should fall in the middle
between them (in order
to reflect the location of
both of them).
But if the third data point
appears on the left hand-side
of the midrange, it should “pull”
the central location to the left.
With one data point
clearly the central
location is at the point
itself.

4. The Arithmetic Mean
This is the most popular and useful measure of central location
Sum of the observations
Number of observations
Mean =

5. The Arithmetic Mean Sample mean Population mean Sample size
Population size

6. The Arithmetic Mean Example 4.1 Example 4.2
The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours. Find the mean time on the Internet. 7 22
11.0
Example 4.2
Suppose the telephone bills of Example 2.1 represent
the population of measurements. The population mean is
42.19
38.45
45.77
43.59

7. The Median
The Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude.
Example 4.3
Find the median of the time on the internet for the 10 adults of example 4.1
Suppose only 9 adults were sampled (exclude, say, the longest time (33))
Comment
0, 0, 5, 7, 8, 9, 12, 14, 22, 33
Even number of observations
Odd number of observations
0, 0, 5, 7, 8, , 12, 14, 22, 33
8.5,
0, 0, 5, 7, 8 9, 12, 14, 22 8

8. The Mode
The Mode of a set of observations is the value that occurs most frequently.
Set of data may have one mode (or modal class), or two or more modes.
For large data sets
the modal class is
much more relevant
than a single-value
mode.
The modal class

9. The Mode
The Mean, Median,
Mode
The Mode
Example 4.5 Find the mode for the data in Example 4.1. Here are the data again: 0, 7, 12, 5, 33, 14, 8, 0, 9, 22
Solution
All observation except “0” occur once. There are two “0”. Thus, the mode is zero.
Is this a good measure of central location?
The value “0” does not reside at the center of this set (compare with the mean = 11.0 and the mode = 8.5).

10. Relationship among Mean, Median, and Mode
If a distribution is symmetrical, the mean, median and mode coincide
If a distribution is asymmetrical, and skewed to the left or to the right, the three measures differ.
A positively skewed distribution
(“skewed to the right”)
Mode
Mean
Median

11. Relationship among Mean, Median, and Mode
If a distribution is symmetrical, the mean, median and mode coincide
If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ.
A positively skewed distribution
(“skewed to the right”)
A negatively skewed distribution
(“skewed to the left”)
Mode
Mean
Mean
Mode
Median
Median

12. The Geometric Mean This is a measure of the average growth rate.
Let Ri denote the the rate of return in period i (i=1,2…,n). The geometric mean of the returns R1, R2, …,Rn is the constant Rg that produces the same terminal wealth at the end of period n as do the actual returns for the n periods.

13. The Geometric Mean = For the given series of rate of
returns the nth period return is
calculated by:
If the rate of return was Rg in every
period, the nth period return would
be calculated by: =
Rg is selected such that…

14. 4.3 Measures of variability
Measures of central location fail to tell the whole story about the distribution.
A question of interest still remains unanswered:
How much are the observations spread out
around the mean value?

15. 4.3 Measures of variability
Observe two hypothetical
data sets:
Small variability
The average value provides
a good representation of the
observations in the data set.
This data set is now
changing to...

16. 4.3 Measures of variability
Observe two hypothetical
data sets:
Small variability
The average value provides
a good representation of the
observations in the data set.
Larger variability
The same average value does not
provide as good representation of the
observations in the data set as before.

17. The range
The range of a set of observations is the difference between the largest and smallest observations.
Its major advantage is the ease with which it can be computed.
Its major shortcoming is its failure to provide information on the dispersion of the observations between the two end points.
But, how do all the observations spread out? ? ? ?
The range cannot assist in answering this question
Range
Smallest
observation
Largest
observation

18. The Variance
This measure reflects the dispersion of all the observations
The variance of a population of size N x1, x2,…,xN whose mean is m is defined as
The variance of a sample of n observations x1, x2, …,xn whose mean is is defined as

19. Why not use the sum of deviations?
Consider two small populations:
9-10= -1
A measure of dispersion
Should agrees with this
observation.
Can the sum of deviations
Be a good measure of dispersion?
11-10= +1
The sum of deviations is zero for both populations, therefore, is not a good measure of dispersion.
8-10= -2 A
12-10= +2 8 9 10 11 12
Sum = 0
…but measurements in B
are more dispersed
then those in A.
The mean of both
populations is 10...
4-10 = - 6
16-10 = +6 B
7-10 = -3 4 7 10 13 16
13-10 = +3

20. The Variance Let us calculate the variance of the two populations
Why is the variance defined as
the average squared deviation?
Why not use the sum of squared
deviations as a measure of
variation instead?
After all, the sum of squared
deviations increases in
magnitude when the variation
of a data set increases!!

21. Which data set has a larger dispersion?
The Variance
Let us calculate the sum of squared deviations for both data sets
Which data set has a larger dispersion?
Data set B
is more dispersed
around the mean A B 1 2 3 1 3 5

22. The Variance A B SumA = (1-2)2 +…+(1-2)2 +(3-2)2 +… +(3-2)2= 10
SumB = (1-3)2 + (5-3)2 = 8
SumA > SumB. This is inconsistent with the observation that set B is more dispersed. A B 1 2 3 1 3 5

23. The Variance
However, when calculated on “per observation” basis (variance), the data set dispersions are properly ranked.
sA2 = SumA/N = 10/5 = 2
sB2 = SumB/N = 8/2 = 4 A B 1 2 3 1 3 5

24. The Variance Example 4.7 Solution
The following sample consists of the number of jobs six students applied for: 17, 15, 23, 7, 9, 13. Finds its mean and variance
Solution

25. The Variance – Shortcut method

26. Standard Deviation
The standard deviation of a set of observations is the square root of the variance .

27. Standard Deviation Example 4.8
To examine the consistency of shots for a new innovative golf club, a golfer was asked to hit 150 shots, 75 with a currently used (7-iron) club, and 75 with the new club.
The distances were recorded.
Which 7-iron is more consistent?

28. Standard Deviation Example 4.8 – solution
The Standard
Deviation
Standard Deviation
Example 4.8 – solution
Excel printout, from the “Descriptive Statistics” sub-menu.
The innovation club is more consistent, and because the means are close, is considered a better club

29. Interpreting Standard Deviation
The standard deviation can be used to
compare the variability of several distributions
make a statement about the general shape of a distribution.
The empirical rule: If a sample of observations has a mound-shaped distribution, the interval

30. Interpreting Standard Deviation
Example 4.9 A statistics practitioner wants to describe the way returns on investment are distributed.
The mean return = 10%
The standard deviation of the return = 8%
The histogram is bell shaped.

31. Interpreting Standard Deviation
Example 4.9 – solution
The empirical rule can be applied (bell shaped histogram)
Describing the return distribution
Approximately 68% of the returns lie between 2% and 18% [10 – 1(8), (8)]
Approximately 95% of the returns lie between -6% and % [10 – 2(8), (8)]
Approximately 99.7% of the returns lie between -14% and 34% [10 – 3(8), (8)]

32. The Chebysheff’s Theorem
The proportion of observations in any sample that lie within k standard deviations of the mean is at least 1-1/k2 for k > 1.
This theorem is valid for any set of measurements (sample, population) of any shape!!
K Interval Chebysheff Empirical Rule
1 at least 0% approximately 68%
2 at least 75% approximately 95%
3 at least 89% approximately 99.7%
(1-1/12)
(1-1/22)
(1-1/32)

33. The Chebysheff’s Theorem
Example 4.10
The annual salaries of the employees of a chain of computer stores produced a positively skewed histogram. The mean and standard deviation are $28,000 and $3,000,respectively. What can you say about the salaries at this chain?
Solution At least 75% of the salaries lie between $22,000 and $34, – 2(3000) (3000)
At least 88.9% of the salaries lie between $$19,000 and $37, – 3(3000) (3000)

34. The Coefficient of Variation
The coefficient of variation of a set of measurements is the standard deviation divided by the mean value.
This coefficient provides a proportionate measure of variation.
A standard deviation of 10 may be perceived
large when the mean value is 100, but only
moderately large when the mean value is 500

35. 4.4 Measures of Relative Standing and Box Plots
Percentile
The pth percentile of a set of measurements is the value for which
p percent of the observations are less than that value
100(1-p) percent of all the observations are greater than that value.
Example
Suppose your score is the 60% percentile of a SAT test. Then
60% of all the scores lie here
40%
Your score

36. Quartiles Commonly used percentiles
First (lower)decile = 10th percentile
First (lower) quartile, Q1, = 25th percentile
Second (middle)quartile,Q2, = 50th percentile
Third quartile, Q3, = 75th percentile
Ninth (upper)decile = 90th percentile

37. Quartiles
Example
Find the quartiles of the following set of measurements 7, 8, 12, 17, 29, 18, 4, 27, 30, 2, 4, 10, 21, 5, 8

38. Quartiles
Solution
Sort the observations
2, 4, 4, 5, 7, 8, 10, 12, 17, 18, 18, 21, 27, 29, 30
15 observations
The first quartile
At most (.25)(15) = 3.75 observations
should appear below the first quartile.
Check the first 3 observations on the
left hand side.
At most (.75)(15)=11.25 observations
should appear above the first quartile.
Check 11 observations on the
right hand side.
Comment:If the number of observations is even, two observations remain unchecked. In this case choose the midpoint between these two observations.

39. Location of Percentiles
Find the location of any percentile using the formula
Example 4.11
Calculate the 25th, 50th, and 75th percentile of the data in Example 4.1

40. Location of Percentiles
Example 4.11 – solution
After sorting the data we have 0, 0, 5, 7, 8, 9, 22, 33. 1
Location
Values
Location 3
3.75
The 2.75th location
Translates to the value
(.75)(5 – 0) = 3.75
2.75

41. Location of Percentiles
Example 4.11 – solution continued
The 50th percentile is halfway between the fifth and sixth observations (in the middle between 8 and 9), that is 8.5.

42. Location of Percentiles
Example 4.11 – solution continued
The 75th percentile is one quarter of the distance between the eighth and ninth observation that is (22 – 14) = 16.
Eighth
observation
Ninth
observation

43. Quartiles and Variability
Quartiles can provide an idea about the shape of a histogram
Q1 Q Q3
Q Q Q3
Positively skewed
histogram
Negatively skewed
histogram

44. Interquartile range = Q3 – Q1
This is a measure of the spread of the middle 50% of the observations
Large value indicates a large spread of the observations
Interquartile range = Q3 – Q1

45. Box Plot
This is a pictorial display that provides the main descriptive measures of the data set:
L - the largest observation
Q3 - The upper quartile
Q2 - The median
Q1 - The lower quartile
S - The smallest observation
1.5(Q3 – Q1)
Whisker S Q1 Q2 Q3 L

46. Box Plot
Example 4.14 (Xm02-01)
Left hand boundary = 9.275–1.5(IQR)=
Right hand boundary= (IQR)=
9.275
119.63
26.905
No outliers are found

47. Box Plot Additional Example - GMAT scores
Create a box plot for the data regarding the GMAT scores of 200 applicants (see GMAT.XLS)
417.5
449
512
537
575
669.5
788
(IQR)
(IQR)

48. Box Plot GMAT - continued Interpreting the box plot results Q1 512 Q2
537 Q3
575
449
669.5
25%
50%
25%
Interpreting the box plot results
The scores range from 449 to 788.
About half the scores are smaller than 537, and about half are larger than 537.
About half the scores lie between 512 and 575.
About a quarter lies below 512 and a quarter above 575.

49. The histogram is positively skewed
Box Plot
GMAT - continued
The histogram is positively skewed Q1
512 Q2
537 Q3
575
449
669.5
25%
50%
25%
50%
25%
25%

50. Box Plot Example 4.15 (Xm04-15) Example 4.15 – solution
A study was organized to compare the quality of service in 5 drive through restaurants.
Interpret the results
Example 4.15 – solution
Minitab box plot

51. Box Plot Jack in the Box Jack in the box is the slowest in service
Hardee’s
Hardee’s service time variability is the largest
McDonalds
Wendy’s service time appears to be the shortest and most consistent.
Wendy’s
Popeyes

52. Box Plot Times are symmetric Times are positively skewed
Jack in the Box
Jack in the box is the slowest in service
Hardee’s
Hardee’s service time variability is the largest
McDonalds
Wendy’s service time appears to be the shortest and most consistent.
Wendy’s
Popeyes

53. 4.5 Measures of Linear Relationship
The covariance and the coefficient of correlation are used to measure the direction and strength of the linear relationship between two variables.
Covariance - is there any pattern to the way two variables move together?
Coefficient of correlation - how strong is the linear relationship between two variables

54. Covariance mx (my) is the population mean of the variable X (Y).
N is the population size.
x (y) is the sample mean of the variable X (Y).
n is the sample size.

55. Covariance Compare the following three sets xi yi (x – x) (y – y)2 6 7 13 20 27 -3 1 -7 21 14
x=5
y =20
Cov(x,y)=17.5 xi yi 2 6 7 20 27 13
Cov(x,y) = -3.5
x=5
y =20 xi yi
(x – x)
(y – y)
(x – x)(y – y) 2 6 7 27 20 13 -3 1 -7
-21
-14
x=5
y =20
Cov(x,y)=-17.5

56. Covariance
If the two variables move in the same direction, (both increase or both decrease), the covariance is a large positive number.
If the two variables move in opposite directions, (one increases when the other one decreases), the covariance is a large negative number.
If the two variables are unrelated, the covariance will be close to zero.

57. The coefficient of correlation
This coefficient answers the question: How strong is the association between X and Y.

58. The coefficient of correlation+1 -1
Strong positive linear relationship
COV(X,Y)>0 or
r or r =
No linear relationship
COV(X,Y)=0
Strong negative linear relationship
COV(X,Y)<0

59. The coefficient of correlation
If the two variables are very strongly positively related, the coefficient value is close to +1 (strong positive linear relationship).
If the two variables are very strongly negatively related, the coefficient value is close to -1 (strong negative linear relationship).
No straight line relationship is indicated by a coefficient close to zero.

60. The coefficient of correlation and the covariance – Example 4.16
Compute the covariance and the coefficient of correlation to measure how GMAT scores and GPA in an MBA program are related to one another.
Solution
We believe GMAT affects GPA. Thus
GMAT is labeled X
GPA is labeled Y

61. The coefficient of correlation and the covariance – Example 4.16
Student x y x y xy
9.6 1
599
358801
92.16
5750.4
cov(x,y)=(1/12-1)[67,559.2-(7587)(106.4)/12]=26.16
Sx = {(1/12-1)[4,817,755-(7587)2/12)]}.5=43.56
Sy = similar to Sx = 1.12
r = cov(x,y)/SxSy = 26.16/(43.56)(1.12) = .5362 2
689
8.8
474721
77.44
6063.2 3
584
7.4
341056
54.76
4321.6 4
………………………………………………….
631 10
398161
100
6310 11
593
8.8
351649
77.44
5218.4 12
683 8
466489 64
5464
Total
7,587
106.4
4,817,755
957.2
67,559.2

62. The coefficient of correlation and the covariance – Example 4
The coefficient of correlation and the covariance – Example 4.16 – Excel
Use the Covariance option in Data Analysis
If your version of Excel returns the population covariance and variances, multiply each one by n/n-1 to obtain the corresponding sample values.
Use the Correlation option to produce the correlation matrix.
Variance-Covariance Matrix
Population
values
Population
values
GPA
GMAT
1.15
23.98
GPA
GMAT
1.25
26.16
Sample
values
Sample
values 12
12-1

63. The coefficient of correlation and the covariance – Example 4
The coefficient of correlation and the covariance – Example 4.16 – Excel
Interpretation
The covariance (26.16) indicates that GMAT score and performance in the MBA program are positively related.
The coefficient of correlation (.5365) indicates that there is a moderately strong positive linear relationship between GMAT and MBA GPA.

64. The Least Squares Method
We are seeking a line that best fits the data when two variables are (presumably) related to one another.
We define “best fit line” as a line for which the sum of squared differences between it and the data points is minimized.
The y value of point i
calculated from the
equation
The actual y value of point i

65. The least Squares MethodY
Errors
Errors X
Different lines generate different errors,
thus different sum of squares of errors.
There is a line that minimizes the sum of squared errors

66. The least Squares Method
The coefficients b0 and b1 of the line that minimizes the sum of squares of errors are calculated from the data.

67. The Least Squares Method
Example 4.17
Find the least squares line for Example 4.16 (Xm04-16.xls)