1. Descriptive Statistics: Numerical Methods
Chapter III
Descriptive Statistics: Numerical Methods

2. Key Learning Objectives and Topics in this Chapter
Measures of Location:
(Mean, Median, Mode, Percentiles, Quartiles)
Measures of Dispersion/Variability
( Range, Variance, Standard Deviation,
Coefficient of Variation)
Measures of distribution shape, and association between two variables

3. Important Note In all cases :
Know the formulas, learn the computation procedures (i.e., apply the formulas) and know the meaning (interpretation) of the measures computed.
Use Excel; Practice! Practice! and Practice!

4. 3.1. Introduction These measures could be computed for
When describing data, usually we focus our attention on two types of measures..
Central location (e.g. average)
Variability or Spread
These measures could be computed for
Population: Parameters
Sample : Statistics

5. 3.2 Measures of Central Location
A center is a reference point. Thus a good measure of central location is expected to reflect the locations of all the other actual points in the data.
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).
if the third data point
appears on the left hand-side
of the center, it should “pull”
the central location to the left.
With one data point
clearly the central
location is at the point
itself.

6. Measures of Location Mean Median Mode Percentiles Quartiles
If the measures are computed
for data from a sample,
they are called sample statistics.
Mean
Median
Mode
If the measures are computed
for data from a population,
they are called population parameters.
Percentiles
Quartiles
A sample statistic is referred to
as the point estimator of the
corresponding population parameter.

7. i) The Arithmetic Mean (µ)
This is the most popular and useful measure of central location
Sum of the observations
Number of observations
Mean =

8. Observations in the data
Sum of the values of
Observations in the data
i) The Arithmetic Mean
Sample mean
Population mean
Number of observations
In the sample
(Sample size)
Number of Observations
In the Population
(Population size)

9. i) The Arithmetic Mean Example 1
Time (hours) spent by 10 adults on the Internet are as follows:
0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours.
Based on this data, compute the mean (average) amount of time spent on the Internet?
Based on this data, the average amount of time spent on the internet
by a typical adult is 11 hours.

10. ii) The Median
The Median of a set of observations is the value that falls in the middle of a data that is arranged in certain order (ascending or descending).
It is the value that divides the observation into two equal halves

11. ii) The Median To find the median:
We Put the data in an array (in increasing or decreasing order).
If the total number of observation in the data set is an ODD number, the median is the middle value.
If the total number of observation contained in the data set is EVEN, then the median is the AVERAGE of the middle two values.

12. Odd Number Observations
iii) The Median
Example 2a
Find the median for the following observations.
0, 7, 12, 5, 14, 8, 0, 9, 22
Step-1: Arrange the data in increasing/ decreasing order
0, 0, 5, 7, 8 9, 12, 14, 22
Odd Number Observations
Median= 8
Step-2: Count the total number of observation in the data (9) …

13. Even number Observations
iii) The Median
Example 2b
Find the median for the following observations.
0, 7, 12, 5, 33, 14, 8, 0, 9, 22
0, 0, 5, 7, 8, 9, 12, 14, 22, 33
Step-1: Arrange the data in increasing/ decreasing order
Step-2: Count the total number of observation in the data (10)…
Even number Observations
Median=(8+9)/2=8.5

14. ii) The Median
Note:
The median (8 in example 2a)of an odd set of data is a member of the data values.
The median (8.5 in example 2b) of an even data set is not necessarily a member of the set of values.
Unlike the mean, the median is not affected by the value of an observation in the data set.

15. III) The Center: Mode The mode is the most frequent value.
The Mode is the value that occurs most frequently in the data. It is the value with the highest frequency
In any data set there is only one value for the mean or the median. However, a data set may have more than one value for the mode.

16. III) The Center: Mode Two modal classes One modal class
Histogram of Income distribution
One modal class
Two modal classes

17. III) The Center: Mode
Example 3: What is the mode for the following data?
0, 7, 12, 5, 33, 14, 8, 0, 9, 22
Solution
All observation except “0” occur once. There are two “0” values. 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 median = 8.5).

18. Comparing Measures of Central Tendency: Mean, Median, Mode
If mean = median = mode, the shape of the distribution is symmetric.

19. Comparing Measures of Central Tendency: Mean, Median, Mode
If mode < median < mean, the shape of the distribution trails to the right, is positively skewed.
If mode > median > mean, the shape of the distribution trails to the left, is negatively skewed.
A positively skewed distribution
(“skewed to the right”)
A negatively skewed distribution
(“skewed to the left”)
Mode
Mean
Mean
Mode
Median
Median

20. Percentiles A percentile provides information about the relative
location and spread of the data between the smallest
to the largest value.
Is a measure of the relative location, but not necessarily
that of the central location
Percentile tells us the proportion of observations
that lie below or above a certain value in the data.
Example: Admission test scores for colleges and universities
are frequently reported in terms of percentiles.

21. Percentiles Definition:
The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.

22. Computing Percentiles
Arrange the data in ascending order.
Compute the ith position of the pth percentile.
If i is not an integer, round up. The p th percentile
is the value in the i th position.
If i is an integer, the p th percentile is the average
of the values in positions i and i +1.

23. Rounding 7.5, we note that the 8th data value is
Compute the 75th percentile of the following data
i = (p/100)n = (75/100)X10 =7.5
Rounding 7.5, we note that the 8th data value is
The 75th Percentile = 435

24. Averaging the 5th and 6th data value, we get
Compute the 50th percentile of the following data
i = (p/100)n = (50/100)X10 =5
Averaging the 5th and 6th data value, we get
5th Percentile = ( )/2 = 435

25. Quartiles Quartiles are specific percentiles.
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = the Median
Third Quartile = 75th Percentile

26. Quartiles
Divide a data set into four equal parts

27. 3.2 Measures of Variability

28. 3.2 Measures of Variability
Measures of central location fail to tell the whole story about the distribution.
A question of interest that remains unanswered even after obtaining measures of central location is how spread out are the observations around the central (say, mean) value?
Variability is Important in business decisions.
For example, in choosing between two suppliers A and B, we might consider not only the average delivery time for each, but also the variability in delivery time for each.

29. Measures of Variability
Range
Inter-Quartile Range
Variance
Standard Deviation
Coefficient of Variation

30. i) The Range Range = largest value – smallest value
The range in a set of observations is the difference between the largest and smallest observations.
The range is the distance between the smallest and the largest data value in the set.
Range = largest value – smallest value
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.
It is also very sensitive to the smallest and largest data values

31. ii) Inter Quartile Range
This is a measure of the spread of the middle 50% of the observations
Large value indicates a large spread of the observations
Is not sensitive to extreme data values
Inter quartile range = Q3 – Q1

32. iii) The Variance
Is the average of the squared differences between each data value and the measure of central location (mean)
Is calculated differently when we use population and when we use a sample
The variance is a measure of variability that utilizes
all the data.

33. iv) The Variance
Variance of a Population
Variance of a sample

34. iii) The Variance Why square the difference?
Sum of deviation from the mean is zero
Why divide by n-1 instead of n ?
Better approximation of the population variance

35. Example- Computing the Variance-Based on a Sample data
Variance of a sample
Find the variance of the following sample observations

36. Computing Variance of a sample
Step-1: Find the mean
9-10= -1
11-10= +1
Step-2: Compute deviations from the mean
8-10= -2
12-10= +2
Step-3: Square the deviations, add them together, and divide the sum of the squared deviations by n-1

37. iv) Standard Deviation
The standard deviation of a set of observations is the square root of the variance .

38. Why Standard Deviation?
The standard deviation
Is often reported in the actual unit of measure in which the data is recorded.
Thus it can be used to compare the variability of several distributions that are measured in the same units,
It can also be used to make a statement about the general shape of a distribution (Kurtosis).

39. Computing the standard deviation
Step-1: Find the mean
9-10= -1
11-10= +1
Step-2: Compute deviations from the mean
8-10= -2
12-10= +2
Step-3: Square the deviations, add them together, and divide the sum of the squared deviations by n-1
step-4: Take the square root of the variance

40. V) Coefficient of Variation
The coefficient of variation is a measure of how large the
standard deviation is relative to the mean.
The coefficient of variation is computed as follows:
CV=
for a
sample
for a
population

41. Why Coefficient of Variation?
Example: Is a standard deviation of 10 large?
A standard deviation of 10 may be perceived large when the mean value is 100, but it is only moderately large if the mean value is 500
Coefficient of Variation can be used to compare variability in data sets
that are measured in different units.

42. Coefficient of Variation
Variance, Standard Deviation,
and Coefficient of Variation
Variance
Standard Deviation
the standard
deviation is
about 11%
of the mean
Coefficient of Variation

43. Compute the Mean, Median, Mode, Range, Variance, Standard Deviation and Coefficient of Variation for income (in $1000) data from the following cities
City
Income
Akron, OH
74.1
Atlanta, GA
82.4
Birmingham, AL
71.2
Cleveland, OH
62.3
Columbia, SC
79.9
Danbury, CT
66.8
Denver, CO
132.3
Detroit, MI
83.4
Lancaster, PA
100.0
Madison, WI
77.0
Minneapolis, MN
67.8

44. Compute every single measure of central location and Variability you have learned in this chapter for the following sample rent data on 70 efficiency apartments