This document provides an overview of descriptive statistics concepts including measures of central tendency (mean, median, mode), measures of variability (range, standard deviation, variance), and how to calculate these measures from both ungrouped and grouped data. It defines key terms, explains how to compute various statistics, and includes example problems and solutions. The learning objectives are to understand and be able to compute different descriptive statistics and apply concepts like the empirical rule and Chebyshev's theorem.

1. Copyright 2010 John Wiley & Sons, Inc. 1
Copyright 2010 John Wiley & Sons, Inc.
Business Statistics, 6th ed.
by Ken Black
Chapter 3
Describing Data
Through Statistics

2. Copyright 2010 John Wiley & Sons, Inc. 2
Learning Objectives
Distinguish between measures of central tendency,
measures of variability, measures of shape, and
measures of association.
Understand the meanings of mean, median, mode,
quartile, percentile, and range.
Compute mean, median, mode, percentile, quartile,
range, variance, standard deviation, and mean
absolute deviation on ungrouped data.
Differentiate between sample and population
variance and standard deviation.

3. Copyright 2010 John Wiley & Sons, Inc. 3
Learning Objectives -- Continued
Understand the meaning of standard deviation as it is
applied by using the empirical rule and Chebyshev’s
theorem.
Compute the mean, median, standard deviation, and
variance on grouped data.
Understand box and whisker plots, skewness, and
kurtosis.
Compute a coefficient of correlation and interpret it.

4. Copyright 2010 John Wiley & Sons, Inc. 4
Measures of Central Tendency:
Ungrouped Data
Measures of central tendency yield information
about “particular places or locations in a group of
numbers.”
Common Measures of Location
Mode
Median
Mean
Percentiles
Quartiles

5. Copyright 2010 John Wiley & Sons, Inc. 5
Mode - the most frequently occurring value in a data
set
Applicable to all levels of data measurement (nominal,
ordinal, interval, and ratio)
Can be used to determine what categories occur most
frequently
Bimodal – In a tie for the most frequently occurring
value, two modes are listed
Multimodal -- Data sets that contain more than two
modes
Mode

6. Copyright 2010 John Wiley & Sons, Inc. 6
Median
Media - middle value in an ordered array of numbers.
For an array with an odd number of terms, the median is
the middle number
For an array with an even number of terms the median is
the average of the middle two numbers

7. Copyright 2010 John Wiley & Sons, Inc. 7
Arithmetic Mean
Mean is the average of a group of numbers
Applicable for interval and ratio data
Not applicable for nominal or ordinal data
Affected by each value in the data set, including
extreme values
Computed by summing all values in the data set and
dividing the sum by the number of values in the data
set

8. Copyright 2010 John Wiley & Sons, Inc. 8
The number of U.S. cars in service by top car rental companies
in a recent year according to Auto Rental News follows.
Company Number of Cars in Service
Enterprise 643,000; Hertz 327,000; National/Alamo 233,000;
Avis 204,000; Dollar/Thrifty 167,000; Budget 144,000;
Advantage 20,000; U-Save 12,000; Payless 10,000; ACE 9,000;
Fox 9,000; Rent-A-Wreck 7,000; Triangle 6,000
Compute the mode, the median, and the mean.
Demonstration Problem 3.1

9. Copyright 2010 John Wiley & Sons, Inc. 9
Demonstration Problem 3.1
Solution
Mode: 9,000
Median: With 13 different companies in this group, N = 13.
The median is located at the (13 +1)/2 = 7th position.
Because the data are already ordered, the 7th term is
20,000, which is the median.
Mean: The total number of cars in service is 1,791,000 = ∑x
μ = ∑x/N = (1,791,000/13) = 137,769.23

10. Copyright 2010 John Wiley & Sons, Inc. 10
 = =
+ + + +
=
+ + + +
=
=
X
N N
X X X XN1 2 3
24 13 19 26 11
5
93
5
18 6
...
.
Population Mean

11. Copyright 2010 John Wiley & Sons, Inc. 11
X
X
n n
X X X Xn
= =
+ + + +
=
+ + + + +
=
=
 1 2 3
57 86 42 38 90 66
6
379
6
63 167
...
.
Sample Mean

12. Copyright 2010 John Wiley & Sons, Inc. 12
Percentiles
Percentile - measures of central tendency that divide
a group of data into 100 parts
At least n% of the data lie below the nth percentile,
and at most (100 - n)% of the data lie above the nth
percentile
Example: 90th percentile indicates that at least 90%
of the data lie below it, and at most 10% of the data
lie above it

13. Copyright 2010 John Wiley & Sons, Inc. 13
Quartiles
Quartile - measures of central tendency that divide a
group of data into four subgroups
Q1: 25% of the data set is below the first quartile
Q2: 50% of the data set is below the second quartile
Q3: 75% of the data set is below the third quartile
25% 25% 25% 25%
Q3Q2Q1

14. Copyright 2010 John Wiley & Sons, Inc. 14
Measures of Variability - tools that describe the
spread or the dispersion of a set of data.
Provides more meaningful data when used with measures
of central tendency
Measures of Variability:
Ungrouped Data

15. Copyright 2010 John Wiley & Sons, Inc. 15
Common Measures of Variability
Range
Inter-quartile Range
Mean Absolute Deviation
Variance
Standard Deviation
Z scores
Coefficient of Variation
Measures of Variability:
Ungrouped Data

16. Copyright 2010 John Wiley & Sons, Inc. 16
Range
The difference between the largest and the smallest
values in a set of data
Advantage – easy to compute
Disadvantage – is affected by extreme values

17. Copyright 2010 John Wiley & Sons, Inc. 17
Interquartile Range
Interquartile Range Q Q= −3 1
Interquartile Range - range of values between the
first and third quartiles
Range of the “middle half”; middle 50%
Useful when researchers are interested in the middle 50%,
and not the extremes
Interquartile Range – used in the construction of box
and whisker plots

18. Copyright 2010 John Wiley & Sons, Inc. 18
Mean Absolute Deviation, Variance,
and Standard Deviation
These data are not meaningful unless the data are at
least interval level data
One way for researchers to look at the spread of data
is to subtract the mean from each data set
Subtracting the mean from each data value gives the
deviation from the mean (X - µ)

19. Copyright 2010 John Wiley & Sons, Inc. 19
Mean Absolute Deviation, Variance,
and Standard Deviation
An examination of deviation from the mean can
reveal information about the variability of the data
Deviations are used mostly as a tool to compute other
measures of variability
The Sum of Deviation from the arithmetic mean is
always zero
Sum (X - µ) = 0

20. Copyright 2010 John Wiley & Sons, Inc. 20
Mean Absolute Deviation, Variance,
and Standard Deviation
An obvious way to force the sum of deviations to
have a non zero total is to take the absolute value of
each deviation around the mean
Allows one to solve for the Mean Absolute Deviation

21. Copyright 2010 John Wiley & Sons, Inc. 21
Mean Absolute Deviation (MAD)
5
9
16
17
18
-8
-4
+3
+4
+5
0
+8
+4
+3
+4
+5
24
X X −  X − 
M A D
X
N
. . .
.
=
−
=
=
 
24
5
4 8
Mean Absolute Deviation - average of the absolute
deviations from the mean

22. Copyright 2010 John Wiley & Sons, Inc. 22
Population Variance
Variance - average of the squared deviations from
the arithmetic mean
Population variance is denoted by σ2
Sum of Squared Deviations (SSD) about the mean of
a set of values (called Sum of Squares of X) is used
throughout the book

23. Copyright 2010 John Wiley & Sons, Inc. 23
Population Variance
5
9
16
17
18
-8
-4
+3
+4
+5
0
64
16
9
16
25
130
X X −  ( )
2
X − 
( )2
2
130
5
26 0


=
=
=
− X
N
.
Variance = average of the squared deviations from
the arithmetic mean
Population variance is denoted by σ2

24. Copyright 2010 John Wiley & Sons, Inc. 24
Sample Variance
2,398
1,844
1,539
1,311
7,092
625
71
-234
-462
0
390,625
5,041
54,756
213,444
663,866
X X X− ( )
2
X X−
( )
67.288,221
3
866,663
1
2
2
=
=
−
=
 −
n
XX
S
Sample Variance - average of the squared deviations
from the arithmetic mean
Sample Variance – denoted by S2

25. Copyright 2010 John Wiley & Sons, Inc. 25
Sample Standard Deviation
( )2
2
2
1
663 866
3
221 288 67
221 288 67
470 41
S
X X
S
n
S
=
−
=
=
=
=
=
−
,
, .
, .
.
Sample Std Dev is the square
root of the sample variance
2,398
1,844
1,539
1,311
7,092
625
71
-234
-462
0
390,625
5,041
54,756
213,444
663,866
X X X− ( )
2
X X−

26. Copyright 2010 John Wiley & Sons, Inc. 26
Empirical Rule
Empirical Rule – used to state the approximate
percentage of values that lie within a given number
of standard deviations from the set of data if the data
are normally distributed
Empirical rule is used only for three numbers of
standard deviation: 1σ, 2σ, and 3σ
1σ = 68% of data;
2σ = 95% of data; and
3σ = 99% of data

27. Copyright 2010 John Wiley & Sons, Inc. 27
Chebyshev’s Theorem
Empirical rule – applies when data are approximately
normally distributed
Chebyshev’s Theorem – applies to all distributions,
and can be used whenever the data distribution
shape is unknown or non-normal

28. Copyright 2010 John Wiley & Sons, Inc. 28
Chebyshev’s Theorem
Chebyshev’s Theorem - states that at least (1 – 1/k2)
values fall within +k standard deviations of the mean
regardless of the shape of the distribution
Example: At least 75% of all values are within +2σ of the
mean regardless of the shape of a distribution
when k = 2, then (1 – 1/k2) = 1- 22 = .75

29. Copyright 2010 John Wiley & Sons, Inc. 29
The effectiveness of district attorneys can be
measured by several variables, including the number
of convictions per month, the number of cases
handled per month, and the total number of years of
conviction per month. A researcher uses a sample of
five district attorneys in a city and determines the total
number of years of conviction that each attorney won
against defendants during the past month, as reported
in the first column in the following tabulations.
Compute the mean absolute deviation, the variance,
and the standard deviation for these figures.
Demonstration Problem 3.6

30. Copyright 2010 John Wiley & Sons, Inc. 30
Demonstration Problem 3.6
X
Solution
The researcher computes the mean absolute
deviation, the variance, and the standard deviation
for these data in the following manner.
x |x- | (x - )2
55 41 1,681
100 4 16
125 29 841
140 44 1,936
60 36 1,296
x = 480
X

31. Copyright 2010 John Wiley & Sons, Inc. 31
Demonstration Problem 3.6
The computational formulas are used to solve for
s2 and s and compares the results.
S2 = (5,770/4) = 1,442.5 and s = Square root of
variance = 37.98
MAD = 154/5 = 30.8

32. Copyright 2010 John Wiley & Sons, Inc. 32
Z Scores
Z score – represents the number of Std Dev a value
(x) is above or below the mean of a set of numbers
when the data are normally distributed
Z score allows translation of a value’s raw distance
from the mean into units of std dev.
Z = (x-µ)/σ

33. Copyright 2010 John Wiley & Sons, Inc. 33
Z Scores
If Z is negative, the raw value (x) is below the mean
If Z is positive, the raw value (x) is above the mean
Between
Z = + 1, are app. 68% of the values
Z = + 2, are app. 95% of the values
Z = + 3, are app. 99% of the values

34. Copyright 2010 John Wiley & Sons, Inc. 34
( )C V. .=


100
Coefficient of Variation (CV) - ratio of the standard
deviation to the mean, expressed as a percentage
useful when comparing Std Dev computed from data
with different means
Measurement of relative dispersion
Coefficient of Variation

35. Copyright 2010 John Wiley & Sons, Inc. 35
( )
( )
1
29
4 6
100
4 6
29
100
1586
1
1
1
1




=
=
=
=
=
.
.
.
. .CV ( )
( )
2
84
10
100
10
84
100
1190
2
2
2
2




=
=
=
=
=
CV. .
.
Coefficient of Variation

36. Copyright 2010 John Wiley & Sons, Inc. 36
Measures of Central Tendency
Mean
Median
Mode
Measures of Variability
Variance
Standard Deviation
Measures of Central Tendency
and Variability: Grouped Data

37. Copyright 2010 John Wiley & Sons, Inc. 37
Measures of Central Tendency
and Variability: Grouped Data
Mean – The midpoint of each class interval is used
to represent all the values in a class interval
Midpoint is weighted by the frequency of values in the
class interval
Mean is computed by summing the products of class
midpoint, and the class frequency for each class and
dividing that sum by the total number of frequencies

38. Copyright 2010 John Wiley & Sons, Inc. 38
Measures of Central Tendency
and Variability: Grouped Data
Median – The middle value in an ordered array of
numbers
Mode – the mode for grouped data is the class
midpoint of the modal class
The modal class is class interval with the greatest frequency

39. Copyright 2010 John Wiley & Sons, Inc. 39
Class Interval Frequency Class Midpoint fM
20-under 30 6 25 150
30-under 40 18 35 630
40-under 50 11 45 495
50-under 60 11 55 605
60-under 70 3 65 195
70-under 80 1 75 75
50 2150
 = = =


fM
f
2150
50
43 0.
Calculation of Grouped Mean

40. Copyright 2010 John Wiley & Sons, Inc. 40
Cumulative
Class Interval Frequency Frequency
20-under 30 6 6
30-under 40 18 24
40-under 50 11 35
50-under 60 11 46
60-under 70 3 49
70-under 80 1 50
N = 50
( )
( )
Md L
N
cf
f
W
p
med
= +
−
= +
−
=
2
40
50
2
24
11
10
40 909.
Median of Grouped Data - Example

41. Copyright 2010 John Wiley & Sons, Inc. 41
Mode of Grouped Data
Class Interval Frequency
20-under 30 6
30-under 40 18
40-under 50 11
50-under 60 11
60-under 70 3
70-under 80 1
Mode =
+
=
30 40
2
35
Midpoint of the modal class
Modal class has the greatest frequency

42. Copyright 2010 John Wiley & Sons, Inc. 42
( )2
2
2



=
=
− f
N
M
Population
( )2
2
2
1S
M X
S
f
n
S
=
−
=
−
Sample
Variance and Standard Deviation
of Grouped Data

43. Copyright 2010 John Wiley & Sons, Inc. 43
Population Variance and Standard
Deviation of Grouped Data
1944
1152
44
1584
1452
1024
7200
20-under 30
30-under 40
40-under 50
50-under 60
60-under 70
70-under 80
Class Interval
6
18
11
11
3
1
50
f
25
35
45
55
65
75
M
150
630
495
605
195
75
2150
fM
-18
-8
2
12
22
32
M− ( )f M
2
− 
324
64
4
144
484
1024
( )−M
2
( )2
2
7200
50
144

= = =
− f
N
M  = = =
2
144 12

44. Copyright 2010 John Wiley & Sons, Inc. 44
Measures of Shape
Symmetrical – the right half is a mirror image of the
left half
Skewness – shows that the distribution lacks
symmetry; used to denote the data is sparse at one
end, and piled at the other end
Absence of symmetry
Extreme values in one side of a distribution

45. Copyright 2010 John Wiley & Sons, Inc. 45
Coefficient of Skewness
( )

 dM
Sk
−
=
3
Coefficient of Skewness (Sk) - compares the mean
and median in light of the magnitude to the standard
deviation; Md is the median; Sk is coefficient of
skewness; σ is the Std Dev

46. Copyright 2010 John Wiley & Sons, Inc. 46
Coefficient of Skewness
Summary measure for skewness
If Sk < 0, the distribution is negatively skewed
(skewed to the left).
If Sk = 0, the distribution is symmetric (not skewed).
If Sk > 0, the distribution is positively skewed (skewed
to the right).
( )

 d
k
M
S
−
=
3