1
Numerical Descriptive Measures

2
Chapter Goals
After completing this chapter, you should be
able to:
•
Compute and interpret the
mean, median,
and
mode
for a set of data
•
Find the
range, variance,
and
standard deviation
and know what these values mean
•
Construct and interpret a
box and whiskers plot
•
Compute and explain the
coefficient of variation
•
Use numerical measures along with graphs,
charts, and tables to describe data

3
Chapter Topics
•
Measures of Center and Location
•
Measures of Variation
•
Measures of Distribution Shape,
Relative Location, and Detecting Outliers
•
Exploratory Data Analysis
•
Measures of Association Between Two Variables
•
The Weighted Mean and Grouped Data

4
Summary Measures

5
Notation Conventions
•
Population Parameters
are denoted with
a letter from the
Greek
alphabet:
µ
(mu) represents the
population mean
(sigma) represents the
population standard
deviation
•
Sample Statistics
are commonly denoted
with letters from the
Roman
alphabet:
(X-bar) represents the
sample mean
s
represents the
sample standard deviation
x

Measures of Center and
Location
•
Mean
•
Median
•
Mode
•
Percentiles
•
Quartiles
6

7
Measures of Center and Location

8
Mean (Arithmetic Average)
•
The
Mean
is the arithmetic average of data
values
Sample mean
Population mean

9
Mean (Arithmetic Average)
•
The most common measure of central tendency
•
Mean = sum of values divided by the number of
values
•
Affected by extreme values (outliers)

Seventy efficiency apartments
were randomly sampled in
a small college town.
The
monthly rent prices for
these apartments are listed
in ascending order on the next slide.
Sample Mean
Example:
Apartment Rents

425
430
430
435
435
435
435
435
440
440
440
440
440
445
445
445
445
445
450
450
450
450
450
450
450
460
460
460
465
465
465
470
470
472
475
475
475
480
480
480
480
485
490
490
490
500
500
500
500
510
510
515
525
525
525
535
549
550
570
570
575
575
580
590
600
600
600
600
615
615
Sample Mean
i

12
Median
•
In an ordered array, the median is the “middle” number
(50% above, 50% below)
–
If n or N is odd, the median is the middle number
–
If n or N is even, the median is the average of the two middle
numbers
•
Not affected by extreme values

13
Median
•
To find the median, rank the n values in
order of magnitude
•
Find the value in the
(n+1)/2 position
–
If n is an even number, let the median be the
mean of the two middle-most observations.

Median
425
430
430
435
435
435
435
435
440
440
440
440
440
445
445
445
445
445
450
450
450
450
450
450
450
460
460
460
465
465
465
470
470
472
475
475
475
480
480
480
480
485
490
490
490
500
500
500
500
510
510
515
525
525
525
535
549
550
570
570
575
575
580
590
600
600
600
600
615
615
Averaging the 35th and 36th data values:
Median = (475 + 475)/2 =
475

15
Mode
•
A measure of central tendency
•
Value that occurs most often
•
Not affected by extreme values
•
Used for either
numerical
or categorical data
•
There may be no mode
•
There may be several modes

Mode
425
430
430
435
435
435
435
435
440
440
440
440
440
445
445
445
445
445
450
450
450
450
450
450
450
460
460
460
465
465
465
470
470
472
475
475
475
480
480
480
480
485
490
490
490
500
500
500
500
510
510
515
525
525
525
535
549
550
570
570
575
575
580
590
600
600
600
600
615
615
450 occurred most frequently (7 times)
Mode =
450

17
Review Example
•
Five houses on a hill by the beach

18
Summary Statistics
•
Mean:
($3,000,000/5)
=
$600,000
•
Median:
middle value of ranked data
=
$300,000
•
Mode:
most frequent value
=
$100,000

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- F. TAILOKA
- Standard Deviation