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3  1
Chapter 3
Descriptive Statistics: Numerical Methods
Learning Objectives
1.
Understand the purpose of measures of location.
2.
Be able to compute the mean, median, mode, quartiles, and various percentiles.
3.
Understand the purpose of measures of variability.
4.
Be able to compute the range, interquartile range, variance, standard deviation, and coefficient of
variation.
5.
Understand skewness as a measure of the shape of a data distribution. Learn how to recognize when a
data distribution is negatively skewed, roughly symmetric, and positively skewed.
6.
Understand how
z
scores are computed and how they are used as a measure of relative location of a
data value.
7.
Know how Chebyshev’s theorem and the empirical rule can be used to determine the percentage of
the data within a specified number of standard deviations from the mean.
8.
Learn how to construct a 5number summary and a box plot.
9.
Be able to compute and interpret covariance and correlation as measures of association between two
variables.
10.
Be able to compute a weighted mean.
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3  2
Solutions:
1.
x
x
n
i
==
=
Σ
75
5
15
10, 12, 16, 17, 20
Median = 16 (middle value)
2.
x
x
n
i
=
Σ
96
6
16
10,
12,
16,
17,
20,
21
Median =
16 17
16.5
2
+
=
3.
15, 20, 25, 25, 27, 28, 30, 34
20
(8)
1.6
100
i
2nd position = 20
25
(8)
2
100
i
20 25
22.5
2
+
=
65
(8)
5.2
100
i
6th position = 28
75
(8)
6
100
i
28 30
29
2
+
=
4.
Mean
===
Σ
x
n
i
657
11
59 727
.
Median = 57
6th item
Mode = 53
It appears 3 times
5.
a.
1106.4
36.88
30
i
x
x
n
Σ
=
b.
There are an even number of items.
Thus, the median is the average of the 15th and 16th items after
the data have been placed in rank order.
Median =
36.6 36.7
36.65
2
+
=
c.
Mode = 36.4
This value appears 4 times
d.
First Quartile
i
=
F
H
G
I
K
J
=
25
100
30
7 5
.
Descriptive Statistics: Numerical Methods
3  3
Rounding up, we see that
Q
1
is at the 8th position.
Q
1
= 36.2
e.
Third Quartile
i
=
F
H
G
I
K
J
=
75
100
30
225
.
Rounding up, we see that
Q
3
is at the 23rd position.
Q
3
= 37.9
6.
a.
871.74
36.32
24
i
x
x
n
Σ
==
=
Median is average of 10th and 11th values after arranging in ascending order.
39.00 39.95
Median
39.48
2
+
Data are multimodal
b.
491.14
20.46
24
i
x
x
n
Σ
=
19.75 19.95
Median
19.85
2
+
Data are bimodal: 19.95 (3 brokers), 29.95 (3 brokers)
c.
Comparing the measures of central location, we conclude that it costs more to trade 100 shares in a
broker assisted trade than 500 shares online.
d.
From the data we have here it is more related to whether the trade is brokerassisted or online. The
amount of the online transaction is 5 times as great but the cost of the transaction is less.
However, if the comparison was restricted to brokerassisted or online trades, we would probably
find that larger transactions cost more.
7.
a.
x
x
n
i
=
Σ
1380
30
46
b.
Yes, the mean here is 46 minutes.
The newspaper reported an average of 45 minutes.
c.
Median
=
+
=
45 52 9
2
48 95
.
.
d.
Q
1
= 7
(value of 8th item in ranked order)
Q
3
= 70.4
(value of 23rd item in ranked list)
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3  4
e. Find position
40
30 12;
100
i
==
40th percentile is average of values in 12th and 13th positions.
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This note was uploaded on 03/01/2012 for the course ECON 371 taught by Professor Staff during the Spring '08 term at UVA.
 Spring '08
 Staff

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