121
Chapter 3
Numerically Describing Data from One Variable
3.1
Measures of Central Tendency
1.
A statistic is resistant if it is not sensitive to extreme data values.
The median is resistant
because it is a positional measure of central tendency and increasing the largest value or
decreasing the smallest value does not affect the position of the center.
The mean is not
resistant because it is a function of the sum of the data values.
Changing the magnitude of
one value changes the sum of the values, and thus affects the mean.
The mode is a
resistant measure of center.
2.
The men and the median are approximately equal when the data are symmetric.
If the
mean is significantly greater than the median, the data are skewed right.
If the mean is
significantly less than the median, the data are skewed left.
3.
Since the distribution of household incomes in the United States is skewed to the right, the
mean is greater than the median.
Thus, the mean household income is $55,263 and the
median is $41,349.
4.
HUD uses the median because the data are skewed.
Explanations will vary.
One
possibility is that the price of homes has a distribution that is skewed to the right, so the
median is more representative of the typical price of a home.
5.
The mean will be larger because it will be influenced by the extreme data values that are to
the right end (or high end) of the distribution.
6.
10,000 1
5000.5
2
+
=
.
The median is between the 5000
th
and the 5001
st
ordered values.
7.
The mode is used with qualitative data because the computations involved with the mean
and median make no sense for qualitative data.
8.
parameter; statistic
9.
False.
A data set may have multiple modes, or it may have no mode at all.
10.
False.
The formula
1
2
n
+
gives the
position
of the median, not the
value
of the median.
11.
20 13 4 8 10
55
11
55
x
++++
==
=
12.
83 65 91 87 84
420
84
x
=
13.
3 6 10 12 14
45
9
μ
=
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Numerically Summarizing Data
122
14.
11
92
51
21
62
81
36 1
3
5
15
99
μ
++++++++
==
=
15.
142
2.4
59
≈
.
The mean price per ad slot is approximately $2.4 million.
16.
Let
x
represent the missing value.
Since there are 6 data values in the list, the median 26.5
is between the 3
rd
and 4
th
ordered values which are 21 and
x
, respectively.
Thus,
21
26.5
2
21
53
32
x
x
x
+
=
+=
=
The missing value is 32.
17.
420 462 409 236
1527
Mean
$381.75
44
+++
=
Data in order:
236,
409,
420,
462
409 420
829
Median
$414.50
22
+
=
No data value occurs more than once so there is no mode.
18.
35.34 42.09 39.43 38.93 43.39 49.26
248.44
Mean
$41.41
66
+++++
≈
Data in order:
35.34,
38.93,
39.43,
42.09,
43.39,
49.26
39.43 42.09
81.52
Median
$40.76
+
=
No data value occurs more than once so there is no mode.
19.
3960 4090 3200 3100 2940 3830 4090 4040 3780
33,030
Mean
3670 psi
=
Data in order:
2940,
3100,
3200,
3780,
3830,
3960,
4040,
4090,
4090
Median = the 5
th
ordered data value = 3830 psi
Mode = 4090 psi (because it is the only data value to occur twice)
20.
282 270 260 266 257 260 267
1862
Mean
266 minutes
77
++++++
=
Data in order:
257,
260,
260,
266,
267,
270,
282
Median = the 4
th
data value with the data in order = 266 minutes
Mode = 260 minutes (because it is the only data value to occur twice)
21.
(a)
The histogram is skewed to the right, suggesting that the mean is greater than the
median.
That is,
x
M
>
.
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 Spring '11
 ahmad
 Standard Deviation, The American, data value

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