quantitative (numerical) or categorical (labels)? Are there any outliers? Often a graphical dis
play reveals many patterns. There might be clusters or outliers. The data could be symmetric
or skewed; unimodal or multimodal.
If the data are paired, then a scatterplot may reveal a
relationship: linear or otherwise. Also, it is straightforward to compute the summary statistics
(as appropriate): mean, median, mode, standard deviation, interquartile range, maximum, min
imum, range, correlation and/or covariance. Most of this is easily accomplished with a computer
and statistical software such as SAS.
(2.4)
A statistic is a number computed from data, while a parameter is a number associated
with a population. Most often, a parameter is unknown, and a statistic is used to estimate a
particular parameter.
(2.5)
By definition,
s
2
=
1
n

1
n
X
i
=1
(
y
i

y
)
2
.
Squaring, we see that (
y
i

y
)
2
=
y
2
i

2
y
i
y
+
y
2
, and distributing the sum gives
s
2
=
1
n

1
n
X
i
=1
(
y
i

y
)
2
=
1
n

1
"
n
X
i
=1
y
2
i

2
y
n
X
i
=1
y
i
+
n
X
i
=1
y
2
.
#
Since
y
is constant, and since
∑
y
i
=
n
y
, we see that
1
n

1
"
n
X
i
=1
y
2
i

2
y
n
X
i
=1
y
i
+
n
X
i
=1
y
2
#
=
1
n

1
"
n
X
i
=1
y
2
i

2
y
·
n
y
+
n
y
2
#
=
1
n

1
"
n
X
i
=1
y
2
i

n
y
2
#
as required.
(2.13)
Very briefly:
(a)
The average is
M/R
, and each average is exact (having factored the billions and thousands
into the computation). Ordered by state we have:
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 Fall '13
 FAROUKNATHOO
 Statistics, Normal Distribution, Standard Deviation, Variance