Basic HTML Version
Q
uantitative
R
easoning &
P
roblem
S
olving
306
© 2014 Pacific Crest
M
athematical Language
Terms and notation
x
i
— represents the position of the
i
th element in the list of variable
x
1
n
i
i
x
=
∑
— represents the sum of the
x
i
where
i
takes on the values from 1 to
n
Chebyshev’s Theorem
— more than 75% of the data will lie within two standard deviations of the
mean
deviations from the mean
— the difference between a data element,
x
i
, of a variable and its mean,
(e.g.,
x
i
–
x
)
nominal
— categorical values that can be counted but not ordered or measured. Common examples
include sex (male/female) and eye color. These values can be assigned a number (brown = 1, blue =
2, etc.), but the numbers are only labels and cannot be subjected to statistical examination.
standard deviation
— a measure of the amount of variation from the average. Represented with the
Greek letter sigma (σ), it is the square root of the variance:
2
σ
σ
=
variance
— in probability and statistics, variance measures how far a set of
numbers is spread out. Represented as σ
2
, it is computed using the formula:
(
)
2
1
1
n
i
i
x x
n
σ
=
−
=
−
∑
I
nformation
What you need to know
R
eadings
R
esources
M
ethodology
C
onstructing
the
V
ariance and
S
tandard
D
eviation of a
S
ample
S
et of
D
ata
Scenario:
Find the variance and standard deviation for a collection of test scores:
68 72 75 78 79 82 85 88 91 92
Step
Explanation
1.
Describe the data
Is the set a random collection of data from a larger sample or is a
convenience sample? What characteristic is being measured? What is the
relevant unit? Are there any specific conditions? If the data set is nominal, it
is not reasonable to construct the variance or standard deviation.
WATCH
IT WORK!
The set is a random collection of exam scores for ten students enrolled in a college
mathematics class. The maximum possible score was 100 points. The data set is
not nominal so it is reasonable to construct the standard deviation and the variance.
2.
Visualize
The whiskers of the boxplot extend from the minimum score to the
maximum score. The box extends from the 25
th
percentile to the 75
th
percentile. The middle line in the box represents the median. In order to
visualize the data set, create a boxplot.