Greenwood press

126
SQUARE ROOTS
The optimum viewing angle of a painting,
α, is greatest when the distance a person
stands from the painting, d, is equal to the
square root of the product of the distances
from the edges of the painting to the eye-
level height
(d =
√
bt).

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STANDARD DEVIATION
The standard deviation is a number that indicates the variability in a set of
data. It is a measure of the dispersion of data in a sample or population. Standard
deviations are used in quality control in business and industry and in the compu-
tation of standard test scores (such as the SAT and ACT). The concept of stan-
dard deviation provides the basis for widely used statistical techniques.
The start of the computation of standard deviation is the deviation about the
mean, the difference of the actual score and mean score. If a college-placement
test has a national mean of 512, and a student has a score of 650, the deviation is
138. Deviations are negative when the score is below the mean.
Even though each deviation tells something about the spread of data, the sum
of deviations is always zero, which gives no overall information about the spread
of the data. To make sure negative deviations do not cancel with positive, statisti-
cians choose to square each deviation. Then they average the squared deviations to
produce a number that indicates how the data is spread out around the mean. The
average squared deviation is called the variance. The square root of the variance is
the standard deviation. There are two formulas for standard deviation. One form  
assumes that the data set is the entire population of cases: 
σ =
Σ(X−µ)
2
N
, where 
µ is the mean of the data, and N is the number of pieces of data. If the numbers
could be considered a sample from the population, then the mean and standard
deviations would represent estimates of the entire season’s scores. The standard
deviation has a different symbol in this case, and a slightly different formula:
s =
Σ(X−X)
2
n−1
, where 
X is the mean of the sample, and n is the sample size.
The standard deviation is used to compute standardized scores for the com-
parison of data from different sets and measures. A standardized score is computed 
as
z =
X−µ
σ
, or the deviation divided by the standard deviation. As a ratio, it has
no units. The standardized score can compare different measures of the same per-
son. Suppose a student had a score of 540 on the SAT-Math and 24 on the ACT
Mathematics. On which did he or she do better? The national mean for SAT-Math 
is 514, with a standard deviation of 113. So 
z
SATM
=
540−514
113
≈ 0.23. The 
national mean for ACT Mathematics is 20.7, with a standard deviation of 
5.0. So 
z
ACTM
=
24−20.7
5.0
≈ 0.66. Therefore she did relatively better on the ACT Mathe-
matics, because she had a greater standardized 
z score.
Z scores have been used to compare baseball players from different eras.
Does Ty Cobb’s batting average of .420 in 1922 represent better batting than
George Brett’s .390 in 1980? It has been argued that it is difficult for a player
today to hit over .400, because the general quality of players is much higher than
it was in the early days of professional baseball. If you use the standard scores
based on means and standard deviations of baseball players in their respective
eras, Cobb has a 
z score of about 4.15 and Brett, 4.07. The two stars were equally
outstanding in performance during their respective eras.
SAT and ACT scores are normally distributed, which means that a frequency