Statistics Review - Mathematics and Statistics for the Environment - page 115

190.0 290.0 390.0 490.0 590.0 690.0 790.0

68%
95%
99.7%

FIGURE 7.2 Sample Scholastic Aptitude Test (SAT) math percentages given by the empirical rule.

Solution:

1. From Figure 7.2, about 68% of seniors scored between 390 and 590 on this SAT test.

2. Because about 99.7% of the scores are between 190 and 790, a score of 795 is excellent.
This is one of the highest scores on this test.

3. Because about 68% of the scores are between 390 and 590, this leaves 32% of the scores
outside the interval. Because a bell-shaped curve is symmetric, one-half of these scores, or
16%, are on each end of the distribution.

7.5 STANDARD DEVIATION
The standard deviation, s or σ (sigma), is often used as an indicator of precision. The standard devia-
tion is a measure of the variation (spread) in the set of observations; that is, it gives us some idea as to
whether most of the individuals in a population are close to the mean or spread out. In order to a gain
better understanding of the benefits derived from using statistical methods in safety engineering, it is
appropriate to consider some of the basic theory of statistics. In any set of data, the true value (mean)
will lie in the middle of all of the measurements taken. This is true, providing the sample size is large
and only random error is present in the analysis. In addition, the measurements will show a normal
distribution as shown in Figure 7.1. In Figure 7.1, 68.26% of the results fall between M + s and M - s ,
95.46% of the results lie between M + 2 s and M - 2 s , and 99.74% of the results lie between M + 3 s and
M - 3 s . Therefore, if they are precise, then 68.26% of all the measurements should fall between the
true value estimated by the mean, plus the standard deviation and the true value minus the standard
deviation. The following equation is used to calculate the sample standard deviation:

( )

2

∑

XX
n

−

s =

− 1

where
s = Standard deviation.
∑ = Means to sum the values from X to X n .
X = Measurements from X to X n .
X = Mean.
n = Number of samples.