CP Prob & Stats
Section 2.6 – Interpreting the Standard Deviation
In this section, we learn how the standard deviation provides a measure of variability for a single
sample. To understand how the standard deviation provides a measure of variability of a data set, we
need to consider a specific data set and answer the following questions:
1. How many measurements are within 1 standard deviation of the mean?
2. How many measurements are within 2 standard deviations of the mean?
As our data set, let’s use the average SAT scores for each of the 50 states and the District of
Columbia in 2000.
State
Alabama
Alaska
Arizona
Arkansas
California
Colorado
Connecticut
Delaware
D.C.
Florida
Georgia
Hawaii
Idaho
Illinois
Indiana
Iowa
Kansas
SAT Score
1114
1034
1044
1117
1015
1071
1017
998
980
998
974
1007
1081
1154
999
1189
1154
State
SAT Score
Kentucky
1098
Louisiana
1120
Maine
1004
Maryland
1016
Massachusetts 1024
Michigan
1126
Minnesota
1175
Mississippi
1111
Missouri
1149
Montana
1089
Nebraska
1131
Nevada
1027
N. Hampshire 1039
N. Jersey
1011
N. Mexico
1092
New York
1000
N. Carolina
988
State
SAT Score
N. Dakota
1197
Ohio
1072
Oklahoma
1123
Oregon
1054
Pennsylvania
995
Rhode Island 1005
S. Carolina
966
S. Dakota
1175
Tennessee
1116
Texas
993
Utah
1139
Vermont
1021
Virginia
1009
Washington
1054
W. Virginia 1037
Wisconsin
1181
Wyoming
1090
The mean of this data is 1066. The standard deviation of the data is 66. We look at the intervals
X  s to X  s and X  2s to X  2s. These are the intervals 1 standard deviation from the mean and
2 standard deviations from the mean, respectively. With our data set, ( X  s, X  s) = (1000, 1132)
and ( X  2s, X  2s) = (934, 1198). Within 1 standard deviation of the mean are 34 measurements or
66.7 % of the data. All 51 measurements, or 100 % of the data, fall within 2 standard deviations of
the mean. These observations identify criteria for interpreting a standard deviation that apply to any
set of data, whether a population or a sample.
There are 2 rules for interpreting the standard deviation, Chebyshev’s Rule and the Empirical Rule.
Chebyshev’s Rule:
This applies to any set of data, regardless of the shape of the frequency distribution of the data.
a. No useful information is provided on the fraction of measurements that fall within 1 standard
deviation of the mean.
3
of the measurements will fall within 2 standard deviations of the mean;
4
( X  2s, X  2s) for samples and (   2 ,   2 ) for populations
b. At least
c. At least
8
of the measurements will fall within 3 standard deviations of the mean;
9
( X  3s, X  3s) for samples and (   3 ,   3 ) for populations
d. Generally, for any number k greater than 1, at least (1  1
k2
) of the measurements will fall
within k standard deviations of the mean;
( X  ks, X  ks) for samples and (   k ,   k ) for populations
Note: This rule gives the smallest percentages that are mathematically possible. In reality, the true
percentages can be much higher than those stated.
Empirical Rule:
This applies only to data sets with normal distributions, that is, the frequency distributions are
symmetric (i.e. mean equals median), unimodal, and bell-shaped, or mound-shaped.
a. Approximately 68 % of the measurements fall within 1 standard deviation of the mean;
( X  s, X  s) for samples and (    ,    ) for populations
b. Approximately 95 % of the measurements fall within 2 standard deviations of the mean;
( X  2s, X  2s) for samples and (   2 ,   2 ) for populations
c. Approximately 99.7 % (essentially all) of the measurements fall within 3 standard deviations
of the mean;
( X  3s, X  3s) for samples and (   3 ,   3 ) for populations
Note: These percentages are only approximations. The real percentages could be higher or lower
depending on the data set.
We can use Chebyshev’s Rule and the Empirical Rule as a “check” when calculating the standard
deviation, since sometimes we forget to take the square root of the variance. From the two Rules, we
know that most of the data is within 2 standard deviations of the mean, and almost all of the data is
within 3 standard deviations of the mean. Therefore, it’s expected that the range of measurements is
between 4 and 6 standard deviations in length. Sometimes, can use s ≈ range/4 to obtain a crude, and
usually conservatively large, approximation for s. (No substitute for actually calculating s.) Larger
data sets typically have extreme values, in which case the range of the data may exceed 6 standard
deviations.