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Chapter 4 Evaluating Analytical Data 57

The relative standard deviation and percent relative standard deviation are
0 051
.
.
s r = =0 016
.
3 117
s r (%) = 0.016 ´100% = 1.6%
It is much easier to determine the standard deviation using a scientific
calculator with built-in statistical functions.*

Variance Another common measure of spread is the square of the standard devia-
tion, or the variance. The standard deviation, rather than the variance, is usually re- variance
2
ported because the units for standard deviation are the same as that for the mean The square of the standard deviation (s ).
value.

EXAMPLE .4
4
What is the variance for the data in Table 4.1?
SOLUTION

The variance is just the square of the absolute standard deviation. Using the
standard deviation found in Example 4.3 gives the variance as
2
2
Variance = s = (0.051) = 0.0026

4 B Characterizing Experimental Errors

Realizing that our data for the mass of a penny can be characterized by a measure of
central tendency and a measure of spread suggests two questions. First, does our
measure of central tendency agree with the true, or expected value? Second, why are
our data scattered around the central value? Errors associated with central tendency
reflect the accuracy of the analysis, but the precision of the analysis is determined by
those errors associated with the spread.

4 B.1 Accuracy
Accuracy is a measure of how close a measure of central tendency is to the true, or
†
expected value, m. Accuracy is usually expressed as either an absolute error
–
E = X – m 4.2
or a percent relative error, E r .

X -m
E r = ´100 4.3
m

*Many scientific calculators include two keys for calculating the standard deviation, only one of which corresponds to
equation 4.3. Your calculator’s manual will help you determine the appropriate key to use.
†The standard convention for representing experimental parameters is to use a Roman letter for a value calculated from
experimental data, and a Greek letter for the corresponding true value. For example, the experimentally determined
–
mean is X, and its underlying true value is m. Likewise, the standard deviation by experiment is given the symbol s, and
its underlying true value is identified as s.