Page 73 - Modern Analytical Chemistry

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56 Modern Analytical Chemistry

measurements in the same direction, by adding or subtracting a constant value,
changes the mean or median, but will not change the magnitude of the spread. Three
common measures of spread are range, standard deviation, and variance.

range Range The range, w, is the difference between the largest and smallest values in
The numerical difference between the the data set.
largest and smallest values in a data set
(w). Range = w = X largest – X smallest
The range provides information about the total variability in the data set, but does
not provide any information about the distribution of individual measurements.
The range for the data in Table 4.1 is the difference between 3.198 g and 3.056 g;
thus
w = 3.198 g – 3.056 g = 0.142 g

standard deviation Standard Deviation The absolute standard deviation, s, describes the spread of
A statistical measure of the “average” individual measurements about the mean and is given as
deviation of data from the data’s mean
value (s). å n ( X i – X) 2
s = i=1 4.1
n -1
–
where X i is one of n individual measurements, and X is the mean. Frequently, the
relative standard deviation, s r , is reported.
s
s r =
X
The percent relative standard deviation is obtained by multiplying s r by 100%.

4 3
EXAMPLE .
What are the standard deviation, the relative standard deviation, and the
percent relative standard deviation for the data in Table 4.1?
SOLUTION
To calculate the standard deviation, we obtain the difference between the mean
value (3.117; see Example 4.1) and each measurement, square the resulting
differences, and add them to determine the sum of the squares (the numerator
of equation 4.1)
2
2
(3.080 – 3.117) = (–0.037) = 0.00137
2
2
(3.094 – 3.117) = (–0.023) = 0.00053
2
2
(3.107 – 3.117) = (–0.010) = 0.00010
2
2
(3.056 – 3.117) = (–0.061) = 0.00372
2
2
(3.112 – 3.117) = (–0.005) = 0.00003
2
2
(3.174 – 3.117) = (+0.057) = 0.00325
2
2
(3.198 – 3.117) = (+0.081) = 0.00656
0.01556
The standard deviation is calculated by dividing the sum of the squares by
n – 1, where n is the number of measurements, and taking the square root.
. 0 01556
.
s = = 0 051
7 -1