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Chapter 4 Evaluating Analytical Data 55

4
EXAMPLE .1
What is the mean for the data in Table 4.1?

SOLUTION
To calculate the mean, we add the results for all measurements
3.080 + 3.094 + 3.107 + 3.056 + 3.112 + 3.174 + 3.198 = 21.821
and divide by the number of measurements

21 821
.
.
X = =3 117 g
7
The mean is the most common estimator of central tendency. It is not consid-
ered a robust estimator, however, because extreme measurements, those much
larger or smaller than the remainder of the data, strongly influence the mean’s
2
value. For example, mistakenly recording the mass of the fourth penny as 31.07 g
instead of 3.107 g, changes the mean from 3.117 g to 7.112 g!

Median The median, X med , is the middle value when data are ordered from the median
smallest to the largest value. When the data include an odd number of measure- That value for a set of ordered data, for
ments, the median is the middle value. For an even number of measurements, the which half of the data is larger in value
–
and half is smaller in value (X med ).
median is the average of the n/2 and the (n/2) + 1 measurements, where n is the
number of measurements.
4
EXAMPLE .2
What is the median for the data in Table 4.1?
SOLUTION
To determine the median, we order the data from the smallest to the largest
value
3.056 3.080 3.094 3.107 3.112 3.174 3.198
Since there is a total of seven measurements, the median is the fourth value in
the ordered data set; thus, the median is 3.107.

As shown by Examples 4.1 and 4.2, the mean and median provide similar esti-
mates of central tendency when all data are similar in magnitude. The median,
however, provides a more robust estimate of central tendency since it is less sensi-
tive to measurements with extreme values. For example, introducing the transcrip-
tion error discussed earlier for the mean only changes the median’s value from
3.107 g to 3.112 g.

4 A.2 Measures of Spread
If the mean or median provides an estimate of a penny’s true mass, then the spread of
the individual measurements must provide an estimate of the variability in the masses
of individual pennies. Although spread is often defined relative to a specific measure
of central tendency, its magnitude is independent of the central value. Changing all