The latest version of this document is available from www.consol.ca (Teaching link).
The fundamental hypothesis in statistics is the Null Hypothesis. The null hypothesis states that random
error is sufficient to explain differences between two values. Statistical tests are designed to test the
null hypothesis. Passing a statistical test means that the nullhypothesis is retained: there is insufficient
evidence to show that there is a difference between the samples.
It is impossible to show that two values are the same; it is only possible to show they are different.
Some values are known or defined to be exact. For example:
• the ½ and 2 in EK = ½ m v2
• the stoichiometric coefficients and molecular formulae in chemicalreactions such as
C3H8 + 5O2
3CO2 + 4H2O
• the speed of light in a vacuum, c, is defined as 2.99792458·108 m/s
There is error in every observation. Error arises due to limitations in the measuring device (ruler, pH
meter, balance, etc.) and problems with equipment or methodology. The former are ‘indeterminate’ or
‘random’ errors and cannot be eliminated. Random errors limit the precision withwhich the final value
can be reported. The latter are ‘determinant’ or ‘systematic’ errors and affect the accuracy of the final
value. Analytical chemists continuously monitor for systematic errors in procedures.
‘Sig-figs’ are a simple, easy to apply, quick-and-dirty method of getting approximately the correct
number of decimal places in a value. The correct, but moredifficult, method is to statistically determine the uncertainty and thus the reportable number of decimal places. This approach considers the
uncertainty associated with every observation and its importance in the overall uncertainty. It is
possible to gain or lose decimal places compared with the sig-figs method.
Instructors may use the term ‘sig-figs’ when they mean ‘statistically calculatednumber of significant
digits’. This often confuses the students and instructor. Interestingly, some instructors demand the
uncertainty have one sig-fig; others accept up to two; still others use a ‘3-30’ rule.1 Any of these methods is acceptable as long as it is consistently applied.
To report the statistical uncertainty in the final value, the text could take the form, “Sample 123A has a
leadcontent of (9.53 ± 0.22) ppm at the 95 % confidence level.” The final value has the same number
of decimal places as the uncertainty. (Remember the leading zero for all numbers between -1 and 1!)
Units in calculations
Inclusion of units in calculations ensures that the final answer is not in error by a simple units conversion: joules « kilojoules, grams « milligrams « micrograms, R = 8.314 J/(molK) = 0.08206 L atm/(mol K), etc.
Critically evaluate every answer. If you react 5 g of A with 7 g of B, is it reasonable to expect the theoretical yield be 39 g? or 240 µg? If you repeat a titration three times, each with 5.00 mL of the
unknown, is it reasonable that the required volumes of titrant are 14.27 mL, 9.54 mL, and 9.61 mL?
Applied Statistics in Chemistry.doc
© Roy Jensen,2002
Several rules for rounding are taught; you have probably met more than one in your courses. Everyone
is adamant their rules are correct. The National Institute of Science and Technology (NIST) policy on
rounding numbers is presented here.2 (It is correct. J)
First, keep all the digits from intermediate calculations. Round the final value as follows:
If the digits to
less than 5
Round the last
digit to be kept
3.7249999 rounded to two decimal places is 3.72.
greater than 5
3.7250001 rounded to two decimal places is 3.73.
exactly 5 (followed only by zeros)
3.72500… rounded to two decimal places is 3.72.
When manipulating data, keep all digits through intermediate calculation. Round the final value...