Accuracy and Precision

Hope College: General Chemistry Laboratories
Fall: Chem 127 & Chem 132 - Spring: Chem 128

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• As mentioned in the discussion of significant figures, there is always some uncertainty in a measured quantity because the last significant figure is based upon estimation.
• If the uncertainty is due to random errors in the measuring process, then a set of replicate measurements will have several different values depending on the estimate that is made for each measurement.
• Since random errors will be statistically distributed in both the high and low direction to an equal extent, making several measurements and reporting the average value tends to reduce the influence of random error when compared to just a single measurement.
• While a large number of experimental measurements effectively reduces (but never eliminates) the uncertainty due to random errors, it is usually impractical in terms of time and money to make a large number of measurements.
• However, even as few as three measurements can greatly reduce the impact of random errors. This is the reason that you are generally asked to perform measurements in triplicate and report the average value.

Table of Contents:

1. Accuracy
2. Precision

 

 

Accuracy

In reporting experimental results two important issues are: 1) how close is the experimentally reported average to the true value and 2) how reproducible or closely grouped was the data used in determining the average? The first issue deals with the concept of accuracy while the second deals with precision. To illustrate these two concepts, consider an experiment in which the density of the nitrogen gas sample mentioned in the previous section (Rounding Off) was determined three times. Because each of the volume and mass measurements have some uncertainty due to estimation of the last digit, the reported densities could have values of 0.1093, 0.1102 and 0.1089 g L−1. The average density is then 0.1095 g L−1. If the true density of this sample is actually 0.1100 g L−1, then the error is

 

0.1095 g/L−0.1100 g/L = −0.0005 g/L

 

in the reported value. A common way of reporting errors is to express the error relative to the true value, either in terms of percent or parts per thousand (ppt). Parts per thousand is similar to percent, since percent is just a relative measurement in terms of parts per hundred. Thus the relative error in this measurement is

 

relative error (in ppt) =

(experimental−true)
|true|

×1000 ppt

=
−0.0005 g/L
0.1100 g/L

×1000 ppt = −5 ppt

 

Since measured or true values may sometimes be negative (such as a freezing point with a negative Celsius value), the general expression for reporting absolute error is (experimental value – true value), where the values are written with the appropriate negative or positive sign. The general expression for reporting the relative error uses the absolute value of the true value in the denominator so that the correct sign in the final relative value is obtained.

 

Precision

The precision is a measure of how closely grouped the individual values are around the average value. A common way to express precision is through the parameter called the standard deviation. While it is beyond the scope of this manual to explain the statistical significance of standard deviation, the formula for calculating it is quite straight forward and is given below

��N2 s= ∑i=1(xi−x ̄)

N−1

where s is the standard deviation,x ̄ is the average, xi the individual values and N the total number of values being considered. While the value of s can be obtained by finding x, calculating each individual difference and its square, summing the squares, dividing by (N-1) and taking the square root, you will find it much easier to learn to use the appropriate statistical function on your calculator or spreadsheet (the standard deviation function is available in Microlab and Excel).

 

The important aspect of standard deviation is that there will always be some deviation due to random errors. A small standard deviation typically indicates that random errors were within expected limits, while a large standard deviation indicates that the random errors were larger than expected. Larger than expected random errors are usually due to poor experimental procedure and improper use or malfunction of equipment and instruments. Thus the standard deviation is an important tool in helping us evaluate your laboratory skills. Always express the standard deviation with two sig figs.

 

For the data in this example the standard deviation is 0.00067 g L−1 (do your own calculations to check this). This number tells us the amount of uncertainty that is observed in our answer, and that any sub- sequent digits become even more uncertain. Using the standard deviation as a guide, the least significant digit of the associated average value should match the second digit in the standard deviation (the “7” is this example). Thus the proper average value for this example would be 0.10947 g L−1, where the “7” is the least significant digit since it has the same decimal placement as the “7” in the standard deviation. The proper way to report this result would be 0.10947 ± 0.00067 g L−1.