Measurement Uncertainty
Precision and Accuracy
When we make a measurement in the laboratory we need to know how good it is. To this end, we introduce two concepts: Precision and Accuracy.
Precision indicates degree of reproducibility of a measured number, and
Accuracy indicates how close your measurements are to the true value.
Let's look at throwing darts and trying to hit the bullseye as an illustration of these two concepts.
When you make measurements in science you want them to be both precise and accurate.
Two students, Raffaella and Barbara, measured the temperature of boiling water, which by definition should be 100°C under 1 atmosphere of pressure. Each student made 10 temperature measurements, shown below as red (Raffaella) and blue (Barbara) dots.
The average of Raffaella's temperature measurements is 100.1°C and the average of Barbara's is also 100.1°C. So, the accuracy of their measurements is identical. On the other hand, you can see from the figure that the precision of Raffaella's measurements was better than Barbara's. The way this is expressed in science is to include an uncertainty with measured values. In this case Raffaella would report a boiling point of 100.1 ± 0.3°C, and Barbara would report 100.1 ± 1.4°C. This uncertainty is also called a random error, and is different from a systematic error, which is the difference between the average value and the true value. Here, both Raffaella and Barbara had systematic errors of 0.1°C, since the true boiling point of water is 100°C.
If, for whatever reason, the measurement uncertainty cannot be specified, then at the very least, the precision in a measured number can be approximately specified through the number of significant figures. In the example above, Rafaella would report a boiling point of 1.001 x 102 °C, whereas Barbara would report 1.00 x 102 °C. That is, Rafaella's result has four significant figures while Barbara's has only three.
Significant Figures
Significant Figures are the number of digits that express the result to the true measured precision. For example, let's consider the number:
Determining the number of significant figures in this number is very simple. Start from the left and count the digits. In this case we find that there are 5 significant figures in this number.
Zeroes, however, are special digits. Sometimes they count as a significant digit, and sometimes they do not count and simply act as place holders. For example, the number
has 5 significant figures. The first two zeroes are not significant, they are just place holders. Therefore, when you're counting significant digits always start from the left and don't start counting until you come to the first non-zero digit, then everything after that counts, including zeroes!
A good way to avoid the problem of significant and non-significant zeroes it is to use scientific notation. For example,
| 0.092067 | is written | 9.2067 x 10-2 | and has 5 sig. figs. |
| 0.092 | is written | 9.2 x 10-2 | and has 2 sig. figs. |
| 0.0920 | is written | 9.20 x 10-2 | and has 3 sig. figs. |
When using scientific notation all digits, including zeroes, are significant.
Rounding to the Correct Number of Significant Figures
In lab you were asked to weigh a crucible and cover three times. Let's say you measured
12.4337 grams, 12.4334 grams, and 12.4335 grams.These numbers have six significant figures, but clearly the precision is poorer for the last digit. If I calculate the average I get
(12.4337 + 12.4334 + 12.4335) / 3 = 12.4335333333 ← calculator displayWhen I report this number it makes no sense to write down all the numbers on my calculator display because the average can have no more significant figures than what I get from a single measurement. We need to round off the number to the correct number of significant figures. In this case
12.4335333333 rounded to six sig. figs. is 12.4335
Generally, when insignificant digits are dropped from a number, the last digit retained should be rounded for the best accuracy. For example,
To determine how the last digit should be rounded, the digits to be dropped are turned into a decimal fraction. In the example above, our decimal fraction would be 0.333333. We can now follow the rules for rounding numbers:
Rules for Rounding numbers to the correct number of significant figures.
- If the decimal fraction is greater than 1/2, then add one to the last digit retained.
- If the decimal fraction is less than 1/2, then leave the last digit retained alone.
- If the decimal fraction is exactly 1/2, then add one to the last digit retained only if it is odd.
Let's look at some examples.
Round the numbers 9.473, 9.437, 9.450, and 9.750 to two significant figures.
- For 9.473 the last digit retained is 4, and the decimal fraction is 0.73. So we use rule #1 above and 9.473 is rounded to 9.5
- For 9.437 the last digit retained is 4, and the decimal fraction is 0.37. So we use rule #2 above and 9.437 is rounded to 9.4
- For 9.450 the last digit retained is 4, and the decimal fraction is 0.50. So we use rule #3 above and 9.450 is rounded to 9.4
- For 9.750 the last digit retained is 7, and the decimal fraction is 0.50. So we use rule #3 above and 9.750 is rounded to 9.8
Using Significant Figures in Calculations
Generally, the more rigorous approach for determining the precision of a calculated result is to propagate the uncertainty in all your measured quantities through the calculation. The topic of error propagation through calculations, however, is outside the scope of this course. Thus, we will present the simpler rules below for determining the approximate number of significant figures in a calculated result.
Addition and Subtraction
- Rule:
- With Addition and Subtraction keep only the number of decimals in the result that occur in the least precise number.
Note that only the final answer is rounded. There is no rounding numbers in the intermediate steps of the calculation.
Multiplication and Division
- Rule:
- With Multiplication and Division, the final result should only have as many significant figures as the term with the least number of significant figures.
Mixing Operations
It can be tricky keeping track of the number of significant figures in a calculation that combines addition, subtraction, multiplication, and division. For example,
My calculator gives 0.013099698, but how many significant figures should be in the final answer? All numbers in the calculation have 3 significant figures, but if you break up the calculation into steps
Exact numbers
Exact numbers are known with infinite precision. For example if there are 10 students in a classroom, that number is an exact number. We treat exact numbers as numbers known with infinite precision
What is the average of 10.2, 11.4, and 10.9?
Powers of 10 and Logarithms
When taking 10 to the power of a number the final answer will have the same number of significant figures as the fractional part of the number.
Calculate 10-9.2067 with the correct number of significant figures.
We split the calculation into the integer and fractional part:
10-9.2067 = (10-9)(10-0.2067) = (10-9)(0.62129806352) → 0.6213 x 10-9 = 6.213 x 10-10
When taking the logarithm of a number the final answer will have the same number of significant figures as the logarithm argument.
Express the logarithm of 6.213 to the correct number of significant figures.
log(6.213) = 0.7933013536 → 0.7933
When taking the logarithm of numbers expressed in scientific notation, remember that the exponent of 10 is an exact number and therefore has infinite precision.
Express the logarithm of 6.213 x 10-10 to the correct number of significant figures.
Recalling that the log(AB) = log(A) + log(B) we can write:
log(6.213 x 10-10) = log(6.213)+log(10-10) = 0.7933013536 + (-10) = -9.2066986464 → -9.2067
You should only round off numbers when reporting your final result. Do not round off numbers in the middle of a calculation.