Significant Figures and Units – ISP209L Spring 2004

When one performs a measurement on something, the measured value is only known to within the limits of the experimental uncertainty. The value of the uncertainty can depend on various factors such as the quality of the apparatus, the skill of the experimenter, and the number of measurements performed. This experimental uncertainty is often referred to as experimental error. Note that in this context error does not mean mistake; it just means uncertainty.

Suppose that we have to measure the area of a table using a meter stick. The smallest division on the meter stick is 1 mm or 0.001 m. So, we may argue that it would be difficult to measure any dimension of the table to accuracy better than 1 mm. Thus, if we measure the length to be 90.3 cm, it might in reality be anywhere between 90.2 cm and 90.4 cm. (Suppose that you line up the meter stick incorrectly with the edge of the table and measure the length of the table to be 85.2 cm. In this class we would say that this is not experimental error; that’s just a mistake.) We can write our measurement as 90.3 +/-0.1 cm. It would not be accurate to write our measurement as 90.300000 cm, since that would imply accuracy greater than what we know is possilbe. The number of significant figures in our measurement is 3 (90.3). Note that the number of significant figures includes the first digit that is estimated.

Suppose that we measure the width of the table to be 49.5 +/-0.1 cm and we want to calculate the area of the table. If I multiply 49.5 by 90.3, I end up with 4469.85 cm². This answer is not justifiable since it contains 6 significant figures, which is more than either of the two individual measurements. A good “rule of thumb” to use as a guide in determining the number of significant figures is as follows: when multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the least accurate of the quantities being multiplied. (Here the “least accurate” means the one having the lowest number of significant figures.) The same rule applies to division.

Thus, we should quote the area of the table above as 4470 cm². Note that the answer has 4 digits but only 3 of them are significant. When the answer is given in this way, it’s ambiguous as to how many significant figures are intended. Scientific notation can help here. We would write the number as 4.47 x 10³ cm². This is the same as 4470, but is in a form that one can immediately see that there are 3 significant figures. In most cases in this course, 3-4 signficant figures should be sufficient.

In grading the labs, we won’t require strict adherence to the above procedures, but we will penalize lab reports in which far too many digits are used in the answers. Most calculators will display 10 digits and a common situation is for students to dutifully write down all 10 digits onto their lab report, even though only 3 of the digits may be significant.

IMPORTANT PARAGRAPH 1: In many of the labs, you will be asked to compare the answers that you obtain with the expected results. In most cases your results should be fairly close to those expected, within the experimental uncertainty of the measurements. If you find your answer to be far from the expected value, it is not sufficient to just chock it up to human error. You should investigate and repeat the measurement. If the discrepancy continues, you should ask the instructor. We will grade the lab reports at least partially on the quality of the results obtained.

IMPORTANT PARAGRAPH 2: Now, some words on units. If we asked in a lab for a measurement of the length of the table given as an example above, the correct answer would be 90.3 cm. An answer of 90.3 would be incorrect since this answer (like most in this course) requires that the correct units be given. You will lose points if you give answers without the correction units.