Reliability of Digits and Calculations
Determining the reliability of digits and calculations is important, because physical measurements have mathematical limitations. For example, suppose a ruler marked with tenths of inches is used to measure the length of an object. If the measurement is reported as 1.54 inches, the first two digits are certain, and the last digit (4) is an estimate. This represents the most reliable length that can reasonably be obtained for that object using that measuring device. Estimating any additional digits following the 4 would misrepresent the accuracy of the measurement and reduce its reliability.
If a calibrated gravimetric scale is used to measure a weight, and provides a digital answer of 2.328 grams, the measurement is assumed to be precise to one thousandth of a gram and can be reported with reliability. In this case, the reliability of the answer is not limited by the estimating ability of the measurement taker, but by the precision (and correct calibration) of the measuring device.
When measured values are added or subtracted, the result can be no more precise than the precision of the least precise measurement. This means that the answer should contain no more digits to the right of the decimal point than does the value with the least number of digits to the right of the decimal point.
For example, assume that a second digital scale, which is calibrated and precise only to a tenth of a gram, is used to weigh a second object and indicates a weight of 1.2 grams. The sum of the two weights, 2.328 grams + 1.2 grams, should be reported as 3.5 grams, not 3.528 grams.
When measured values are multiplied or divided, the number of significant figures is a determining factor regarding the reliability of the answer. Significant figures are numbers that are known with some degree of reliability. They include:
- Nonzero digits (e.g., 3.758 contains four significant figures)
- Zeroes between nonzeroes (e.g., 10.004 contains five significant figures)
- Zeroes to the right of nonzeroes that are right of the decimal point (e.g., 12.50 contains four significant figures)
For values less than 1, zeroes to the left of a decimal point and zeroes to the left of the first nonzero are NOT significant figures (e.g., 0.8 and 0.0000008 each contain only one significant figure). These zeroes indicate the location of the decimal point. They do not provide information about the reliability of the value.
For measurements reported in whole numbers, zeroes to the right of nonzeroes may or may not be significant (e.g., a weight of 65,000 pounds could have several different numbers of significant figures, depending on the precision of the scale). This uncertainty is eliminated by using exponential or scientific notation to specify the number of significant figures. Thus, 65,000 pounds reported as 6.5 x 104 pounds contains two significant figures and indicates that the scale is reliable to the nearest 1000 pounds. Likewise, 65,000 written as 6.50 x 104 pounds contains three significant figures, and indicates reliability to the nearest 100 pounds, and so forth.
The result of multiplication or division can contain no more significant figures than does the value with the least number of significant figures. For example, the product of 1.2 meters and 1.867 meters should be reported as 2.2 square meters, not 2.2404 square meters. The answer must contain only two significant figures. The product of 13.0 feet and 0.487 feet should be reported as 6.33 square feet, rather than 6.331 square feet, because 13.0 contains three significant figures.
One exception to these calculation rules applies when known exact numbers are used, e.g., 12 inches to a foot or 35 people in a room. Exact numbers, which are usually integers, are assumed to have an infinite number of significant figures. Therefore, they place no limiting factors on the calculation results, (e.g., 12 inches/foot multiplied by 3.456 feet should be reported as 41.47 inches, rather than 41 inches).
Extreme care must be taken when using a calculator or computer to calculate an answer based on measured observations. These devices can provide answers that include many more decimal places or significant figures than are actually reliable. The final answer must be adjusted according to the rules given above.
A slightly more sophisticated technique for indicating the reliability of measurements is called error analysis, and is based on statistical analysis of the random errors in the measurements. Random errors are deviations from the true value introduced by the observer's operation and reading of the instrument. For example, assume that 20 length measurements of the same object are taken and calculated to have a mean of 11.7 inches and a standard deviation of 2.683 inches. The standard error of the mean is calculated from 2.683/20 = 0.6 inches. Therefore, the final answer would be reported as 11.7 ±0.6 inches. If this point were shown on a graph, error bars would be placed on both sides of the point to indicate the extent of its standard error. If several values! containing standard errors are used in numerical calculations, (e.g., 11.7 ±0.6 inches * 25.3 ±1.1 inches), error propagation formulas are used to calculate the standard error in the final answer.
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