■ What are the inherent limitations of my measuring equipment? How much is the measurement affected by environmental factors, user error, etc.?
• The plastic ruler you buy at the discount store for considerably less than a dollar will measure lengths up to 12 inches with a precision of better than 0.1 inch, but not as good as 0.01 inch. On the other hand, you can spend a few hundred dollars for a high-quality micrometer and measure lengths up to perhaps 6 inches to a precision of 0.0001 inch.
• Pumps at gas stations all over the United States often display their gas price and the amount of gas pumped to three decimal places. When gas prices are high, it is extremely important to consumers that the pumps are correctly calibrated. The National Institute of Standards and Technology (NIST) requires that for every 5 gallons pumped, the amount must not be off by more than 6 cubic inches. To deter-mine if a gas pump is calibrated correctly, you need to be able to see to three deci-mal places the amount pumped since 6 cubic inches is approximately 0.026 gallons.
When considering if the precision of a numeric value is reasonable, always ask yourself the following questions:
■ How many significant figures do I need in my design param-eters? The more precision you specify in a design, the more it will cost and the less competitive it will be unless the extra precision is really needed.
• You can buy a really nice 16-ounce hammer for about
$20. If you wanted a 16 { 0.0001-ounce hammer, it would probably cost well over a hundred dollars, possibly thou-sands.
When Is an Answer Reasonably Precise?
First, we need to differentiate between the two terms: accurate and precise . Although laypersons tend to use these words interchangeably, an aspiring engineer should under-stand the difference in meaning as applied to measured or calculated values.
Accuracy is a measure of how close a calculation or measurement (or an average of a group of measurements) is to the actual value. For measured data, if the average of all measurements of a specific parameter is close to the actual value, then the mea-surement is accurate, whether or not the individual meamea-surements are close to each other. The difference between the measured value and the actual value is the error in the measurement. Errors come about due to lack of accuracy of measuring equipment, poor measurement techniques, misuse of equipment, and factors in the environment (e.g., temperature or vibration).
Repeatability is a measure of how close together multiple measurements of the same parameter are, whether or not they are close to the actual value.
Precision is a combination of accuracy and repeatability, and is reflected in the number of significant figures used to report a value. The more significant figures, the more precise the value is, assuming it is also accurate.
To illustrate these concepts, consider the distribution of hits on a standard “bulls-eye”
target. The figure shows all four combinations of accuracy and repeatable.
NOTE
You should report values in engineering calculations in a way that does not imply a higher level of accuracy than is known . Use the fewest number of decimal places without reducing the usefulness of the answer. Several examples, given below, illustrate this concept. number. It seems reasonable to give our answer as 5.4 or 5.43 square centimeters since the original data of diameter is given to two decimal places. Most of the time, reporting answers with two to four significant digits is acceptable and reasonable.
If we keep only two decimal places, our answer would be 0.00 square centime-ters, which has no meaning. Consequently, when reporting numerical results, particu-larly those with a magnitude much smaller than 1, we use significant figures, not decimal places. It would be reasonable to report our answer as 0.00044 or 4.4 * 10-4 square centimeters.
■ We want to determine a linear relationship for a set of data, using a standard soft-ware package such as Excel ® . The program will automatically generate a linear rela-tionship based on the data set. Suppose that the result of this exercise is
y = 0.50236x + 2.0378
While we do not necessarily have proof that the coefficients in this equation are nice simple numbers or even integers, a look at the equation above suggests that the linear relationship should probably be taken as
y = 0.5x + 2
■ If calculations and design procedures require a high level of precision, pay close attention to the established rules regarding significant digits. If the values you gener-ate are small, you may need more significant digits. For example, if all the values are between 0.02 and 0.04 and you select one significant figure, all your values will read 0.02, 0.03, or 0.04; going to two significant figures gives values such as 0.026 or 0.021 or 0.034.
■ Calculators are often set to show eight or more decimal places.
• If you measure the size of a rectangle as 2 1>16 inches by 51>8 inches, then the area is calculated to be 6.4453125 square inches since the calculator does not care about how many significant digits result. It is unreasonable that we can determine the area of a rectangle to seven decimal places when we made two measurements, the most accurate of which was 0.0625 inches, or four decimal places.
• If a car has a mass of 1.5 tons, should we say it has a mass of 3010.29
■ Worksheets in Excel often have a default of six to eight decimal places. Two impor-tant reasons to use fewer are that: (1) long decimal places are often unreasonable and (2) columns of numbers to this many decimal places make a worksheet difficult to read and unnecessarily cluttered.
COMPREHENSION
CHECK 5-3 In each of the cases below, a value of the desired quantity has been determined in some way, resulting in a number displayed on a calculator or computer screen.
Your task is to round each number to a reasonable number of significant digits—
up if a higher value is conservative, down if a lower value is conservative, and to the nearest value if it does not make a difference. Specify why your assumption is conservative.
(a) The mass of an adult human riding on an elevator 178.8 pounds (b) The amount of milk needed to fill a cereal bowl 1.25 cups (c) The time it takes to sing Happy Birthday 32.67 seconds
Increasingly, engineers are working at smaller and small-er scales. On the left, a vas-cular clamp is compared to the tip of a match. The clamp is made from a bio-absorb-able plastic through the pro-cess of injection molding.
Photo courtesy of E. Stephan
5.8 NOTATION
When discussing numerical values, there are several different ways to represent the values. To read, interpret, and discuss values between scientists and engineers, it is important to understand the different styles of notation. For example, in the United States a period is used as the decimal separator and a comma is used as a digit group LEARN TO: Report calculated numbers in standard, scientifi c, and engineering notation
separator, indicating groups of a thousand (such as 5,245.25). In some countries, how-ever, this notation is reversed (5.245,25) and in other countries a space is used as the digit group separator (5 245.25). It is important to always consider the country of origin when interpreting written values. Several other types of notations are dis-cussed below.