How many significant digits (SD's) should you report when you make a measurement? The answer is the full number that is warranted by the equipment you use. You do not throw away your accuracy. You probably have used this principle in some of your physics labs. A brief discussion of how SD's relate to measurement tools is provided below, before we get to the more general question of how many SD to report in general engineering calculations. Professional grade measurement equipment is usually designed so that the smallest division marks on the reading scales are spaced so that the true value should be within the +1 in the least SD of the reading. A micrometer is a good example. A common micrometer usually has marks in intervals of 0.001 inches (0.01 mm in the metric case). If the micrometer is used correctly (calibrated and the jaws tightened with the built in clutch) the true value should be within +0.001 inches of your measured value. You would report a typical micrometer measurements as 0.327, or 0.817, or 0.480 inches. Whenever possible, measurement equipment is designed so that the reading you get follows the plan that the least significant digit you read is within 1 digit of the true value. Unfortunately this is not always possible and other approaches are also used such as putting a % error on the measurement. You have to look at each piece of measurement equipment separately to find out how accuracy (measurement errors) are specified. Using SD's is a great approximate way to get an idea of how close the true value is to your measured value. Some general rules have been adopted in engineering practice on the reasonable number of SD that should be included in the results of calculations. We will explore how rounding off to some number of SD's contributes to the error range in calculations. We will also use this topic to explore and discuss some of the exceptions to the general rule our textbook uses to round answers off to 3 SD's. Answers in our text book are rounded off to 3 significant digits (commonly called SD's). The reason for this is that in most engineering calculations we do not know the actual values of the quantities accurately enough to warrant more significant digits. If you assume that the true value of a quantity is within t-one digit in the least significant digit, you can calculate the maximum amount of error you will add by round off an answer to a specific number of digits. As long as the error you may add by rounding off an answer is significantly less that the potential error in the measurement quantities used in the calculation, you can make the round off. For example, a 0.4% round off error does not significantly affect the overall accuracy of a measurement that already has a 3% error. Remember that these % errors are only an approximation and do not represent an actual limit, probabilities are involved so we only get a rough estimate. Never the less, using SD's is the most common and easiest way to get some indication of the probably range of the real value around a measurement There are two other major approaches to addressing how to calculate the range of the true value based on the uncertainties of our measurements. We will not be studying them in this course but they are worth mentioning (as done in the following two paragraphs). The first is just to specifying the possible % error (called the uncertainty in the measurement) for each piece of data used in a calculation and then add these % values up while considering how they are affected by the mathematical process (there are a number of rules for specific processes). This approach is sometimes called the Propagation of Error. The second method is to apply Probability and Statistics to the data, usually taught in an upper division course. This approach requires repeated measurements to determine variability. The result of this approach might be a statement like "The true value is within 4.5% of the measured value with a probability of 99%". What this means is that if you repeated the set of measurements 1000 times, then roughly 990 times the actual value would be within 4.5% of the measured value and about 10 results would be outside that range. You have to do a lot of usually expensive repeated testing to prove these type of statements. This procedure is used only when the task warrants the expense. Your Task: Calculate the maximum and minimum error you would add by rounding off a number to a specific number of SD's. Hint, pick numbers like 1 versus 9 (1 SD), 10 versus 99 (2sd), 100 versus 999 (3 SD), etc. Make a post with your results to the thread titled "Discussion 2" and then see what others have found. Make response posts to at least 2 other people during the week. How many significant digits (SD's) should you report when you make a measurement? The answer is the full number that is warranted by the equipment you use. You do not throw away your accuracy. You probably have used this principle in some of your physics labs. A brief discussion of how SD's relate to measurement tools is provided below, before we get to the more general question of how many SD to report in general engineering calculations. Professional grade measurement equipment is usually designed so that the smallest division marks on the reading scales are spaced so that the true value should be within the +1 in the least SD of the reading. A micrometer is a good example. A common micrometer usually has marks in intervals of 0.001 inches (0.01 mm in the metric case). If the micrometer is used correctly (calibrated and the jaws tightened with the built in clutch) the true value should be within +0.001 inches of your measured value. You would report a typical micrometer measurements as 0.327, or 0.817, or 0.480 inches. Whenever possible, measurement equipment is designed so that the reading you get follows the plan that the least significant digit you read is within 1 digit of the true value. Unfortunately this is not always possible and other approaches are also used such as putting a % error on the measurement. You have to look at each piece of measurement equipment separately to find out how accuracy (measurement errors) are specified. Using SD's is a great approximate way to get an idea of how close the true value is to your measured value. Some general rules have been adopted in engineering practice on the reasonable number of SD that should be included in the results of calculations. We will explore how rounding off to some number of SD's contributes to the error range in calculations. We will also use this topic to explore and discuss some of the exceptions to the general rule our textbook uses to round answers off to 3 SD's. Answers in our text book are rounded off to 3 significant digits (commonly called SD's). The reason for this is that in most engineering calculations we do not know the actual values of the quantities accurately enough to warrant more significant digits. If you assume that the true value of a quantity is within t-one digit in the least significant digit, you can calculate the maximum amount of error you will add by round off an answer to a specific number of digits. As long as the error you may add by rounding off an answer is significantly less that the potential error in the measurement quantities used in the calculation, you can make the round off. For example, a 0.4% round off error does not significantly affect the overall accuracy of a measurement that already has a 3% error. Remember that these % errors are only an approximation and do not represent an actual limit, probabilities are involved so we only get a rough estimate. Never the less, using SD's is the most common and easiest way to get some indication of the probably range of the real value around a measurement There are two other major approaches to addressing how to calculate the range of the true value based on the uncertainties of our measurements. We will not be studying them in this course but they are worth mentioning (as done in the following two paragraphs). The first is just to specifying the possible % error (called the uncertainty in the measurement) for each piece of data used in a calculation and then add these % values up while considering how they are affected by the mathematical process (there are a number of rules for specific processes). This approach is sometimes called the Propagation of Error. The second method is to apply Probability and Statistics to the data, usually taught in an upper division course. This approach requires repeated measurements to determine variability. The result of this approach might be a statement like "The true value is within 4.5% of the measured value with a probability of 99%". What this means is that if you repeated the set of measurements 1000 times, then roughly 990 times the actual value would be within 4.5% of the measured value and about 10 results would be outside that range. You have to do a lot of usually expensive repeated testing to prove these type of statements. This procedure is used only when the task warrants the expense. Your Task: Calculate the maximum and minimum error you would add by rounding off a number to a specific number of SD's. Hint, pick numbers like 1 versus 9 (1 SD), 10 versus 99 (2sd), 100 versus 999 (3 SD), etc. Make a post with your results to the thread titled "Discussion 2" and then see what others have found. Make response posts to at least 2 other people during the week