BIOL 1406
PreLab 1.10
How do I use a handheld calculator to determine the mean, percent error of the mean, and standard deviation of a random variable?

Uncertainty, or experimental error, is always involved when making a measurement. If two people independently took several measurements of a physical quantity such as the weight of an object, it is unlikely that both would come up with exactly the same results every time. Perhaps the instrument used to make the measurements is out of calibration or is influenced by variations in temperature and line voltage. Or perhaps the instrument can’t discriminate well between two very similar values. Or maybe personal error, carelessness, or bias is involved. For reasons like these, measurements are never perfect; they are only approximations of some true value that is being measured. The accuracy of a measurement refers to how close the measured values come to the true value. In other words, it is a measure of the correctness of the result. The precision of a measurement refers to how variable the measured values are when the same quantity is measured several times. A very precise measurement would be one that did not vary much over several trials, although it may or may not be accurate. In this section we will examine three statistics that scientists commonly use to evaluate the accuracy and precision of their measurements.

Graphing Calculator 
MeanThe mean is a statistic that is used to express the “average” or “center” of a set of numbers. When the same quantity is measured several times, the mean of the replicated measurements is a more precise measure of the quantity than any of the individual measurements. You should have a scientific calculator that automatically calculates the mean for a list of numbers. Use the instructions that came with your calculator to learn how to calculate the mean. If you no longer have the instructions, you can use the instructions in Appendix B of this Manual, provided you are using the TI36X. If you are using a different calculator, it is fairly easy to find instructions for most calculators on the Internet (check the manufacturer’s Web site or do a Google search). If you are still having problems, get help from one of the math tutors located in the Tutoring Labs at any ACC campus.

Accuracy refers to how close a series of measurements are to the true or target value. It is often expressed in terms of the percent error of the mean. In general, the closer the measurements are to the true or target value, the lower the percent error of the mean, and the better the accuracy of the measurements. The percent error of the mean is calculated using the following formula:
Percent error of mean = [│calculated mean – true or target value │ ∕ true or target value ] x 100 
In the formula above, the notation │number│ means “the absolute value of the number.” The absolute value refers to the numerical value of a number without regard to sign. Thus, 7 is the absolute value of both +7 and 7.
For example, suppose you want to know if you have a realistic sense of how long one minute is. You check your sense of time by saying “start” and “stop” while a lab partner measures the actual time that passes with a stop watch. Here are your results: 45 sec, 52 sec, 50 sec, 47 sec, and 53 sec. The mean number of seconds that you thought was one minute is 49 sec (note that the calculated mean cannot have more decimal places than the original measurements.) The target value is 60 sec. Using the formula for calculating percent error of the mean:
[│49  60│ sec ∕ 60 sec ] x 100 = 18%
So, in this case your sense of the length of one minute is 18% inaccurate.

Standard Deviation – used to express precisionPrecision refers to how close replicate measurements are to each other. It is often expressed in terms of the standard deviation. In general, the less variation there is among the values, the lower the standard deviation, and the better the precision of the measurements. Use the instructions that came with your calculator to learn how to calculate standard deviation. If you no longer have the instructions, you can use the instructions in Appendix B of this Manual, provided you are using the TI36X. If you are using a different calculator, you can find instructions for most calculators on the Internet (check the manufacturer’s Web site or do a Google search). If you are still having problems, get help from one of the math tutors located in the Tutoring Labs at any ACC campus.

How can I determine how accurately a given volume of water has been measured?
Pure water has a density of 1.00 g/mL at room temperature. Therefore, we can easily convert a given volume of water into its corresponding weight and vice versa. For example, 5 mL of deionized water weighs 5g, and 5g of water has a volume of 5 mL. Because weight can be easily measured to the nearest 0.01g on the electronic balances in lab, the weight of a given volume of water can be used to check how accurately a the volume was measured to the nearest 0.01 mL.
Your Turn  

NOTE: Make sure you use the correct units and the correct number of significant digits in all of your answers below. A student uses a 100 mL beaker to measure 80.0 mL of deionized water (dH_{2}O). She then weighs the water with an electronic balance. Following this procedure 6 times, she obtains the following results: 82.89 g, 80.23 g, 81.27 g, 81.92 g, 82.03 g, and 80.17 g.


Check your answers: 


During the second part of her experiment, the student repeats her measurements using a 100 mL graduated cylinder instead of a beaker. This time she obtains the following results: 80.79 g, 79.33 g, 80.25 g, 79.82 g, 80.13 g, and 80.19 g.


Check your answers: 

Which measuring device seems to be more accurate? Explain your answer. 

Which measuring device seems to be more precise? Explain your answer.  
Notice the number of significant
digits in the percent error answers. Because the target value (80.00 mL) is
not a measured value it does not affect the number of significant digits in
the answer. Therefore, the number of signification digits is determined
entirely by the numbers in the numerator of the percent error calculation:

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