significant figures and unit

Experimental Error and Uncertain No measurement is perfect so how do you know how good the measurement is? Can you trust the value if you continue to calculate with it? These are questions that prompted the study of experimental error. There are several topics involved in our study today – Determine Random or Statistical Error vs. Systematic Error Know the difference between Accuracy vs. Precision and use them correctly Explain the Accuracy of Measuring Devices Use Percent Error and Percent Difference in correct circumstances Find the Average or Mean Take the Standard Deviation from the Mean Review and use Significant Digits Correctly Random vs. Systematic Error (also referred to as indeterminate error and determinate error) Observational error   (or   measurement error ) is the difference between a   measured   value of quantity and its true value.   In   statistics , an error is not a "mistake". Variability is an inherent part of things being measured and of the measurement process. When either  randomness  or uncertainty modeled by  probability theory  is attributed to such errors, they are "errors" in the sense in which that term is used in  statistics ; see  errors and residuals in statistics . Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results. The common  statistical model  we use is that the error has two additive parts: 1. systematic error which always occurs, with the same value, when we use the instrument in the same way and in the same case, and 2. random error which may vary from observation to observation. Systematic error is sometimes called statistical bias. It may often be reduced by very carefully standardized procedures. Part of the education in every   science   is how to use the standard instruments of the discipline. The random error (or  random variation ) is due to factors which we cannot (or do not) control. It may be too expensive or we may be too ignorant of these factors to control them each time we measure. It may even be that whatever we are trying to measure is changing in time (see  dynamic models ), or is fundamentally probabilistic (as is the case in quantum mechanics—see  Measurement in quantum mechanics ). Random error often occurs when instruments are pushed to their limits. For example, it is common for digital balances to exhibit random error in their least significant digit. Three measurements of a single object might read something like 0.9111g, 0.9110g, and 0.9112g. The section on Random and Systematic Errors is taken from Wikipedia, page https://en.wikipedia.org/wiki/Observational_error . Accuracy vs. Precision

Accuracy and precision are similar but not the same ideas. Accuracy means that you are close to a target place while precision means that all your shots are located in a small pattern which may or may not be close to the target. Here are some examples taken from National Ocean Service | National Oceanic and Atmospheric Administration | U.S. Department of Commerce | USA.gov http://celebrating200years.noaa.gov/magazine/tct/accuracy_vs_precision.html Accuracy of Measuring Devices The measuring device you use determines the accuracy of the measurement you will get. The smallest unit of marking on the device plus one estimated value beyond that creates the number of significant digits that will be measured by that device. One can estimate to the tenths between the smallest two markings on the measuring scale. As an example a meter stick with mm markings on it can be read to the tenth of a millimeter or to an accuracy of .0001 meters. Percent Error and Percent Difference These values are distinct and used in different situations. Percent error, Percent Error ( % Error ) = | Scientifically Accepted Value − Laboratory Measured Value Scientifically Accepted Value | × 100% Is used when a value that is known and agreed upon by the scientific community is used or compared in an experiment. Percent Difference (% Difference) = | Measured Value 1 − Measured Value 2 1 2 ( MeasuredValue 1 + MeasuredValue 2 ) | × 100 % should be used when comparing two values that were measured or calculated during the experiment and supposed to be for the same

item. An example would be a measurement of the acceleration of the ball down an incline comparing it to the calculated value of the acceleration of the ball down the incline. In this case one of the values is a calculation and one is a measurement. The percent difference is used when no scientifically accepted value is known. Finding the Average (or Mean), the deviation from the mean and the standard deviation Finding the average (the mathematical mean) is no different than what you do to calculate the average of your test scores. If you have 5 tests add up all the individual scores and divide by 5. Finding the Standard Deviation from the mean simply means taking each individual test score from this example and subtracting the average score from it. Here is an example with 5 tests included. T1 = 60; T2=80; T3 = 77; T4=82; T5=74 Test Average = T 1 + T 2 + T 3 + T 4 + T 5 5 = 60 + 80 + 77 + 82 + 74 5 = 74.6 To find the deviation of each test from the mean subtract the mean from the test. d T 1 = T 1 − Mean = 60 − 74.6 =− 10.6 d T 2 = T 2 − Mean = 80 − 74.6 = 5.4 d T 3 = T 3 − Mean = 77 − 74.6 = 2.4 d T 4 = T 4 − Mean = 82 − 74.6 = 7.4 d T 5 = T 5 − Mean = 74 − 74.6 =− 0.6 If you are required to find the standard deviation at this point all you need to do is take the average of the individual deviations that you just calculated. In this case the standard deviation is found taking − 10.6 + 5.4 + 2.4 + 7.4 − 0.6 ¿ 5 = 0.8 ¿ . Rules for Significant Digits The  significant figures  of a number are  digits  that carry meaning contributing to its  measurement resolution . This includes all digits  except :  All  leading zeros ;  Trailing zeros  when they are merely placeholders to indicate the scale of the number (exact rules are explained at  identifying significant figures ); and  Spurious  digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports. Significance arithmetic  are approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as  propagation of uncertainty . Numbers are often  rounded  to avoid reporting insignificant figures. For example, it would create  false precision  to express a measurement as 12.34500 kg (which has seven significant figures) if the scales only measured to the nearest gram and gave a reading of 12.345 kg (which has five significant figures). Numbers can also be rounded