For multiplication by an exact number , multiply the uncertainty by the same exact number. Derivation : We can derive the relation for multiplication easily. Using Eq. The calculation of the uncertainty in is the same as that shown to the left. Then from Eq. Answer i 0. The actual error in a quantity, having the same units as the quantity. See Relative Error.
How close a measurement is to being correct. See Precision. When several measurements of a quantity are made, the sum of the measurements divided by the number of measurements. The average of the absolute value of the differences between each measurement and the average.
See Standard Deviation. The fraction of measurements that can be expected to lie within a given range. A measure of range of measurements from the average. Also called error oruncertainty. Also called deviation or uncertainty. An uncertainty estimated by the observer based on his or her knowledge of the experiment and the equipment.
This is in contrast to ILE , standard deviation or average deviation. The familiar bell-shaped distribution. Simple statistics assumes that random errors are distributed in this distribution.
Also called Normal Distribution. Changing the value of one variable has no effect on any of the other variables. Propagation of errors assumes that all variables are independent. The smallest reading that an observer can make from an instrument. This is generally smaller than the Least Count. The size of the smallest division on a scale. Also called Gaussian Distribution. The number of significant figures in a measurement. Greater precision does not mean greater accuracy! See Accuracy. Given independent variables each with an uncertainty, the method of determining an uncertainty in a function of these variables.
Deviations from the "true value" can be equally likely to be higher or lower than the true value. See Systematic Error. See C onfidence Level. This may also be called percentage error or fractional uncertainty. See Absolute Error. All non-zero digits plus zeros that do not just hold a place before or after a decimal point. An advanced statistical measure of the effect of large numbers of measurements on the range of values expected for the average or mean. A situation where all measurements fall above or below the "true value".
Recognizing and correcting systematic errors is very difficult. Also called deviation or error. The significant figure rules are important to know and use in all chemistry calculations, but they are limited in that they assume an uncertainty in the measured quantities.
So while the significant figure rules are always to be used in any calculation, when precision matters a propagation of error analysis must also be performed to obtain an accurate prediction of the uncertainty arising from the precision of the measured quantities.
We know that , and , and can then make these substitutions in Eqn. Dividing both sides by V gives Eqn. Note that there are several implications of Eqn. First, if one side has a large uncertainty relative to the length of that side such as when one side is very short , then this side will dominate the uncertainty. Second, when the volume is large and the uncertainty in measuring a dimension is small compared to the uncertainty in the measurement, then the uncertainty in the volume will be small.
You have measured the volume and mass of a set of regular wooden blocks and have fit a graph of their volume as a function of their mass to a straight line using the regression package in Excel.
Note that you have also seen this equation before in the CHEM Determination of Density exercise, but now you can derive it. The relationship between volume and mass is. Note that b does not affect.
We could have also have used Eqn. First we need to find the first derivative of the density with respect to the slope, which is Substituting this into Eqn. Recognizing the relationship between s and d , this simplifies to.
This problem is the simplest example of how one determines the uncertainty in a quantity extracted from a best-fit line. In general you will have the uncertainty in the slope and intercept and the relationship between each of these to the desired quantities. It is then a simple process to apply Eqn. Propagation of Uncertainty through a Calibration Curve. A situation that is often encountered in chemistry is the use of a calibration curve to determine a value of some quantity from another, measured quantity.
For example, in CHEM you created and used a calibration curve to determine the percent by mass of aluminum in alum. In that exercise, we did not propagate the uncertainty associated with the absorbance measurement through the calibration curve to the percent by mass.
However, in most quantitative measurements, it is necessary to propagate the uncertainty in a measured value through a calibration curve to the final value being sought.
The general procedure is quite straight-forward, and is covered in detail in CHEM Uncertainty in measurement comes about in a variety of ways: instrument variability, different observers, sample differences, time of day, etc. Anytime a calculation requires more than one variable to solve, propagation of error is necessary to properly determine the uncertainty. Since at least two of the variables have an uncertainty based on the equipment used, a propagation of error formula must be applied to measure a more exact uncertainty of the molar absorptivity.
This example will be continued below, after the derivation. Suppose a certain experiment requires multiple instruments to carry out.
These instruments each have different variability in their measurements. The results of each instrument are given as: a, b, c, d For simplification purposes, only the variables a, b , and c will be used throughout this derivation. A relationship between the standard deviations of x and a, b, c, etc In the first step, two unique terms appear on the right hand side of the equation: square terms and cross terms.
Square terms, due to the nature of squaring, are always positive, and therefore never cancel each other out. By contrast, cross terms may cancel each other out, due to the possibility that each term may be positive or negative.
If da, db, and dc represent random and independent uncertainties, about half of the cross terms will be negative and half positive this is primarily due to the fact that the variables represent uncertainty about a mean.
However, if the variables are correlated rather than independent, the cross term may not cancel out. Thus, the end result is achieved. In the next section, derivations for common calculations are given, with an example of how the derivation was obtained. Addition, subtraction, and logarithmic equations leads to an absolute standard deviation, while multiplication, division, exponential, and anti-logarithmic equations lead to relative standard deviations. Starting with a simple equation:.
Continuing the example from the introduction where we are calculating the molar absorptivity of a molecule , suppose we have a concentration of If you are given an equation that relates two different variables and given the relative uncertainties of one of the variables, it is possible to determine the relative uncertainty of the other variable by using calculus. In problems, the uncertainty is usually given as a percent.
0コメント