Accuracy and Precision s1

Accuracy and Precision

Accuracy: this is how close your result is to the correct, known, or actual value. E.g. if you are to determine the value of the acceleration on objects due to the Earth’s gravitational field, then 9.80 m/s2 is the correct, known, and actual result. Reporting other values indicates error in your experiment. The margin of safety in all of the labs is 10%. Beyond that something has been amiss in the determination of the final answer. Points are deducted for values reported beyond 10%.

Precision: this is how small your error is in determining the final answer. E.g. if your precision depends on length measurements made with a meter stick then your best precision is ½ of the smallest increment for the meter stick. Meter sticks are always made with mm marks. Thus your precision in length is ± 0.5 mm, or in SI units, ± 5 x 10 -4 m. If no precision is reported, then at most 1 point is deducted from the lab.

How to determine precision: the equation used to determine the correct, known, or actual value (heretofore referred to as the “correct” value) is used along with the precision for the lab equipment. In all cases the precision for the lab equipment is ½ the smallest increment measurable indicated. Each factor in the equation, which is not a constant (constant multipliers are dropped) can be treated in the following manor:

 Factors raised to positive powers: if the variable is x, and the power is n, then the contribution to the overall precision is nx(n-1)x. If the power of the factor is 1, then the factor’s contribution to the precision is just x.  Factors raised to negative powers: if the variable is x, and the power is –n (note the minus sign is separate from the value n, which is positive), then the contribution to the overall precision is nx-(n+1)x.  Factors which are trigonometric functions: o sin()  cos()  o cos()  sin()  o tan()  [1 + tan2()]   Factors which are exponentials (eikx) are not present in Phys I

Each factor is then added to the other factors and that sum is divided by the original equation. An example follows:

Let F= x2 * y / z, then what is the precision in F, F?

F = (2xx y /z + x2 y /z + x2 y z/z2)/ F = (2xx y /z + x2 y /z + x2 y z/z2)/ (x2y/z) = 2x + y/y + z/z,

x, y, and z are the precision of the lab equipment used to measure those quantities.

What is the precision in a if a = g cos()? Accuracy and Precision (cont’d.)

Significant Figures: these are the number of digits which surround the decimal point. E.g. in the above description of precision, the error is reported with only 1 significant figure. In lab, almost always only 3 sig figs are available. On rare occasions only 2, and on one occasion at most 4. To level out this minutia of lab work, only 3 sig figs are to be reported. Any other number of sig figs, inappropriately used will warrant loss of points.

Percent Error: this is a comparison of the result to the correct value. NB: it is not the same as the precision of the experiment. To determine the percent error, simply take the ratio of the deviation to the correct value. This represents the fraction of error. To determine the percent error, then simply multiply the ratio by 100. E.g.

x% = (xmeas – xknown)/xknown * 100

Deviation: simply, the result minus the correct value: xmeas – xknown

Absolute deviation: simply, the absolute value of the deviation: |xmeas – xknown|