MEASUREMENT AND ERROR. A Short Cut to all your Error Analysis Needs Fall 2011 Physics Danielle McDermott

MEASUREMENT AND ERROR A Short Cut to all your Error Analysis Needs Fall 2011 Physics 31210 Danielle McDermott

Estimating Errors in Measurements The term error refers to the uncertainty in our measurements. Analog Devices: For a measurement with a ruler, triple-beam balance, or other device with a continuous scale, the precision and accuracy depend on the user. YOU will have to judge how accurately you can measure based on the given scale. Digital Devices: For a measurement with an instrument with a digital readout, the reading error is "± one-half of the last digit." We would write the temperature as 12.80 ± 0.05 degrees C.

Estimating Errors in Measurements The term error refers to the uncertainty in our measurements. For rulers, assume an uncertainty of half the smallest mark. Half of 0.1 cm = 0.05 cm L = 24.21 ± 0.05 cm

Absolute Error Value Absolute Error Relative Error L L 24.21 cm 0.05 cm L = 24.21 ± 0.05 cm Absolute Error has the same units as the value.

Relative Error Value Absolute Error Relative Error L L L/ L 24.21 cm 0.05 cm 0.2 % L = 24.21 ± 0.05 cm Relative Error is a percentage of the value. R= L/ L=0.05 /24.21 = 0.00206 = 0.2 %

Accuracy vs. Precision Accuracy: How well does your measurement match the predicted value? Precision: Do your data points fall w/in a small range?

Rounding Errors + Sig Figs AFTER you've calculated the error, you need to take care of significant figures in your value: 1) Conventionally,.499 rounds down (to 0) but.500 rounds up (to 1). 2) Your error value should be rounded to one significant figure* L=0.537 L=0.5 L=0.137 L=0.14 L=6.537 L=7 3) *THE EXCEPTION If the number starts with a one, conventional rounding lowers your precision too much, so keep two sig figs L=157 L=160

Statistics To improve the precision and accuracy of measurements it is often best to repeat a measurement many times and study the spread of results. 1) Take a repeated measurement of the same (or similar) quantity 2) Take a Mean: Average your results 3) Take a Standard Deviation: Indication of how much the individual measurements in your sample defer from the mean value. 4) Calculate Standard Deviation of the Mean: The error in the mean value of your sample 5) Present your Best Value: Your data set can now be represented as the mean +/- the error (standard deviation of the mean).

Take N Measurements N = 3 x1 = 42, x2 = 45, x3 = 40 Count = 42 Count = 45 Count = 40

Mean Value N X = 1 N n=i x i The Σ means SUM, so add up all the M&M counts X = 1 3 42 45 40 =42.333333 X = 42.333333 Just the average of 3 numbers

Standard Deviation = 1 N N 1 n=i x i 2 Individual deviations x i = x i X x 1 = 40 42.333 = 2.333 x 3 = 42 42.333 = 0.333 x 2 = 45 42.333 = 2.667 Get ready to Σ these!

Standard Deviation = 1 N N 1 n=i x i 2 Just Σ the deviations you calculated = 1 3 1 x 1 2 x 2 2 x 3 2 = 1 2 2.333 2 0.333 2 2.667 2 = 2.5166 3 M & M ' s Round error values to one sig fig

Standard Deviation of the Mean Notice this value is unrounded x = N = 2.5166 3 M & M ' s x = 2.5166 3 = 1.453 1.5 M & M ' s x = 1.5 M & M ' s Keep two sig figs because the first digit is a one (yes, with M&M's that is strange)

Best Value x = x ± x x = 1.5 M & M ' s Round your average value X = 42.3333 42.3 Present your best value 42.3±1.5 M & M ' s

Statistics Formula Summary TAKE DATA Take N Measurements N MEAN Take an average N X = 1 N n=i x i STANDARD DEVIATION STD DEV OF THE MEAN BEST VALUE Find the error in an individual measurement Find the error in the mean Report your value = 1 N N 1 n=i x = N x = x ± x x i 2

Propagation of Error I Single Quantity So you've measured A w/ some error in that measurement δa... Multiplying or Dividing by a Constant c c(a ± δa) = ca ± cδa Adding or Subtracting a Constant c c + (A ± δa) = (c + A) ± δa The absolute error is unchanged. The relative error must be recalculated. Exponentiation (Powers and Roots) If a quantity is raised to the power n, the relative (fractional) error is multiplied by n. B = A n, δb/b = n δa/a.

Propagation of Error I Single Quantity So you've measured A w/ some error in that measurement δa... Multiplying or Dividing by a Constant c c(a ± δa) = ca ± cδa Example: You measure the diameter of a circle D = 5.03 ± 0.05 cm, To calculate the circumference C = π * D, So the error will be δc = π * δd C = π * 5.03 = 15.802 cm^2 ΔC = π * 0.05 = 0.157 which rounds to 0.16 So C = 15.80 ± 0.16 cm^2

Propagation of Error II Multiple Quantities w/ independent errors! The Basic Principle Independent Errors Add Quadratically (or Add in Quadrature) Here you've measured A, B, C, D and each has an error δa, δb, δc, δd... Adding and Subtracting or absolute errors add quadratically. X = A ± B± C X = A 2 B 2 C 2 Multiplying and Dividing or relative errors add quadratically. X = A x B C x D X X = A A 2 B B 2 C C 2 D D 2

Propagation of Error II Multiple Quantities w/ independent errors! Example: You've measured the Length and Width of your table. L = 3.25 ± 0.05 m, W = 1.32 ± 0.05 m Calculate the perimeter: P = 2L+2W X = A ± B± C X = A 2 B 2 C 2 P = 2 L 2 2 W 2 P = 0.14m

Propagation of Error III The General Formulas The General Rule for a Single Quantity If A is a quantity whose measured value is Ao with error δa, then the error in some calculated quantity B which can be expressed as a function f (A) is approximated as X = f A X = df da A 0 A So, take a derivative with respect to A and plug in A 0. Multiply this by δa. The General Rule for Several Measured Quantities X = f A,B,C... 2 X = [df da A ] 0 A 2 [ df 2 db B ] 0 B 2 [ df 2 dc C ] 0 C 2...