Uncertainties in Measurement Laboratory investigations involve taking measurements of physical quantities. All measurements will involve some degree of experimental uncertainty. QUESTIONS 1. How does one express the uncertainty in an experimental measurement ? 2.How does one determine the uncertainty in an experimental measurement ? 3. How does one compare an experimental measurement with an accepted (or published) value ? 4.How does one determine the uncertainty in a quantity that is computed from uncertain measurements ?

Expressing Uncertainty We will express the results of measurements in this laboratory as (measured value  uncertainty) units For example g   g = (9.803  0.008) m/s 2

Types of Experimental Uncertainty Random, Indeterminate or Statistical Results from unknown and unpredictable variations that arise in all experimental situations. Repeated measurements will give slightly different values each time. You cannot determine the magnitude (size) or sign of random uncertainty from a single measurement. Random errors can be estimated by taking several measurements. Random errors can be reduced by refining experimental techniques.

Types of Experimental Uncertainty Systematic or Determinate Associated with particular measurement instruments or techniques. The same sign and nearly the same magnitude of the error is obtained on repeated measurements. Commonly caused by improperly “calibrated” or “zeroed” instrument or by experimenter bias.

Accuracy and Precision Accuracy Is a measure of how close an experimental result is to the “true” (or published or accepted) value. Precision Is a measure of the degree of closeness of repeated measurements.

Accuracy and Precision Consider the two measurements: The length of a 1 inch long object was found to be … A = (2.52 ± 0.02) cm B = (2.54 ± 0.05) cm Which is more precise? Why? Which is more accurate? Why?

Accuracy and Precision _____ accuracy _____ precision _____ accuracy _____ precision _____ accuracy _____ precision Answer with GOOD or POOR...

Implied Uncertainty The uncertainty in a measurement can sometimes be implied by the way the result is written. Suppose the mass of an object is measured using two different balances. Balance 1 Reading = 1250 g Balance 2 Reading = kg Which balance is more precise? Explain.

Significant Figures In a measured quantity, all digits are significant except any zeros whose sole purpose is to show the location of the decimal place. How many SF for each, and answer the last one g m cm s cm 1.23 x 10 2 g x 10 2 g 1.2 x m x s x s 1 x 10 3 cm x 10 3 cm ? _________

Rounding  If the digit to the right of the position you wish to round to is < 5 then leave the digit alone.  If the digit to the right of the position you wish to round to is >= 5 then round the digit up by one.  For multiple arithmetic operations you should keep one or two extra significant digits until the final result is obtained and then round appropriately.  Proper rounding of your final result will not introduce uncertainty into your answer. IN COLLEGE SCIENCE, ROUNDING DURING CALCULATIONS IS NOT A VALID SOURCE OF ERROR, SO BE CAREFUL!

Expressing Uncertainty When expressing a measurement and its associated uncertainty as (measured value  uncertainty) units  Round the uncertainty to one significant digit, then  round the measurement to the same precision as the uncertainty. For example, round  m/s 2 to g   g = (9.803  0.008) m/s 2 1 SF

Significant Figures in Calculations When multiplying or dividing physical quantities, the number of significant digits in the final result is the same as the factor (or divisor…) with the fewest number of significant digits N  m * 5.5m  21C° N·m  m/C° ________N·m_________  m/C° Multiplication and Division

Significant Figures in Calculations When adding or subtracting physical quantities, the precision of the final result is the same as the precision of the least precise term (The one with the fewest decimal places) cm 0.823cm cm cm --> _______ cm Addition and Subtraction

Comparing Experimental and Accepted Values E ±  E = An experimental value and its uncertainty. A = An accepted (published) value. Percent Discrepancy quantifies the __________ of a measurement. Percent Uncertainty quantifies the __________ of a measurement.

Comparing Two Experimental Values E 1 and E 2 = Two different experimental values.

Average (Mean) Value Let x 1, x 2,… x N represent a set of N measurements of a quantity x. The average or mean value of this set of measurements is given by

Variance and Standard Deviation  Variance (aka. Deviation): the difference between a single result and the mean of many results.  Standard Deviation: the smaller the standard deviation the more precise the data Large sample size  Small sample size (n<30) ○ Slightly larger value 

The bell curve which represents a normal distribution of data shows what standard deviation represents. One standard deviation away from the mean in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99 percent of the data.