Introducing Uncertainties For Physics Labs Intro Mechanics Prof. Adam Wright Important Concepts • Types of measurement error/uncertainty • Statistical/random • Systematic • Statistical uncertainty • Populations and Samples • Mean, Standard deviation, Sample standard deviation, Error on the mean • Systematic uncertainty • Warnings on notation, propagation, and exact formulas Types of measurement error/uncertainty Statistical Errors • Determined by the amount of data points you have • Want smaller statistical errors? Just take more data • There are formulas for stat error Systematic Error • Systemic to your experiment • Want smaller systematic errors? You’re going to have to improve your experiment • No formulas for sys error • Hard to quantify, a best guess will have to do in intro physics. Example: the smallest increment of your measuring device is an approximate uncertainty (e.g. the tick marks on a ruler) Q: Intuition tells you that taking more data leads to smaller uncertainties, but what if human error introduces a bias into your measurement? Statistical Errors Populations vs. Samples • Your data is a “sample” distribution • In theory, you could take more data... • Think of the limiting case of an infinite number of data points taken just like yours, that’s the “population” • Your sample is a subset of the true population • The larger your sample is, the more likely it is to be shaped like the population 0 1 2 3 4 5 75 76 77 78 79 80 81 82 83 84 85 86 number of times measured time between bounces (1/100 s) Time between 1 st and 2 nd bounce of a ping - pong ball • Populations tend to be Gaussian (aka Normal) distributed (often missing high or low equally likely) • t = time between two bounces of a ping pong ball dropped from a height of 1 meter • Gaussian distribution of the measured value t is described by two parameters: • mean: the location of the peak • standard deviation: the width of the hump Expect 68% of data points to fall within one σ of the mean ( i.e ҧ 𝑡 − 𝜎 ≤ 𝑡 𝑖 ≤ ҧ 𝑡 + 𝜎 for ~ 2/3 of your measurements 𝑡 𝑖 ) It’s ~95% at 2 σ , at 3 σ it’s ~99%. Statistical Errors • But we don’t have population, we have a sample, so what to do about this? • Mean of sample = good estimation of pop. mean: • 𝜎 𝑡 won’t work, since 𝑡 𝑡𝑟𝑢𝑒 is unknown Statistical Errors Instead we use the sample standard deviation ( 𝑠 𝑡 ). It has a slightly different functional form: 68% within ҧ 𝑡 − 𝑠 𝑡 ≤ 𝑡 𝑖 ≤ ҧ 𝑡 + 𝑠 𝑡 . It’s ~95% at 2 𝑠 𝑡 , at 3 𝑠 𝑡 it’s ~99%. 0 1 2 3 4 5 75 76 77 78 79 80 81 82 83 84 85 86 number of times measured time between bounces (1/100 s) Time between 1 st and 2 nd bounce of a ping - pong ball • Standard practice to take th e stat uncertainty/error/error bar on a measurement to be that interval which encompasses 68% of the data. • We use Δ 𝑡 to denote the uncertainty on t (syntax can be confusing) • The sample standard deviation ( 𝑠 𝑡 ) is the stat uncertainty on each specific measurement ( 𝑡 𝑖 ) • Example: Let’s say I’ve got 𝑠 𝑡 = 0 02 𝑠 for my sample standard deviation, and I’m going to ignore sys errors, then for a specific measurement 𝑡 5 = 0 82 𝑠 , I can say: • the stat uncertainty is Δ 𝑡 = 0 02 𝑠 • the proper way to write the measurement is 𝑡 5 = 0 82 ± 0 02𝑠 • there is a 68% chance that the true value of 𝑡 5 is between 0 80 𝑠 and 0 84 𝑠 • If another student had measured 𝑡 5 = 0 8 4 𝑠 , you would say they’re consistent since that’s within the uncertainty • If the student had measured 𝑡 5 = 0 96 𝑠 , you would say that they’re inconsistent since that’s a 7𝜎 discrepancy. Someone must’ve done something wrong. • If the student had measured 𝑡 5 = 0 8 6 𝑠 , you would conclude that it’s outside of your error bars, but only by 2 𝜎 (it’s within the 95% confidence interval), so it’s a bit ambiguous if they agree or not. Statistical Errors • We know that the sample standard deviation ( 𝑠 𝑡 ) is the stat uncertainty on each specific measurement ( 𝑡 𝑖 ), now what about the uncertainty on the average of a number of measurements? Statistical Errors • We know that the sample standard deviation ( 𝑠 𝑡 ) is the stat uncertainty on each measurement ( 𝑡 𝑖 ), now what about the uncertainty on the average of several measurements? • Δ 𝑡 → Δ ҧ 𝑡 and 𝑠 𝑡 → 𝑠 ҧ 𝑡 • For that we use the error on the mean ( 𝑠 ҧ 𝑡 ): • Example : Let ’ s say I ’ ve got 30 measurements in my sample, the average time is ҧ 𝑡 = 0 79 7 𝑠 , and my sample standard deviation i s 𝑠 𝑡 = 0 02 5 𝑠 , then 𝑠 ҧ 𝑡 = 0 025 𝑠 30 = 0 005 𝑠 . Ignoring sys errors, I can say: • the stat uncertainty is Δ ҧ 𝑡 = 0 0 05 𝑠 • the proper way to write the measurement is ҧ 𝑡 = 0 797 ± 0 0 05 𝑠 • note that writing 0 797 334 ± 0 005 𝑠 wouldn ’ t make sense, who cares about those digits • there is a 68 % chance that the true ҧ 𝑡 is between 0 792 𝑠 and 0 8 02 𝑠 • If another student had measured ҧ 𝑡 = 0 8 09 ± 0 009 𝑠 , you would say they ’ re consistent since the 68 % confidence intervals overlap. Statistical Errors Systematic uncertainty • Systemic to your experiment. More data won’t help you. • Can only mitigate by designing a better experiment • Hard to quantify, important to think about the sources of sys error and roughly estimate using some strategies: • Smallest increment on measuring device a decent lower bound (e .g. the tick marks on a ruler ) • Comparison between groups (only captures the group - to - group systematics) Warnings on notation, propagation, & exact formulas • *Total error: Δ ҧ 𝑡 𝑇𝑜𝑡𝑎𝑙 = Δ ҧ 𝑡 𝑆𝑦𝑠 + Δ ҧ 𝑡 𝑆𝑡𝑎𝑡 (in other notation 𝑈 𝑡 𝑇𝑜𝑡𝑎𝑙 = 𝑈 𝑡 𝑆𝑦𝑠 + 𝑈 𝑡 𝑆𝑡𝑎𝑡 ) • Propagation of error: when combining uncertainties from multiple sources of error coming together (e.g. getting the error on a velocity measurement from errors on the distance and time measurements it’s derived from), it’s a bit more complicated than you might expect, we’ll talk more about that later in the term. * Technically ( Δ ҧ 𝑡 𝑇𝑜𝑡𝑎𝑙 ) 2 = ( Δ ҧ 𝑡 𝑆𝑦𝑠 ) 2 + ( Δ ҧ 𝑡 𝑆𝑡𝑎𝑡 ) 2 Quantity Lab Manual Dr. Wright’s Other Syntax Statistical Error Δ ҧ 𝑡 = 𝑠 𝑡 𝑈 𝑡 𝑆𝑡𝑎𝑡 = 𝑠 𝑡 Systematic Error systematic error 𝑈 𝑡 𝑆𝑦𝑠 Total Error Δ𝑡 = 𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑒𝑟𝑟𝑜𝑟 + Δ ҧ 𝑡 𝑈 𝑡 𝑇𝑜𝑡𝑎𝑙 = 𝑈 𝑡 𝑆𝑦𝑠 + 𝑈 𝑡 𝑆𝑡𝑎𝑡 *Note: Excel ® has thi s built - in. Use “= STDEV.S( A1 : A30)” to get the sample standard deviation of cells A1 through A30. Measurement Uncertainties Conclusion • Statistical Uncertainty Quantities: • Sample standard deviation ( 𝑠 𝑡 ) • Error on the mean ( 𝑠 ҧ 𝑡 ): • Curve Fitting uncertainties (we ’ ll talk about this more in a later video) • Systematic Uncertainties usually dominate the overall error bar, we ’ ll invoke different strategies to quantify them when we can.