Statistics and Uncertainties in the Laboratory

Statistics and Uncertainties in the Laboratory

A lecture course on handling numbers in the laboratory

• Precise and Imprecise numbers

• Counting and Measurement

• Recipes for the first couple of experiments

• Later Lectures – Justification and Explanation

Precise and imprecise numbers


Maths give precise numbers, such as 1, 2, 3, 0, -1, ½, ⅓, 0.333 . . . , , e, etc

Counting gives precise numbers, the integers.

Measurement gives imprecise numbers.

Sampling gives imprecise numbers.

How many cm marks? – 21How long is the pencil – 75 ± 2 mm (or ~75 mm)







How many spikes (counts) in this ten-second interval? We see 8How many counts per ten-second sample? We report 8 ± 3.







How many counts per ten-second sample? We say 8 ± 3What is the count rate? Our measurement is 0.8 ± 0.3 cps

Recipes for imprecise numbers


First Recipe

If you count N in a sample, you have measured 𝑁 ± 𝑁 per sample


First Recipe

If you count N in a sample, you have measured 𝑁 ± 𝑁 per sample

Same Recipe, Different Situation

If you count ~N in each of two samples, you have measured ~2𝑁 ± 2𝑁 per sample.

Say 2N came out at 20. The counts per 20 seconds are 20 ± 4½, count rate is 1 ± 0.2 cps.

Note this is consistent with, but more accurate than, our previous result of 0.8 ± 0.3 cps



Second Recipe

Take samples of different lengths, and say you get 9 counts in one and 16 counts in the other, so you record get 9 ± 3 and 16 ± 4. Add the two samples together and you have 25 ± 5. Recognise this 5 as the hypotenuse of a 3-4-5 right-angled triangle.

For addition and subtraction: Add the absolute uncertainties by Pythagoras.

This recipe applies to all quantities, not just counts, e.g. two lengths:

100 ± 5 mm + 50 ± 12mm = 150 ± 52 + 122 mm = 150 ± 13 mm

For multiplying and dividing: Add the relative uncertainties by Pythagoras.

Area = 100 ± 5 mm 40 ± 10mm = 100 mm ± 5% 40 mm ± 25% = 4000 mm2 ± 25.5% = 4000 ± 1019.8 mm2 which should be written as 4000 ± 1000 mm2. Note that adding by Pythagoras largely discounts the smaller of two errors.



Second Recipe continued

Take ten samples, and say you get about ten counts in each, so you record about 10 ± 3 ten times. Add the ten samples together and you have about 100 ± 10. Recognise this 10 as the hypotenuse of a 3-3-3-3-3-3-3-3-3-3 right-angled triangle in 10 dimensions.

For averaging: To average, you add up as above and then divide by the number of

samples. This give about 10 ± 1. The uncertainty has been reduced by 10Note that the uncertainty in the mean of N repeated measurements is smaller than the

uncertainty of a single measurement by the factor 𝑁. This also applies to any measurements, not just sampling.



Third Recipe

log(x ± x): How to get the error bar on the logarithm of an imprecise number?

One way: Consider log10(1000 ± 30). You can take the logs of the ends of the error bars, i.e. log10 970 =2.987 or log10 1030 = 3.013. Then log10(1000 ± 30) = 3 ± 0.013.

Another way: More mathematically, sketch a diagram to convince yourself that if

y = f (x), then f (x ± x) = f (x) ±d𝑦

d𝑥x which for log10(1000±30) is

3 ±log10𝑒

𝑥x = 3 ±

0.43

100030 = 3 ± 0.013


Questions? Don’t hesitate to ask in the 2-4pm sessions Mon and Thur, or e-mail me:

[email protected]

Documents

Statistics and Uncertainties in the Laboratory