Environmental Chemistry Selected topics in statistics and metrology

The Plan Measurement errors The types and sources of errors Errors and uncertainty of measurement Estimation of measurement uncertainty Using a spreadsheet for estimating measurement uncertainty Significant digits Numbers' rounding Presentation of the measurement end-result 2

Measurement errors 3

Types of errors Gross errors Systematic errors Random errors 8

Gross errors They occur from time to time, sporadically. Most commonly the human error is their source, eg. scattering a part of the sample or its accidental contamination. Results flawed with gross errors can usually be detected using appropriate statistical tests (eg. Dixon Q-Test). 9

Systematic errors The most common causes are the restrictions of analytical reactions used, improper calibration or misuse of laboratory equipment. They have a reproducible size and specific reason. Usually easy to detect and eliminate. They can be constant (independent of sample size) or proportional (depending on sample size). Affect the accuracy of the results. 10

Constant systematic error Constant systematic error 11

Constant systematic error Constant systematic error 12

Proportional systematic error Proportional systematic error 13

Proportional systematic error Proportional systematic error 14

Sources of systematic errors Instrumental errors (bad quality measuring vessels, improper calibration of measuring instruments). Analytical errors (incomplete reaction and/or side reactions, the instability of the reaction products, the difference between the equivalence and end points of titration). Personal errors (subjective scale or color reading). 15

Detection and elimination of systematic errors Instrumental and personal errors: regular calibration of the equipment, choosing better end-point detection methods, eg. replacing the color indicator with a ph meter. Errors introduced by analytical methods: analysis of certified reference materials, analysis of blank samples, using a different methodology, change in the size of the sample. 16

Random errors They occur always. Their value, sign and causes are random. There is no possibility of their total elimination. Affect the precision of the results, i.e. their spread. 17

Errors and uncertainty of measurement The result is always given together with the accompanying uncertainty of the measurement. mean uncertainty The value of the uncertainty depends on the sum of all types of errors made during the measurement. 18

Estimation of measurement uncertainty Estimation of the uncertainty is a complex subject. The two easiest ways to express measurement uncertainty is to provide: standard deviation of the sample or confidence interval for the mean. 19

Presentation of uncertainty As an absolute value Δ x = x μ expressed in the same units as the measured value. As a relative value Δx Δx Δ x rel = μ or Δ x rel = x dimensionless, usually expressed in per cent, ppm, ppb and such. 20

Using a spreadsheet for estimating measurement uncertainty 1) COUNT(range) counts the number of cells that contain numbers, n. 2) AVERAGE(range) returns the average (arithmetic mean) of the arguments, 3) STDEV(range) estimates standard deviation based on a sample, s. 4) TINV( ;n-1) returns the two-tailed inverse of the Student's t-distribution (t parameter for given statistical significance. and number of degrees of freedom n-1). 21

Significant figures The significant figures of a number are digits that carry meaning contributing to its measurement resolution. Example: Certainly we can say that a burette in the figure on left shows a volume of more than 11.2 and less than 11.3 ml. Additionally, we can estimate the level of fluid between scale marks, with an accuracy of approx. 0.02 ml and say that a burette indicates a volume of 11.24 ml. We say that a reading of 11.24 has four significant digits, as the first three digits (11.2) this reading are a certain and the last (4) is uncertain. 22

Identifying significant figures 1. All non-zero digits are significant. 2. Leading zeros (ie. those defining the position of the decimal point) are never significant. 3. Zeros between non-zero digits are significant. 4. In a number with a decimal point, trailing zeros, those to the right of the last nonzero digit, are significant. 5. In a number without a decimal point, trailing zeros may or may not be significant. In order to avoid confusion, such numbers must be written in scientific notation. Eg. the number 1200 can be written as 1.2 103 if hundreds are uncertain or 1.20 103 if uncertainty applies to the tens, or 1,200 103 if uncertainty applies to units. 23

Identifying significant figures - examples In the following examples not-significant figures are marked in red. 1584,26 (6 significant figures) 1235 (4 significant figures) 0,00215 (3 significant figures) 0,235 (3 significant figures) 1002 (4 significant figures) 0,02005 (4 significant figures) 125,10 (5 significant figures) 45,00 (4 significant figures) 3500 (? significant figures) 3,5 103 (2 significant figures) 3,50 103 (3 significant figures) 3,500 103 (4 significant figures) 24

Rounding numbers to n significant figures (in this example rounding to three significant digits) We reject all of the digits located to the right of the n-significant figure (if it is on fractional position ) or replace them with zeros. a) If the first rejected digit is smaller than 5, the digits on left does not change. 0,004524 ---> 0,00452 2892654 ---> 2890000 (better 2,89 106) b) If the first discarded digit is greater than 5, we add 1 to the last digit that was left. 0,2526 ---> 0,253 186835 ---> 187000 (better 1,87 105) 25

Rounding numbers to n significant figures (in this example rounding to three significant digits) c) If the first discarded digit is equal to 5, and: 1. among other rejected numbers, at least one is greater than 0, we add 1 to the last digit that was left. 0,0239501 ---> 0,0240 154510 ---> 155000 (better 1,55 105) 2. other discarded digits are zero, we add 1 to the last digit that was left if it is odd or leave it unchanged if it is even or equal to 0. 1,2350 ---> 1,24 98750 ---> 98800 (better 9,88 104) 1,2250 ---> 1,22 98050 ---> 98000 (better 9,80 104) 26

Presentation of the meaurement end-result The result given without the associated uncertainty is worthless. The uncertainty accompanying the results of the analysis, can be present in different ways. It could be: confidence interval for the mean, standard deviation, uncertainty due to the error propagation. When providing uncertainties, you must clearly indicate how and on the basis of how many results it was calculated. The uncertainty of the result is sometimes given in the form of significant digits. However, it is ambiguous and not recommended way of reporting uncertainty. 27

Presentation of the meaurement end-result In general, the uncertainty should be presented with one significant digit. This is due to the fact that the standard deviation in itself is burdened with uncertainty. In some cases, however, it is required to provide two significant digits of uncertainty. The end-result is always written with the same number of decimal places as the value of uncertainty. 1 significant figure (US) 2 significant figures (EU) 42 ± 1 mg/ml 42,2 ± 1,3 mg/ml 2,22 ± 0,03 mg/ml 2,225 ± 0,028 mg/ml (7,2 ± 0,2) 102 mg/ml 723 ± 15 mg/ml 28

Numbers' rounding Once again, I remind. Numbers' rounding is the last arithmetic operation. We perform it only on the final result. Too early rounding leads to accumulation of calculation errors. 29