Metrology

metrology

-the science of measurement

-rely on probability and statistics for random events

-rely on chemometrics for many chemical measurements

combining independent methods

-allows the precision and the class to be narrowed down

sufficient precision

-when the ultimate goal of individualization has been achieved

The importance of reliable data

-why we have chain of custody

-mislabeled samples = gross error

-modern methods with serial numbers and bar coding prevents bias because analysts may be unaware of what is in the sample

precision

-can be implicitly defined using the significant digit convention (the least sig fig is uncertain)

-precision is usually defined using standard deviation (common in forensics, not great because the range of +- 1 standard deviation spans 68% of the variance)

-confidence level is better for expressing precision (95% CI)

rounding

-when rounding to the digit round to the nearest even number

-43.55000 with 3 sf is rounded to 43.6

-42.59 when rounded to 2 sf is 42

-43.59 with 2 sig figs is rounded to 44

Sig figs

-in 0.1920 = 4 sig figs, 0 furthest to the right is known

-in 1920 = do not know if last digit is known, to fix the fourth digit can use scientific notation (1.920 × 10³)

indeterminant errors

-all measurements have an associated error or precision

-sums of random numbers have a normal or Gaussian distribution

-when distribution is not normal, mean median and mode may not be equal

-the width of a distribution is defined by the standard deviation

standard deviation

-distance from mean to the inflection point (where there is a major change in the slope of the curve)

width of a peak

-number of theoretical plates and resolution (N)

-want N as big as possible to give better chromatography

American society for testing and materials (ASTM)

-formed to address the frequent rail breaks plaguing the railroad industry, established a standard for the steel used to fabricate rails

-voluntary organization but many of the standards defined by ASTM are enforced by International Organization for Standardization (ISO) certification used by all US government crime labs and agencies

-ASTM publishes the Journal of Forensic Sciences

ASTM standards (6 categories)

-The standard specification

-the standard test method

-the standard practice

-the standard guide

-the standard classification

-the terminology standard

The standard specification

-defines the requirement to be satisfied by subject of the standard

The standard test method

-defines the way a test is performed

-the result of the test may be used to assess compliance with a specification

the standard practice

-defines a sequence of operations that, unlike a test, does not produce a result

-used for SOPs

The standard guide

-provides an organized collection of info or series of options that does not recommend a specific course of action

The standard classification

-provides an arrangement or division of materials, products, systems, or services into groups based on similar characteristics such as origin, composition, properties, or use

The terminology standard

-provides agreed definitions of terms used in other standards

-IUPAC (international union of pure and applied chemistry) = the gold book

a sample

-is a collected piece of evidence or a standard reference material submitted for analysis

a replicate

-is a repeated analysis applied to the same sample

Precision

-is a measure of the repeatability of a method and defined by the standard deviation

-methods that are precise have small StDev’s

Accuracy

-is a measure of agreement between a measured value and the true value

-true value (μ) is defined by a standard reference material (SRM)

Absolute error

-E_i = x_i - μ

-E_i: error of result i

-x_i: result i

-μ: true value

Relative error

-expressed as a percent

-E_r = [(x_i - μ)/ μ] x 100%

-can be expressed in ppt or ppm

-%= per hundred, ‰ = per mille (thousand)

bias

-another way to express accuracy

-bias = x̄ - μ

types of errors

-random (bias)

-systemic (constant or proportional)

-constant (error stays the same)

-proportional (as sample increases, error increases)

systematic error

-has definite value, assignable cause, and affects all replicates similar

-can be usually corrected by subtracting or dividing by the error

origins of systemic error

-instrumental error caused by faulty calibration

-method of error caused by: unrepresentative sampling, incomplete chemistry, impure/unstable reagents (expired)

-operator error caused by: misreading, poor techniques (mouth pipetting)

propagation of error

-equation on slide 2-17 on slideshow

-used to determine the precision of a method or calculation

-can only add terms with the same units

-should resemble the expressions that you learn in quantitative analysis

to determine sig figs

-must now use propagation of error

-KNOW THIS

-example on slide 2-18

survey

-every object is measured, in many cases making a complete measurement is impractical or impossible

sample

-a subset of a survey collected that represents the distribution of objects in the complete population

-collecting representative samples is very important to obtain accurate measurements of the underlying population

-samples are usually collected randomly

outliers

-non-representative samples usually caused by gross measurement errors or transcription errors

-two tests recognized by the American society for testing and materials (ASTM) for outlier detection: Dixon Q test, Grubs test

-KNOW EQUATIONS slide 2-20

Q-test

Q_calc = abs ((x_out - x_nearest)/ (x_max - x_min))

-find outlier and find its closest value, divide by the range

-compare Qcalc to Qtrue on Q table

Grubs test

G_calc = abs ((x_out - x̄) / (s))

-compare value to Grubbs G-table

Calibration curve

-it is impractical to survey by measuring every concentration

-a subset of concentrations is obtained usually by a constant distribution across the working range of analysis

-it is essential to run these concentrations of standards randomly (Not lowest to highest)

-the replicates should be blocked in that a complete series of the standards is run in a random order (random block design) before the next group of replicates

-this approach minimizes the effect of time or loss in sensitivity on the calibration line

properties of Gaussian curves

-defines normal distribution

-plot of relative frequency vs. deviation from mean

-mean occurs at central point (no deviation from mean, occurs at highest relative frequency)

-symmetrical

-exponential decrease in frequency as d increases

-area under Gaussian curve = type 1 error

probability of gaussian curves

-if calculate area under curve as 0.954, there is a 95.4% probability that a measurement will be within +-2σ of the mean

95% CI

-if you ran the test 100x, 95 times the average will fall in that range

applications of statistics

-the probability of a random event x is calculated from the cumulative distribution function (CDF), ex.) area under the curve

-the range that a random event may occur within a probability (the CI)

-statistical testing: do a s

-let of random variables belong to a distribution? are parameters such as the mean or variance the same or different?

-always use double sided statistics when you do not know if the mean is greater or less than the true value

Null hypothesis

-a random variable belongs to the tested distribution

Alternative hypothesis

-a random variable does not belong to the tested distribution

type 1 error (a)

-null hypothesis is rejected when it was true

-is not random

type 2 error (B)

-null hypothesis is accepted when it is false

-is random

optimal separation

-for two distributions the optimal separation that minimizes a+B is where the 2 PDFs intersect

-if more false +, more right

-if more false -, more left

-want low false negative and high false positive if looking for weapons in airplanes cuz bombs = bad

-if the critical point is increased a will decrease but B will increase and bias the result

-if the critical point is decreased a will increase more than B decreases

Calibration

-if we know the true concentration of a standard μ₀ and its true standard deviation σ₀

-The error σ₀ arises from errors in preparing or storing the standard

-if t_calc is greater than t_ref then we know that we are out of calibration

tcalc = ((abs ( μ_{0 -} x̄ )) / s) * √n

how do more measurements lower error rate and increase precision?

-standard deviation approaches the normal distribution

-s/√n = σ

-because n is sqrt in denominator, as n increases standard deviation decreases

comparing 2 means from diff samples

-t-test

-are the results from two methods the same?

-compare two samples using the F-test; if they are the same use the t-test

comparing two means with the same sample

-better statistical test; more powerful

-the same samples are run using two different methods so differences can be factored out

-the null hypothesis is the average of the differences equals zero

-only works when samples match

-take avg’s of differences

-if the average is small they’re not significantly different

confidence intervals and t-statistics

-the normal distribution is fine when a large number of measurements are made

-for small measurements (less than 30) the t-distribution is used which accounts for the uncertainty in the standard deviation

confidence interval

-ci_95% = t(0.05,df)s/√n

-the confidence interval becomes more precise as the square root of the number of measurements increases

-at 7 measurements it will be less than the standard deviation and span 95% of the range instead of 68%

how is independence good or bad?

-good: accurate and reliable data, reproducibility, avoid measurement bias

-bad: loss of info, reduce sensitivity

gain of information

-conditional probabilities allow us to gain information from observing a measurement A, if the property B is not independent of A. Otherwise, we would not gain any info from the experiment

-info is measured as the reduction of uncertainty

Final ideas

-standardization methods and certified protocols reduce forensic testing errors

-to maintain SRMs should be routinely measured

-representative sampling is important but difficult in many forensic cases

-more samples and measurements yield improved precision, but cost more with respect to time and money

statistics important for testing

-calibration

-outlier detection

-comparing samples