Reading –lots to revise and learn • Chapter 3 • Chapter 4 • Chapter 5-1 and 5-2 • Chapter 5-3 will be necessary background for the AA lab • Chapter 5-4 we will use later
Data Analysis • Most data quantitative - derived from measurements • Never really know error • With more measurements you get a better idea what it might be • Don’t spend a lot of time on an answer -where only 20% accuracy is required -or where sampling error is big - although you don’t want to make the error worse
Significant Figure Convention • Final answer should only contain figures that are certain, plus the first uncertain number • eg 45.2% • error less than 1% or we would only write 45% • error larger than 0.05% or would write 45.23%
Remember • Leading zeros are not significant • Trailing zeros are significant • 0.06037 - 4 significant figures • 0.060370 - 5 significant figures • 1200 ???? • 12 x 102 - 2 significant figures
Rounding Off • Round a 5 to nearest even number • 4.55 to 4.6 • Carry an extra figure all through calculations • BUT NOT 6 EXTRA • Just round off at the end
Adding • Absolute uncertainty of answer must not exceed that of most uncertain number • Simple rule: Decimal places in answer = decimal places in number with fewest places 12.2 00.365 01.04 13.605 goes to 13.6
When errors are known • Rr =(A a) + (B b) + (C c) • where r2 = a2 + b2 + c2 • Example: Calculate the error in the MW of FeS from the following atomic weights: • Fe:55.847 0.004 S:32.064 0.003 • r = (0.0042 + 0.0032)1/2 • MW = 87.911 0.005
Multiplication and Division • Simplest rule: Sig figs in answer = smallest number of sig figs in any value used • This can lead to problems - particularly if the first digit of the number is 9. • 1.07400 x 0.993 = 1.07 • 1.07400 x 1.002 = 1.076 • Error is ~ 1/1000 therefore 4 significant figs in answer
Multiplication and Division • The relative uncertainty of the answer must fall between 0.2 and 2.0 times the largest relative uncertainty in the data used in the calculation. • Unless otherwise specified, the absolute uncertainty in an experimental measurement is taken to be +/- the last digit
Multiplication and Division • With known errors - add squares of relativeuncertainties • r/R = [(a/A)2 + (b/B)2 +(c/C)2]1/2
Logs • Only figures in the mantissa (after the decimal point) are significant figures • Use as many places in mantissa as there are significant figures in the corresponding number • pH = 2.45 has 2 sig figs
Definitions • Arithmetic mean, (average) • Median -middle value • for N=even number, use average of central pair
Accuracy • Deviation from true answer • Difficult to know • Best way is to use Reference standards • National Bureau of Standards • Traceable Standards
Precision • Describes reproducibility of results • What is used to calculate the confidence limit • Can use deviation from mean • or relative deviation • 0.1/5 x 1000 = 20ppt (parts per thousand) • 0.1/5 x 100% = 2%
Precision of Analytical Methods • Absolute standard deviation s or sd • Relative standard deviation (RSD) • Standard deviation of the mean sm • Sm = s/N½ • Coefficient of variation (CV) s/x x 100% • Variance s2
Standard Curve Not necessarily linear. Linear is mathematically easier to deal with.
Correlation coefficients • Show how good a fit you have. • R or R2 • For perfect correlation, R = 1, R2 = 1
LINEST • Calculates slope and intercept • Calculates the uncertainty in the slope and the intercept • Calculates R2 • Calculates s.d. of the population of y values • See page pp 68-72, Harris.
Use these values to determine the number of sig figs for the slope and intercept
Indeterminate Error • Repeating a coarse measurement gives the same result • eg weighing 50 g object to nearest g - only error would be determinate - such as there being a fault in the balance • If same object was weighed to several decimal places -get random errors
How many eggs in a dozen? • How wide is your desk? • Will everyone get the same answer? • What does this depend on?
With a few measurements, the mean won’t reflect the true mean as well as if you take a lot of measurements
Random errors • With many measurements, more will be close to the mean • Various little errors add in different ways • Some cancel - sometimes will all be one way • A plot of frequency versus value gives a bell curve or Gaussian curve or normal error curve • Errors in a chemical analysis will fit this curve
If z is abscissa (x axis) • Same curve is always obtained as z expresses the deviation from the mean in units of standard deviation
Statistics • Statistics apply to an infinite number of results • Often we only do an analysis 2 or 3 times and want to use the results to estimate the mean and the precision
Standard deviation • 68.3% of area is within ± 1 of mean • 95.5% of area is within ± 2 of mean • 99.7% of area is within ± 3 of mean • For any analysis, chances are 95.5 in 100 that error is ± 2 • Can say answer is within ± 2 with 95.5% confidence
For a large data set • Get a good estimate of the mean, • Know this formula -but use a calculator • 2 = variance • Useful because additive
Small set of data • Average (x ) • An extra uncertainty • The standard deviation calculated will differ for each small set of data used • It will be smaller than the value calculated over the larger set • Could call that a negative bias
s • For use N in denominator • For s use N-1 in denominator (we have one less degree of freedom - don’t know ) • At end, round s to 2 sig figs or less if there are not enough sig figs in data
Confidence Interval • We are doing an analysis to find the true mean - it is unknown • What we measure is x but it may not be the same as • Set a confidence limit eg 4.5 ± 0.3 g • The mean of the measurements was 4.5 g • The true mean is in the interval 4.2-4.8 with some specified degree of confidence
Confidence limit • A measure of the reliability (Re) • The reliability of a mean (x ) increases as more measurements are taken • Re = k(n)1/2 • Reliability increases with square root of number of measurements • Quickly reach a condition of limiting return
Reliability • Would you want a car that is 95% reliable? • How often would that break down?
Confidence Interval • For 100 % confidence - need a huge interval • Often use 95 % • The confidence level chosen can change with the reason for the analysis
Confidence Interval when s ~ s • µ ± xi = 1.96 for 95 % confidence • z = (xi - µ)/ =1.96 • Appropriate z values are given as a table • This applies to a single measurement • The confidence limit decreases as (N)1/2 as more measurements are taken
Confidence Interval • In the lab this year I will make you go home before you can get enough data for s to = • Therefore we will have to do a different kind of calculation to estimate the precision.
Student’s t-test The Student's t-Test was formulated by W. Gossett in the early 1900's. His employer (brewery) had regulations concerning trade secrets that prevented him from publishing his discovery, but in light of the importance of the t distribution, Gossett was allowed to publish under the pseudonym "Student". The t-Test is typically used to compare the means of two populations
t-test • t depends on desired confidence limit • degrees of freedom (N-1)
For practical purposes • Assume = s if you have made 20 measurements • Sometimes can be evaluated for a particular technique rather than for each sample • Usually too time consuming to do 20 replicate measurements on each sample