Error Analysis, Statistics, Graphing and Excel Necessary skills for Chem V01BL

Accuracy and Precision In a quantitative experiment we need to report how accurate and how precise our results are – Accuracy – is how close our value is to a reference (accepted value) - it is normally reported as the % error and measures systematic error – Precision – how repeatable is our result from trial to trial? It is reported in terms of the standard deviation this is a measure of random error

Accuracy and Precision Accurate but not precise Random error > systematic error Precise but not accurate Systematic error > random error

Error Analysis Percent Error (How accurate?) Good way to determine how good your experimental design/technique is, is by calculating the percent error Requires you measure some observable for which there is an accepted “true” value Accuracy: the % error measures closeness of agreement between a measured and true value. Accuracy is a measure of systematic errors (biases)

Error Analysis Mean Value In experiments random errors exist. When measuring some observable to assess these random errors we repeat measurements in independent trials We report the mean value We generally then report the % error for the mean value % error measures systematic error

Error Analysis Standard Deviation (Precision) It is important to measure how large the random errors are in an experiment We measure this using the standard deviation σ This normally requires at least 6 independent trials Can be evaluated for a column of data in Excel with the =STDEV(A1:A10) type syntax σ is a measure of precision. The smaller σ is, the more precise the measurement was.

Error Analysis Standard Deviation Continued In particle physics they use the standard of 5σ for the declaration of a discovery At this level there is only a chance of 1 in 2,000,000 that the value was arrived at was a random fluctuation about the true value

Error Analysis Standard Deviation Continued

Error Analysis Rejecting Outliers: Q Test In our experiments we often won’t have enough time to take at least 6 independent trials The Q test is a way of deciding if any given trial is an outlier The Q test tells us with 90%, 95% or 99% confidence that a trial point is in error and can be thrown out If so the average and standard deviation can be recalculated without that point

Error Analysis Q Test If Q > Q x% then the point is rejected at the x% confidence level N: Q 90% : Q 95% : Q 99% :

Error Analysis Q Test With the data as it stands the average is 0.255±0.057 Q 90% = 0.642, Q 95% = 0.710, and Q 99% = If Q > Q x% then the point is rejected at the x% confidence level Here Q = 0.8 which means it can be rejected at both the 90% and 95% level but not at the 99% level If we reject it the average becomes 0.230±0.031 Nxd^2xgapQ E av0.2546sum d^ range0.135 σ0.0565

Graphing and Excel Graphing is a widely used technique for determining mathematical relationships (correlations) between a dependent and independent variable To determine if two variables are correlated can be measured by calculating the coefficient of determination R 2 Here y(x i ) is the value of the independent value for the dependent value and f(x i ) is the function attempting model y=f(x i ) As we shall see if R 2 >0.99 data fits the model We will routinely use Excel to plot two variables which are linearly correlated and use Excel to determine the equation for the line The slope of that line, and the intercept will both correspond to scientific observables that are of considerable value to us

Graphing and Excel Correlated data: ideal gas law Find the relationship between the volume of 1 mole of O 2 and its temperature at 1 atmosphere The equation for the straight line where y = V and x = T gives The R 2 =1 means there is a perfect linear correlation between T and V V(L)T( o C)

Graphing and Excel Correlated data: ideal gas law Find the relationship between the volume of 1 mole of O 2 and its Pressure at 273K. P is the independent variable (x) and Volume the dependent variable (y) P(torr)V(mL) Black line is a linear trendline y = mx + b Orange line is a power trendline y = mx b

Graphing and Excel Correlated data: ideal gas law Find the relationship between the volume of 1 mole of O 2 and its Pressure at 273K. P is the independent variable (x) and Volume the dependent variable (y) When we plot P vs 1/V R 2 = , when we plot P vs V R 2 = So P(torr)1/V(mL)

Graphing and Excel Uncorrelated data House value versus Street Address

Graphing and Excel Often data from a trendline fit is used to obtain certain observables. With any experimentally measured quantity we need to estimate the random error In kinetics σ Ea and σ ln(A) are needed! can be calculated with LINEST function in Excel

Summary using LINEST In a free cell type =LINEST(Y Values,X Values,TRUE,TRUE) enter Select 2 columns x 5 rows with the cell which you types LINEST in being the top left In the formula window hit Command Enter on a mac In the formula window hit Control Shift Enter on windows Slope intercept σ slope σ intercept R 2 s(y) s(y) error bar in y F degrees freedom (terms calculating R) σ reg σ residual (terms calculating R)

Tonight Answer the problem set on pages Each student must individually make a graphs of V vs T, V vs P, 1/V vs P. Each graph curve should be fitted using a linear trendline where the equation and R 2 value is shown. Use the graphing procedure given on page 9 For question 4 calculate the standard deviations for the slope and the intercept and attempt to put error bars using dy