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

Accuracy and Precision In experiments there are errors. We measure how big these errors are with 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 is. 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 Mean Value Repeat measurements xi in N independent trials Report the mean value This equation adds up all the values and divide the sum by the number of values Reduces random errors from multiple trials

Measuring Accuracy Percent Error Accuracy: the % error measures closeness of agreement between a measured and true value. Accuracy is a measure of systematic error (biases)

Measuring Precision Standard Deviation s σ is a measure of precision. The smaller σ is, the more precise the measurement was This normally requires at least 6 independent trials Can be evaluated for a column of data in Excel® (or Numbers®) with the =STDEV(A1:A10) type syntax

Precision Continued If an observable is subject to random error, we get a binomial (Gaussian) distribution about the mean value. s is a measure of the width of the distribution, and therefore the magnitude of the random error Reporting 𝑥±𝜎 describes the dark blue region (68.2% or the trials fell in this range) Reporting 𝑥±2𝜎 include the next shade of blue as well and spans 95.4% of all trials. In Science we generally report answer in this way. 68.2% of trials 95.4% of trials We use this

Random Error Standard Deviation Continued More precise

Rejecting Outliers: Q Test 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

Rejecting Outliers: Q Test If Q > Qx% then the point is rejected at the x% confidence level

Rejecting Outliers: Q Test x gap Q 0.219 0.004 0.0296 0.223 0.008 0.0593 0.231 0.015 0.111 0.246 0.354 0.108 0.8 average 0.255 0.135 range σ = 0.057 𝑄= 0.354−0.246 0.354−0.219 = 0.108 0.135 =0.8 The average is 0.255±0.057 (68% confidence level) Q90% = 0.642, Q95% = 0.710, and Q99% = 0.821 If Q > Qx% then the point is rejected at the x% confidence level Q is largest for the last point (Q = 0.8 > Q95%) which means it can be rejected at the 95% level If we reject the last value, the average becomes 0.230±0.031

Graphing and Excel Graphing is used to see mathematical relationships (correlations) between a dependent and independent variable The correlation is measured by calculating the coefficient of determination R2 𝑅 2 =1− 𝑖=1 𝑁 𝑦 𝑥 𝑖 −𝑓 𝑥 𝑖 2 𝑖=1 𝑁 𝑦 𝑥 𝑖 − 𝑦 2 y(xi) is the value of the independent value for the dependent value and f(xi) is the function attempting model y(xi)=f(xi) Use Excel to plot two variables which are linearly correlated and use Excel to determine the equation for the line as well as R2 As we shall see if R2>0.99 data fits the model

Graphing and Excel Correlated data: ideal gas law Find the relationship between the volume of 1 mole of O2 and its temperature at 1 atmosphere The equation for the straight line where y = V and x = T gives The R2=1 means there is a perfect linear correlation between T and V V(L) T(oC) 25 31.49 30 92.38 35 153.28 40 214.18 45 275.08 50 335.97

Graphing and Excel Correlated data: ideal gas law Find the relationship between the volume of 1 mole of O2 and its Pressure at 273K. P is the independent variable (x) and Volume the dependent variable (y) P(torr) V(mL) 400 42.6 500 34.1 600 28.4 700 24.3 800 21.3 900 18.9 1000 17 1100 15.5 1200 14.2 𝑉 𝑚𝐿 = 17119 (𝑇𝑜𝑟𝑟.𝑚𝐿) 𝑃(𝑇𝑜𝑟𝑟) Black line is a linear trendline y = mx + b Orange line is a power trendline y = mxb

Graphing and Excel Uncorrelated data House value versus Street Address Number

Group Activity Pair up with your neighbor Measure the mass of an empty cylinder and record it on the sheet given (Part A) Add 10 mL of DI water and re-measure the mass of the cylinder on the sheet (Part A) Measure the temperature of the water to 0.1oC and record on the sheet Look up and record the density for water at that temperature Calculate the mass of the 10 mL of water (Part A) Record your answer on the board Copy all the board data onto your sheet in Part B Calculate the volume for each trial from the board Calculate the mean volume and the standard deviation Complete the sheet