Data Analysis II. CU- Boulder CHEM-4181 Instrumental Analysis Laboratory. Prof. Jose-Luis Jimenez Spring 2007

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1 Data Analysis II CU- Boulder CHEM-48 Instrumental Analysis Laboratory Prof. Jose-Luis Jimenez Spring 007 Lecture will be posted on course web page based on lab manual, Skoog, web links Summary of Last Lecture Treat data in your lab reports and student choice ep. in a professional way Topics covered in lecture I Significant figures Precision vs. accuracy Errors E a and RE Gross errors -> outliers Random errors Treat with statistics (Gaussian distribution) Systematic errors Identify and get rid of them Today: treatment of random errors & Ecel 3

2 Population and Sample Mean Sample Mean ( ) Average of a finite set of data In general not the same as µ, because of finite error AVERAGE() in Ecel i= = i Population Mean (µ) Also limiting mean It is the true value of the quantity being measured µ i= = lim i 4 Standard Deviation and Variance I Population Standard Deviation (σ) Measure of the precision of a population of data STDEVP() in Ecel Variance (σ ) Std. dev. has same units of, variance as units of Variance from different effects is often additive σ = σ +σ +σ 3 + Std. Dev. is not! VAR() in Ecel σ (IDEPEDET effects) i= = lim ( µ ) i 5

3 Standard Deviation and Variance II Sample Standard Deviation (s) s instead of σ instead of µ (-) instead of umber of degrees of freedom, ν = - Because is used in the calculation, only - values are independent, the last one can be calculated from the mean and the other values STDEV() in Ecel s = i= ( i ) 6 RSD and CV Relative Standard Deviation (RSD) Often more informative than absolute SDs s z RSD = 0 z = => percent z = 3 => ppth Coefficient of Variation (CV) RSD epressed as a percent CV s = 0 7

4 Standard Error of the Mean Standard deviation estimate of the probable error of a single measurement Standard error of the mean Estimate of the probable error of the mean of measurements σ s σ m = s m = More generally The mean of measurements has a distribution (µ,σ m ) This is true in the limit even if error is OT Gaussian Central limit theorem of probability 8 Ecel Tutorial Part 9

5 Entering Data In Ecel data are entered in cells Cells can be empty or contain data or formulas Every cell has coordinates A, B, etc. Absolute coordinates: $A or A$ or $A$ Very important distinctions! CQ: Do you know how to use relative and absolute references in Ecel? A. yes B. a little C. no Demo: pasting a series of data Useful to create regularly spaced data 0 Formulas and Equations I umerical operators: Task Multiplication Division Eponent Order of Operations Power of ten Operator * / ^ (..) e or E Eample *3 4/ ^3 *3+5 or *(3+5) 3.e+ or 3.e- CQ: 0e4 in computer notation equals: A.,000 B. 0,000 C. 00,000 D. either E. I don t know Result 6 8 or 6 30 or 0.03

6 Formulas and Equations II Enter a formula which is calculated based on other cells Drag or Copy / Paste ote that the result is different if you use absolute or relative references Pre-Programmed Functions Ecel has lots of pre-programmed functions E.g. normal distribution Click f symbol to get a menu Also look up the help files 3

7 Useful Pre-Programmed Functions AVERAGE() STDDEV() STDEVP() MEDIA() MAX() MI() VAR() ORMDIST() ORMDISTIV() TDIST() TTEST() CQ: I have used A. all of these B. most of these C. a few of these D. none of these E. what are these? These are only a few of the statistical functions, there are lots more! 4 Plotting in Ecel Select data Choose Insert -> Chart 5

8 Proper Graph Formatting for Reports Label aes w/ Units (AU or arb. Units if needed) Independent variable on X-ais Dependent variable on Y-ais. Scatter (not line) plot Add a regression line (if appropriate) Descriptive title Remember Sigfigs! Do not plot too many data sets on single graph - make multiple graphs instead 6 The ormal Error Curve Random errors are often distributed according to the normal error law Probability Density ormal Probability Density Function Absorbance (unitless) CQ: the probability that the absorbance is eactly 0.49 is: A B. 5 C. infinity D. zero E. I don t know 7

9 The ormal Error Law I The fraction of a population of observations whose values are between and +d is: d = σ I.e. the probability that an observation is between and +d is: ( µ ) P d e / σ (, + ) = d σ π Probability density: ( µ ) / σ e π d P( ) = e σ π ( µ ) / σ 8 The ormal Error Law II Cumulative probability The probability that has a value between and is: ( µ ) = / σ P(, ) e d σ π ormalized distribution: In Ecel Can type the whole formula (prone to errors) ORMDIST(,µ,σ,FALSE) for PDF E.g. =ORMDIST(3,,0.3,FALSE) ORMDIST(,µ,σ,TRUE) for PDF z = µ σ 9

10 PDF vs CDF Probability Density vs. Cumulative Probability Probability Density ormal Probability Density Function Cumulative Probability ormal Cumulative Probability Function Absorbance (unitless) Absorbance (unitless) CQ: the probability that the absorbance is between 0.48 and 0.49 is: A. zero B. It is not defined C D E. I don t know 0 ormal Error Curves With units: ormalized:

11 Areas under Gaussian Curve Also know that +/- σ has 67% Q: calculation w/ Ecel? Linear Regression in Ecel I Easy way (in graph) More comple way & more information (Analysis ToolPak) Eample: calibration curve for fluorescence Input data from table: Fluorescence Intensities Concentration (pg/ml)

12 Linear Regression in Ecel II Right-Click on data point: Steps & 3: 4