ANALYTICAL CHEMISTRY - CLUTCH 1E CH STATISTICS, QUALITY ASSURANCE AND CALIBRATION METHODS

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2 CONCEPT: MEAN EVALUATION The measures how close data results are in relation to the mean or average value. s = i (x i x) n 1 = Individual Measurement = Average or Mean = variance = Number of Measurements = Degrees of Freedom = Relative Standard Deviation (Coefficient of Variation) EXAMPLE: Data below gives the volumes obtained by a chemist from the use of a pipet. Determine the standard deviation. 4.9 ml, 5.0 ml, 4.8 ml, 4.6 ml, 4.6 ml, 4.3 ml Volume (xi) Difference from the mean (x i x) Difference from the mean squared (x i x) i (x i x) Page

3 CONCEPT: THE GAUSSIAN DISTRIBUTION Performing an experiment numerous times with no systematic error results in a smooth curve called the Gaussian Distribution. f(x) µ σ x f(x) = e = σ π e (x µ) /σ In terms of the Gaussian Distribution curve, increasing the number of measurements in the experiment: Changes the mean,, to mu, to represent the population mean. Changes the standard deviation,, to sigma,, to represent the population standard deviation. The shape of the Gaussian Distribution curve can occur by: Changing, which will shift the distribution curve to the left or right. Changing, which will increase or decrease the broadness of the distribution curve. Normally each distributed variable has its own mean and standard deviation. The standard normal distribution simplifies this by setting the mean at and standard deviation in units of. f(x) Standard Normal Distribution Formula y = e z / π z = Abscissa (Z-Score) Value mean Standard Deviation = X µ σ of the data falls between the -1 to +1 area Standard Deviation Z-Score Cumulative % σ %.3% 15.9% 50% 84.1% 97.7% 99.9% x of the data falls between the - to + area of the data falls between the -3 to +3 area Page 3

4 PRACTICE: THE GAUSSIAN DISTRIBUTION & Z-TABLES The use of Z-Tables is essential in the determination of probabilities. Probability -3 z 0 +3 Probability -3 0 z +3 Page 4

5 PRACTICE: THE GAUSSIAN DISTRIBUTION & Z-TABLES CALCULATIONS 1 EXAMPLE 1: Suppose there are 100 students in your analytical lecture and at the end of the semester the class average is an 80 with a standard deviation of 5.3, determine the distribution and probability of grades based on your understanding of the Gaussian distribution curve. f(x) x Standard Deviation EXAMPLE : From EXAMPLE 1, determine the percentage of final grades that would lie below 71. f(x) x Standard Deviation PRACTICE: From EXAMPLE 1, determine the percentage of final grades that would lie between 88 to 9. f(x) x Standard Deviation Page 5

6 CONCEPT: CONFIDENCE INTERVALS A confidence interval is a specific interval estimate of a parameter determined by using data obtained from a sample. For example a 95% confidence interval means we are 95% confident the mean lies within a given interval. = Student's t = standard deviation Confidence int erval = x ± ts n = # of measurements = average or mean A Student s t is a statistical table used in our understanding of confidence intervals and in the comparative data from different experiments. EXAMPLE: Construct a 95% confidence interval for an experiment that found the mean temperature for a given city in July as o C with a standard deviation of 1.8 from 10 measurements. Page 6

7 PRACTICE: CONFIDENCE INTERVALS CALCULATIONS 1 EXAMPLE 1: The barium content of a metal ore was analyzed several times by a percent composition process. Calculate the mean, median and mode EXAMPLE : From EXAMPLE 1, calculate the standard deviation. PRACTICE: From the examples given above, find the 90% confidence interval. Page 7

8 CONCEPT: T-TEST The t-test is used to test the of two populations, one of which could be a standard. In order to test the similarities and differences between these two populations you can utilize the t-score. Use the t score formula when we don t know the population standard deviation and have a sample size less than. t = x µ 0 s n = sample average = population average = sample standard deviation = number of samples The larger the t-score then the more the populations. The smaller the t-score then the more the populations. t-calculated (for equal variance) t Calculated = x1 x s pooled n s pooled = s 1 ( 1)+s (n 1) + n n 1 + n t-calculated (for unequal variance) Degrees of freedom = + n t calculated = x1 x s 1 + s n Degrees of freedom = s 1 + s n s 1 s 1 + n n 1 t-calculated (paired data) t Calculated = d s n s = Σ (d i d) n 1 Page 8

9 PRACTICE: T-TEST CALCULATIONS 1 EXAMPLE: A student wishing to calculate the amount of arsenic in cigarettes decides to run two separate methods in her analysis. The results (shown in ppm) are shown below: Sample Method 1 Method Is there a significant difference between the two analytical methods under a 95% confidence interval? Page 9

10 PRACTICE: T-TEST CALCULATIONS EXAMPLE: You want to determine if concentrations of hydrocarbons in seawater measured by fluorescence are significantly different than concentrations measured by a second method, specifically based on the use of gas chromatography/flame ionization detection (GC-FID). You measure the concentration of a certified standard reference material (100.0 µm) with both methods seven (n=7) times. Specifically, you first measure each sample by fluorescence, and then measure the same sample by GC-FID. The concentrations determined by the two methods are shown below. [fluorene (µm)] Sample Fluorescence GC-FID Calculate the appropriate t-statistic to compare the two sets of measurements. Page 10

11 PRACTICE: T-TEST CALCULATIONS 3 EXAMPLE: A sample of size n = 100 produced the sample mean of 16. Assuming the population deviation is 3, compute a 95% confidence interval for the population mean. PRACTICE: The average height of the US male is approximately 68 inches. What is the probability of selecting a group of males with average height of 7 inches or greater with a standard deviation of 5 inches? Probability -3 0 z +3 Page 11

12 CONCEPT: F-TEST The f-test is used to test the of two populations, which recall is equal to the standard deviation. FCalculated represents the quotient of the squares of the standard deviations: F Calculated = s 1 s When calculating the f quotient always set the larger standard deviation as the numerator so that F 1 If FCalculated FTable then the difference will not be significant. t Calculated = x1 x s pooled n s pooled = s 1 ( 1)+s (n 1) + n + n If FCalculated FTable then the difference will be significant. t calculated = x1 x s 1 + s n s 1 + s n s 1 s 1 + n n 1 Degrees of Degrees of Freedom for s 1 Freedom for s Page 1

13 PRACTICE: F-TEST CALCULATIONS 1 EXAMPLE 1: In the process of assessing responsibility for an oil spill, two possible suspects are identified. To differentiate between the two samples of oil, the ratio of the concentration for two polyaromatic hydrocarbons is measured using fluorescence spectroscopy. These values are then compared to the sample obtained from the body of water: Mean Standard Deviation # Samples Suspect Suspect Sample From the above results, should there be a concern that any combination of the standard deviation values demonstrates a significant difference? EXAMPLE : Can either (or both) of the suspects be eliminated based on the results of the analysis at the 99% confidence interval? Page 13

14 PRACTICE: F-TEST CALCULATIONS EXAMPLE 1: You are measuring the effects of a toxic compound on an enzyme. You expose five (test tubes of cells to 100 µl of a 5 ppm aqueous solution of the toxic compound and mark them as treated, and expose five test tubes of cells to an equal volume of only water and mark them as untreated. You then measure the enzyme activity of cells in each test tube; enzyme activity is in units of µmol/minute. The following are the measurements of enzyme activity: Activity (Treated) Activity (Untreated) Tube (µmol/min) Tube (µmol/min) Average: 3.84 Average: 6.15 Standard Standard Deviation: 0.36 Deviation: 0.9 Is the variance of the measured enzyme activity of cells exposed to the toxic compound equal to that of cells exposed to water alone? EXAMPLE : Is the average enzyme activity measured for cells exposed to the toxic compound significantly different (at 95% confidence level) than that measured for cells exposed to water alone? Page 14

15 CONCEPT: DETECTION OF GROSS ERRORS Grubbs test is used to detect a single outlier in a single variable data set that follows some type of normal distribution. Grubbs Test G Calculated = Questionable value x s Number G Table or G Critical of Observations (90% Confidence) (95% Confidence) (99% Confidence) G Table < G Calculated G Table > G Calculated Disregard Value Hold Value The Q-Test is another method used in finding outliers in very small, normally distributed, data sets. The number of measurements is normally between 3 to 7 values. Q-Test Q Calculated = Gap Range = x 1 x n+1 r x 1 = x n+1 = r = range (largest smallest value in data set) Q Table < Q Calculated Disregard Value Number Q Table or Q Critical of Observations (90% Confidence) (95% Confidence) (99% Confidence) Q Table > Q Calculated Retain Value Page 15

16 PRACTICE: DETECTION OF GROSS ERRORS CALCULATIONS 1 EXAMPLE 1: Wishing to measure the amount of caffeine in a cup of coffee you pour ten cups. From the data provided perform a Q-test to determine if the outlier can be retained or disregarded. Caffeine per cup of coffee Cup of Coffee ppm of coffee EXAMPLE : White blood cells are the defending cells of the human immune system and fight against infectious diseases. Provided below is the normal white blood cell counts for a healthy adult woman. Determine if the current white blood cell count is reasonable by Grubbs test. "Normal" Days Today cells /µl cells /µl cells /µl cells /µl cells /µl cells /µl cells /µl Page 16