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# Chapter 04

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### Chapter 04

1. 1. Chapter 4 Statistics
2. 2. " normal" days Today s count 5.1Is my red blood cell 5.3 count high today?  4.8 ×106 cells 5.6 ×106 cells μL μL 5.4   5.2
3. 3. 4.1 The Gaussian Distributions -1• Nerve cells  muscle cells (1991 Nobel Prize in Medicine & Physiology) Sakmann & Neher absence neurotransmitter present neurotransmitter
4. 4. 4.1 The Gaussian Distributions -2 922 ion channels response Typical lab measurements: Gaussian distribution 1 − ( x −μ ) y= e 2σ 2 σ 2π
5. 5. 4.1 The Gaussian Distributions -3Gaussian distribution is characterized by3) Mean: x( μ) ∑x 1 i x= i = ( x1 + x 2 +  + x n ) n n7) Standard deviation: S( σ ) ∑ (x ) 2 i −x S= i n −1
6. 6. 4.1 The Gaussian Distributions -5 The smaller the s , ⇒ the more precise the results  reproducible ( )  x, S for a finite set.  ( μ, σ ) for an infinite set.
7. 7. 4.1 The Gaussian Distributions -4 Other terms • Median • Range σ & probability • Table 4.1
8. 8. 4.2 Student’s t -1Student’s t is the statistical tool used to expressconfidence intervals & to compare results fromdifferent experiments.confidence interval: allows us to estimate therange within which the true value (µ) might fall,(given probability = confidence level) defined bymean and standard deviation. ts Confindence interval : μ = x ± n
9. 9. 4.2 Student’s t -3(ex) In replicate analyses, the carbohydrate content of a glycoprotein (a protein with sugars attached to it) is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohydrate per 100 g of protein. Find the 50 % and 90% confidence intervals for the carbohydrate content.
10. 10. 4.2 Student’s t -4 ts μ ( 50% ) = x ± n ts μ ( 90% ) = x ± n
11. 11. 4.2 Student’s t -5 Smaller confidence intervals Better measurementFor 90% sure that a quantity lies in the range 62.3 ± 0.5 vs. 62.3 ± 1.3
12. 12. 4.2 Student’s t -6 tsμ=x± n* improving the reliability of yourmeasurement(1) make more measurements ( n ↑) ∝ 1 n(2) improve expt. procedure ( ↓ S)
13. 13. 4.2 Student’s t -7t test : used to compare one set of measurements with another to decide whether or not they are different.Three ways in which a t test can be used will be described.
14. 14. 4.2 Student’s t -8Case 1 : a. comparing a measured result with a “known” value Sample: 3.19 wt% (known value) a new analytical method : 3.29, 3.22, 3.30, 3.23 wt% X = 3.260 S = 0.041
15. 15. 4.2 Student’s t -9Does answer agree with the known answer ? known value − x t calculate = n s 3.19 − 3.26 = 4 = 3.41 0.041 95% confidence tcalculate > ttable ⇒ result is different from the known value.
16. 16. 4.2 Student’s t -10 Case 2 • comparing replicate measurements. 1904 Nobel Prize by Lord Rayleigh. for discovering Inert gas argon :
17. 17. 4.2 Student’s t -11 1 N2 OCu (s) + O 2 → CuO (s)  ← NO 2 NH NO  4 2
18. 18. 4.2 Student’s t -12t Test for comparison of means : x1 − x 2 n1n 2 t= s pooled n1 + n 2 s1 ( n1 − 1) + s 2 ( n 2 − 1) 2 where s pooled = 2 n1 + n 2 − 2
19. 19. 4.2 Student’s t -13Case 3• Comparing individual differences Cholesterol content (g/L) Sample Method A Method B Different (di) 1 1.46 1.42 0.04 2 2.22 2.38 -0.16 3 2.84 2.67 0.17 4 1.97 1.80 0.17 5 1.13 1.09 0.04 6 2.35 2.25 0.10 d = 0.060
20. 20. 4.2 Student’s t -14 d t calculate = n sd ∑ (d ) 2 i −d sd = n −1 = ( 0.04 − 0.06 ) 2 + ( − 0.16 − 0.06 ) 2 = 0.12 2 6 −1 0.06 0 ∴ t calculate = 6 = 1.20 t cal < t table 0.12 2∴ two techniques are not significant different at the 95% confidence level
21. 21. 4.3 Q test for bad data -1help decide whether to retain or discard a datum gap Q test for discarding : Q = range
22. 22. 4.3 Q test for bad data -2 Qcalculate > Qt discard any datum from a faulty procedure.
23. 23. 4.4 Finding the “Best” straight line -1 calibration methods  prepare calibration curve.
24. 24. 4.4 Finding the “Best” straight line -2Mrthod of least square y = mx + b di = y i - y = y i - (mx + b) ( + or -) di2 = (y i − mx − b) 2 (postive only) n∑ ( x i y i ) − ∑ x i ∑ y iLeast - squares slope : m = DLeast - squares intercept : b= ∑ ( x )∑ y − ∑ ( x y ) ∑ x 2 i i i i i Dwhere the denominato r, D, is given by ( ) D = n∑ x − ( ∑ x i ) 2 i 2
25. 25. 4.5 Constructing a Calibration Curve -1 1) Blank standard soln Spectrophotometer readings for protein analysis by the Table 4-6 Lowry method Sample Absorbance of three Corrected absorbance Range ( after subtracting average (μg) independent samples blank ) blank 0 0.099 0.099 0.100 0.001 -0.0003 -0.0003 0.0007 5 0.185 0.187 0.188 0.003 0.0857 0.0877 0.0887Standard 10 0.282 0.272 0.272 0.010 0.1827 0.1727 0.1727 soln 15 0.392 0.345 0.347 0.047 --- 0.2457 0.2477 20 0.425 0.425 0.430 0.005 0.3257 0.3527 0.3307 25 0.483 0.488 0.496 0.013 0.3837 0.3887 0.3967
26. 26. 4.5 Constructing a Calibration Curve -2 m =  b = 1) Finding the protein in an unknown