1. Unit 3: Variation and the Normal Curve

2. Not normal doesn’t mean abnormal The “normal (Gaussian, bell-shaped) curve” is common, but NOT universal –salaries –small data sets Other common distributions are –uniform: all counts are the same (rolls of a die) –Poisson: life lengths, etc. Sums or averages of big samples from a population often have normal distributions –(We’ll see this later) –IQ’s: sums of many small mental properties –people heights: sums of many small bone lengths

3. Review: Standard Units “z-score” (“std units”): z = ( x –  x ) / σ –the number of σ’s above average –(if negative, below average) Ex: Data 3, 3, 5, 6, 7, 9:  x = 5.5 –differences: -2.5, -2.5, -.5,.5, 1.5, 3.5 –σ = RMS of differences ≈ 2.15 –z ≈ -1.17, -1.17, -.23,.23,.70, 1.63 NOT normally distributed

4. Ex: A list of 100 numbers, already in standard units, begins -5.8, -4.3, 6.1,.2, 10.2, -3.7. Is something wrong? They seem large -- remember, 3σ away from μ, which is ±3 in std units, is very rare Can we check? Well, μ = 0, σ = 1, so sum of their squares should be 1 = 100/100 But (-5.8) 2 + (4.3) 2 + (6.1) 2 +... is adding up to more than 100 fast In fact, (10.2) 2 alone is more than 100 So yes, they are too big to be in std units

5. Normal approximation Valid only IF you believe data is roughly normally distributed... and you need its avg and std dev Normal table in text gives percentages of data within z σ’s of μ, for normally distributed data in std units Switch to and from std units for other normally distributed data Don’t make an algorithm -- draw the bell curve(s) to decide what arithmetic to do

6. Normal table z Area(%) z Area(%) z Area(%) z Area(%) z Area(%) 0.0 0.0 0.9 63.19 1.8 92.81 2.7 99.31 3.6 99.968 0.05 3.99 0.95 65.79 1.85 93.57 2.75 99.4 3.65 99.974 0.1 7.97 1 68.27 1.9 94.26 2.8 99.49 3.7 99.978 0.15 11.92 1.05 70.63 1.95 94.88 2.85 99.56 3.75 99.982 0.2 15.85 1.1 72.87 2 95.45 2.9 99.63 3.8 99.986 0.25 19.74 1.15 74.99 2.05 95.96 2.95 99.68 3.85 99.988 0.3 23.58 1.2 76.99 2.1 96.43 3 99.73 3.9 99.99 0.35 27.37 1.25 78.87 2.15 96.84 3.05 99.771 3.95 99.992 0.4 31.08 1.3 80.64 2.2 97.22 3.1 99.806 4 99.9937 0.45 34.73 1.35 82.3 2.25 97.56 3.15 99.837 4.05 99.9949 0.5 38.29 1.4 83.85 2.3 97.86 3.2 99.863 4.1 99.9959 0.55 41.77 1.45 85.29 2.35 98.12 3.25 99.885 4.15 99.9967 0.6 45.15 1.5 86.64 2.4 98.36 3.3 99.903 4.2 99.9973 0.65 48.43 1.55 87.89 2.45 98.57 3.35 99.919 4.25 99.9979 0.7 51.61 1.6 89.04 2.5 98.76 3.4 99.933 4.3 99.9983 0.75 54.67 1.65 90.11 2.55 98.92 3.45 99.944 4.35 99.9986 0.8 57.63 1.7 91.09 2.6 99.07 3.5 99.953 4.4 99.9989 0.85 60.47 1.75 91.99 2.65 99.2 3.55 99.961 4.45 99.9991

7. Normal approx: Ex 1 Weights in the population of a city follow the normal curve, with  w = 140, σ = 30. About what % of pop weighs over 185? –In std units, 185 is (185-140)/30 = 1.5. Normal table says % > 1.5 or < -1.5 is (100- 86.64)% = 13.36%. We only want right half: 13.36%/2 = 6.68% –Much too “accurate”; this is only approximation: 6.7%, or even 7%

8. Normal approx: Ex 2 Scores on a college entrance exam follow normal curve (odd!), with  x = 120 and σ = 40. –(a) About what score is the 80th %ile? –(b) About what is the IQR? (a)In normal table, we need z that gives percent in center, not 80%, but (80 - (100- 80))% = 60%, which is z =.85. So 80th %ile of scores is [undoing std units] 120 +.85(40) ≈ 154 (b) We need z so that 50% of the data is between z and -z, and that’s z =.70. So the 3rd quartile is 120 + 40(.70), the 1st is 120 + 40(-.70), and their difference is the IQR, 2(40(.70)) = 56

9. Normal approx: Ex 3 Data following the normal curve has avg 80 and std dev 10. –(a) What is the 15th %ile? –(b) What is the 83rd %ile? –(c) What % of data is between 85 and 95? –(d) What % of data is between 60 and 90?

10. Measurement errors (Scientific) measurements are sum of –the true value (an ideal concept) –bias (not prejudgment, but a consistent error built into the method of measurement) e.g., a measuring tape that has stretched too far –chance error (the result of many tiny changes that affect the measuring process) e.g., how you stand, humidity, phase of moon, etc. In the NB10 ex in text, true value is 10g (by law), bias is about 400μg (consistent error), chance error is small differences from that

11. The terms might be grouped as... by an idealist: true value + error –Profs. Valente & Schult have a short note relating this to Plato’s metaphor of the cave by an experimenter: (consistent measurement) + (chance error) –because you can’t distinguish between true value and bias without changing the method of measurement

12. Foreshadowing... The fact that chance errors in measurement are (regarded as) normally distributed is the basis for the significance tests called z-tests that we’ll study at the end of the course.

13. Recall some elementary algebra “Plotting points”: Points in plane ↔ ordered pairs of real numbers Graph of y = mx + b is set of points whose pairs that satisfy it –It’s a line –(y-)intercept b is value where line hits y-axis (i.e., x=0) –m = slope = rise / run = (diff in y-values) / (diff in x-values) for any two points on line –Ex: Line through (1,4) and (3,2) has slope -1, y-intercept 5 Useful form: Point-slope form: –y - (y-value) = (slope) (x - (x-value)) –Ex: Equation of line through (2,3) and (5,7)