1. Probability Density Function (pdf) continuous distributions 1f(x) >= 0 all x 2f(x) is the “likelihood” of x 3integral under f(x) is exactly 1.0

2. 2F(x)=Pr(x<=X) probability X is <= x yarea under pdf from -inf up to x Cumulative Distribution Function (cdf)

3. FREQUENCY TABULATION *IntervalFreq Rel Cum * Freq Prob *40-4422.11.11 *44-4846.23.34 *48-5274.37.71 *52-5652.26.97 *56-606.031.00

4. HISTOGRAM DISTRIBUTION ~ FREQUENCY TABULATION

5. CDF for HISTOGRAM DISTRIBUTION

6. MEDIAN, 1st & 3rd QUARTILE from CDF for HISTOGRAM Q1MQ3

7. INTERPOLATION FORMULA for FRACTILE given cumulative probability x k = left end-point of interval x k+1 = right end-point of interval p k = cum prob at left end-point of interval p k+1 = cum prob at right end-point of interval p = given cum prob between p k and p k+1

8. INTERPOLATION FORMULA for cumulative probability given x x k = left end-point of interval x k+1 = right end-point of interval p k = cum prob at left end-point of interval p k+1 = cum prob at right end-point of interval x = given variable value between x k and x k+1

9. MEAN VALUE FORMULA for HISTOGRAM DISTRIBUTION zInterval Mid-points ym k = (x k + x k+1 )/2 zInterval Probabilities yp k = P k+1 - P k zMean Value yµ=Sum of p k * m k =

10. VARIANCE FORMULA for HISTOGRAM DISTRIBUTION zAVERAGE of x 2 for each interval ym i 2 = (x i 2 + x i x i-1 + x i-1 2 )/3 zSECOND MOMENT ySum of p i m i 2 zVARIANCE yσ 2 =Second Moment - (mean)^2 zSTANDARD DEVIATION ysqrt(Variance)

11. Example Computations zMEAN =.11*42 +.23*46 +.37*50 +.26*54 +.03*58 = 49.48 zSECOND MOMENT zVARIANCE