1. Jan. 20-23 Shapes of distributions… “Statistics” for one quantitative variable… Mean and median Percentiles Standard deviations Transforming data… Rescale: Y = c times X Recenter: Y = X plus a adding variables to each other other transformations

2. Shape of a distribution… Outliers Unimodal --- Bimodal --- Multimodal Symmetrical Skew - right or left?

3. Colleges – Datadesk histogram

4. GE daily changes ($/share)

5. NH polls, 1/26/04 - errors

6. Population vs. Sample

7. A statistic is anything that can be computed from data.

8. STATISTICS of a single quantitative variable MEAN MEDIAN QUARTILES ( Q1, Q3 ) Five-number summary Boxplots Interquartile range PERCENTILES / QUANTILES / FRACTILES (“quantiles” and “fractiles” are synonyms for “percentiles” for people who don’t like the implied multiplication by 100) STANDARD DEVIATION VARIANCE

9. Statistics of one variable… MEAN — Sum of values, divided by n MEDIAN — Middle value (when values are ranked, smallest to largest) (or, average of two middle values)

10. Number of Colleges (ranked) 1126812 11468 11568 1156813 1156814 11578 11579 11579 115710 1157 1157 1167 126812

11. Colleges – Datadesk histogram median — 5 mean — 5.36

12. Salaries 200005000080000 300005000080000 300005000080000 300005000080000 300005000085000 300005000090000 3000060000100000 3000060000100000 3500060000100000 3500060000120000 4000060000125000 4000060000150000 4000065000150000 4000070000150000 4000070000200000 4500070000250000 4500072500400000 5000075000500000 5000075000600000 50000750001000000

13. salaries median — 60,000 mean — 106,875

14. So, which measure of “center” is best? All the measures agree (roughly) when the distribution is symmetrical Mean has attractive mathematical properties Also, the mean is related to the total, if that’s what you care about Median may be more “typical” when the distribution is non- symmetrical A measure is “robust” if it works reasonably well under a wide variety of circumstances Medians are robust

15. Jan. 23 RMS, Geometric mean Percentiles, Quartiles (Q1, Q3), BOX PLOTS Measures of spread: IQR (range containing middle half) Standard deviation ( , s ) Variance Transforming data… Rescale: Y = c times X Recenter: Y = X plus a adding variables to each other other transformations “STANDARDIZING” a variable NORMAL DISTRIBUTIONS

16. Computing percentiles To calculate 20-th percentile: Rank the values from smallest to largest Compute 20% of n… 20% of 72 = 14.4 Count off that many values (from lowest)… The value at which you stop is the 20-th percentile. What if you stop between values ?

17. Number of Colleges 1126812 11468 11568 1156813 1156814 11578 11579 11579 115710 1157 1157 1167 126812

18. QUARTILES Lower quartile (Q1) = 25-th percentile Upper quartile (Q3) = 75-th percentile ( What’s Q2 ? ) INTERQUARTILE RANGE ( IQR ) = Q3 minus Q1

19. Five-number summary — maximum — Q3 — median — Q1 — minimum

20. VARIANCE and STANDARD DEVIATION VARIANCE (s 2 ): STANDARD DEVIATION (s):

21. Linear Transformations If you MULTIPLY or DIVIDE a variable by a constant… Y = c times X Y = X / c then… measures of center are multiplied or divided by c measures of spread are multiplied or divided by |c| If you ADD or SUBTRACT a constant from a variable… Y = X + aY = X – a then… measures of center are increased (decreased) by a measures of spread are UNCHANGED.

22. More transformations ADDING VARIABLES: W = X + Y Mean (W) = Mean (X) + Mean (Y) Standard Deviation of (W) — anything can happen OTHER TRANSFORMATIONS: Y = X squared ? Y = log (X) ? …NO RELIABLE RULES for mean or std. dev.

23. Standardized Variables Write and S for mean, standard deviation of X Then form transformed variable: Z = (X - ) / S Then… mean (Z) = 0 std dev (Z) = 1 Z answers the question: How many standard deviations is this value above (or below) the mean?

24. Jan. 25 More on transforming and standardizing variables More on normal distributions Jan. 27++ Relations among variables --- scatterplots “independent” variables correlations linear regressions (best fit lines)

25. Normal Density Function X ~  ( ,  )  = mean,  = std. dev. (Why Greek? Why not x-bar, s?)  

26. Trying the integral Standard normal: mean = 0, std. dev. = 1 Density curve: …so the area between a and b is: 0 1

27. The core computation If X ~ N( ,  ), what fraction of values are between a and b ? Rule of 68 – 95 – 99.7 Standardizing Tables and computers Reversing the calculation a b

28. Standardizing Same Question: Is X between a and b ? Is (X-  )/  between (b-  )/  and (b-  )/  ? But Z = (X-  )/  is a variable with a standard normal distribution (mean 0, standard deviation 1). So, if we can answer this question for standard normals, we can answer it for all normals.