1. Jan. 29 “Statistics” for one quantitative variable… Mean and standard deviation (last week!) “Robust” measures of location (median and its friends) Quartiles, IQR, five-number summary, Box plots Percentiles Transforming data… Rescale: Y = c times X Recenter: Y = X plus a other transformations adding variables to each other Standardizing data…

2. Population vs. Sample

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

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

5. STATISTICS of a single quantitative variable MEAN MEDIAN QUARTILES ( Q1, Q3 ) Five-number summary Boxplots Interquartile range PERCENTILES / QUANTILES / FRACTILES STANDARD DEVIATION VARIANCE

6. Statistics of one variable… Median --- middle value (when values are ranked, smallest to largest) (or, average of two middle values) “Robust” Trimmed mean Midmean Geometric mean “RMS mean”

7. Number of Colleges 1212121 119115 178611 1015578 161 41 1 577 1514816 11581 1156 6751314 125718 1 69 87186

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

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

10. Mean vs. Median Large tails affect the mean more than the median. So: Right-skewed distribution  Mean right of median Left-skewed distribution  Mean left of median

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

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

13. 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

14. 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 ?

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

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

17. Five-number summary — maximum (or, say, 95 %ile) — Q3 — median — Q1 — minimum (or, say, 5 %ile)

18. 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.

19. 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.

20. 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?