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SIMS Quant Course Lecture 4

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Rashmi Sinha
Rashmi Sinha

The document discusses key concepts in statistics like histograms, mean, median, standard deviation, skewness, and kurtosis. Several histograms of data distributions are presented with calculations of their statistical properties. Key points made are that the mean and standard deviation best summarize normal distributions, while other measures like median and interquartile range are better for skewed distributions.

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Those who don’t know statistics are condemned to reinvent it… David Freedman
All you ever wanted to know about the histogram and more ...
Distribution of No of Graphics on web pages (N=1873) Mean = 17.93 N = 1873 Graphic Count Std. Dev = 17.92 Median = 16.00 1 95.0 90.0 85.0 80.0 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 400 300 200 100 0
Horizontal Scale 2
Distribution of Redundant Link % on web pages (N =1861) Std. Dev = 37.33 Mean = 22.1 N = 1861.00 Median = 14 3 480.0 440.0 400.0 360.0 320.0 280.0 240.0 200.0 160.0 120.0 80.0 40.0 0.0 1000 800 600 400 200 0
Plotting a histogram: endpoint convention, plot frequencies, make equal intervals etc.
Frequency Table convention: include the left endpoint in the class interval 4
Frequency/Probability
No of fonts used on a web-page Frequency /probability 5 0/ 0 200/ .1 400/ .2 600/ .3 800/ .4 1000/ .5 Frequency 110 430 860 280 180 40 20 10 1 3 5 7 9 11 13 15 Probability .06 .22 .45 .15 .09 .02 .01 .01
Cleaning up a histogram: getting rid of outliers
Distribution of word count (N=1903) Std. Dev = 725.24 Mean = 393.2 Maximum = 20,357 Minimum = 0 Median = 223 20000.0 18000.0 16000.0 14000.0 12000.0 10000.0 8000.0 6000.0 4000.0 2000.0 0.0 1600 1400 1200 1000 800 600 400 200 0
Distribution of word count (N=1897) top six removed Std. Dev = 474.04 Mean = 368.0 Maximum = 4132 Minimum = 0 Median = 223 7 4000.0 3600.0 3200.0 2800.0 2400.0 2000.0 1600.0 1200.0 800.0 400.0 0.0 800 600 400 200 0
Distribution of word count (N=1873) Std. Dev = 360.30 Mean = 333.4 Maximum = 4132 Minimum = 0 WORDCNT2 Median = 220 2400.0 2200.0 2000.0 1800.0 1600.0 1400.0 1200.0 1000.0 800.0 600.0 400.0 200.0 0.0 500 400 300 200 100 0
What can histograms tell you
Distribution of link count on good & bad web-pages Good Sites Bad Sites 8 2 8 0 . 0 2 4 0 . 0 2 0 0 . 0 1 6 0 . 0 1 2 0 . 0 8 0 . 0 4 0 . 0 0 . 0 3 0 0 2 0 0 1 0 0 0
Making inferences from histograms: Incidence of riots and temperature 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 temperature 9
Mean and Median ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Mean is arithmetic average, median is 50% point Mean is point where graph balances
The instability of means and standard deviations
Add two numbers: watch the mean, median, & SD
Add one outlier...
Standard Deviation: a measure of spread
Same mean, different spread S D S D 10
The Standard Deviation
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Understanding the standard deviation ,[object Object],0% 25% 50% Histogram is symmetric about 2, 2 is mean, and 50% to left of 2, 50% to right
[object Object],[object Object],[object Object],0% 25% 50% 0% 25% 50% List: 1, 2, 2, 5 Average =2.5 SD = 1.73 0% 25% 50% List: 1, 2, 2, 7 Average =3 SD = 2.71
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Computing the standard deviation
Properties of the standard deviation ,[object Object],[object Object],[object Object]
Properties of the Normal Probability Curve ,[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 SD= 68% 2 SD = 95% 3 SD= 99.7% 11
Distribution of judges ratings for the Webby Awards Std. Dev = 1.98 Mean = 6.3 N = 1867.00 Skewness = -.43 Kurtosis = -.201 Median = 6.3 12 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 500 400 300 200 100 0
It is a remarkable fact that many histograms in real life tend to follow the Normal Curve. For such histograms, the mean and SD are good summary statistics . The average pins down the center, while the SD gives the spread. For histogram which do not follow the normal Curve, the mean and SD are not good summary statistics. What when the histogram is not normal ...
+- 3 SD = (384 * 3) = 1152 Mean - 1152 = about 30% sample had negative number of links Mean = 348.3 Std. Dev = 384.83 Distribution of word count on web pages 13 2800.0 2600.0 2400.0 2200.0 2000.0 1800.0 1600.0 1400.0 1200.0 1000.0 800.0 600.0 400.0 200.0 0.0 500 400 300 200 100 0
Note. A percentile is a score below which a certain % of sample is When SD is influenced by outliers Use inter quartile range 75th percentile - 25th percentile
Measures of Normality ,[object Object],[object Object],Positively Skewed Negatively Skewed Symmetric 14
Kurtosis: Does it cluster in the middle? Large tail Small tail Normal Tail ,[object Object],[object Object],[object Object],[object Object],15
Positively Skewed and Leptokurtic: Word Count Std. Dev = 725.24 Mean = 393.2 N = 1903.00 Kurtosis = 321.84 Skewness = 13.62 Median = 223 20000.0 18000.0 16000.0 14000.0 12000.0 10000.0 8000.0 6000.0 4000.0 2000.0 0.0 1600 1400 1200 1000 800 600 400 200 0
Distribution of word count (N=1897) top six removed Std. Dev = 474.04 Mean = 368.0 N = 1897.00 Skewness = 3.49 Kurtosis = 16.40 Median = 223 4000.0 3600.0 3200.0 2800.0 2400.0 2000.0 1600.0 1200.0 800.0 400.0 0.0 800 600 400 200 0
Degree of Freedom ,[object Object],[object Object],[object Object],[object Object]

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SIMS Quant Course Lecture 4

  • 1. Those who don’t know statistics are condemned to reinvent it… David Freedman
  • 2. All you ever wanted to know about the histogram and more ...
  • 3. Distribution of No of Graphics on web pages (N=1873) Mean = 17.93 N = 1873 Graphic Count Std. Dev = 17.92 Median = 16.00 1 95.0 90.0 85.0 80.0 75.0 70.0 65.0 60.0 55.0 50.0 45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 400 300 200 100 0
  • 5. Distribution of Redundant Link % on web pages (N =1861) Std. Dev = 37.33 Mean = 22.1 N = 1861.00 Median = 14 3 480.0 440.0 400.0 360.0 320.0 280.0 240.0 200.0 160.0 120.0 80.0 40.0 0.0 1000 800 600 400 200 0
  • 6. Plotting a histogram: endpoint convention, plot frequencies, make equal intervals etc.
  • 7. Frequency Table convention: include the left endpoint in the class interval 4
  • 9. No of fonts used on a web-page Frequency /probability 5 0/ 0 200/ .1 400/ .2 600/ .3 800/ .4 1000/ .5 Frequency 110 430 860 280 180 40 20 10 1 3 5 7 9 11 13 15 Probability .06 .22 .45 .15 .09 .02 .01 .01
  • 10. Cleaning up a histogram: getting rid of outliers
  • 11. Distribution of word count (N=1903) Std. Dev = 725.24 Mean = 393.2 Maximum = 20,357 Minimum = 0 Median = 223 20000.0 18000.0 16000.0 14000.0 12000.0 10000.0 8000.0 6000.0 4000.0 2000.0 0.0 1600 1400 1200 1000 800 600 400 200 0
  • 12. Distribution of word count (N=1897) top six removed Std. Dev = 474.04 Mean = 368.0 Maximum = 4132 Minimum = 0 Median = 223 7 4000.0 3600.0 3200.0 2800.0 2400.0 2000.0 1600.0 1200.0 800.0 400.0 0.0 800 600 400 200 0
  • 13. Distribution of word count (N=1873) Std. Dev = 360.30 Mean = 333.4 Maximum = 4132 Minimum = 0 WORDCNT2 Median = 220 2400.0 2200.0 2000.0 1800.0 1600.0 1400.0 1200.0 1000.0 800.0 600.0 400.0 200.0 0.0 500 400 300 200 100 0
  • 15. Distribution of link count on good & bad web-pages Good Sites Bad Sites 8 2 8 0 . 0 2 4 0 . 0 2 0 0 . 0 1 6 0 . 0 1 2 0 . 0 8 0 . 0 4 0 . 0 0 . 0 3 0 0 2 0 0 1 0 0 0
  • 16. Making inferences from histograms: Incidence of riots and temperature 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 temperature 9
  • 17.
  • 18. The instability of means and standard deviations
  • 19. Add two numbers: watch the mean, median, & SD
  • 21. Standard Deviation: a measure of spread
  • 22. Same mean, different spread S D S D 10
  • 24.
  • 25.
  • 26.
  • 27.
  • 28.
  • 29.
  • 30.
  • 31. Distribution of judges ratings for the Webby Awards Std. Dev = 1.98 Mean = 6.3 N = 1867.00 Skewness = -.43 Kurtosis = -.201 Median = 6.3 12 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 500 400 300 200 100 0
  • 32. It is a remarkable fact that many histograms in real life tend to follow the Normal Curve. For such histograms, the mean and SD are good summary statistics . The average pins down the center, while the SD gives the spread. For histogram which do not follow the normal Curve, the mean and SD are not good summary statistics. What when the histogram is not normal ...
  • 33. +- 3 SD = (384 * 3) = 1152 Mean - 1152 = about 30% sample had negative number of links Mean = 348.3 Std. Dev = 384.83 Distribution of word count on web pages 13 2800.0 2600.0 2400.0 2200.0 2000.0 1800.0 1600.0 1400.0 1200.0 1000.0 800.0 600.0 400.0 200.0 0.0 500 400 300 200 100 0
  • 34. Note. A percentile is a score below which a certain % of sample is When SD is influenced by outliers Use inter quartile range 75th percentile - 25th percentile
  • 35.
  • 36.
  • 37. Positively Skewed and Leptokurtic: Word Count Std. Dev = 725.24 Mean = 393.2 N = 1903.00 Kurtosis = 321.84 Skewness = 13.62 Median = 223 20000.0 18000.0 16000.0 14000.0 12000.0 10000.0 8000.0 6000.0 4000.0 2000.0 0.0 1600 1400 1200 1000 800 600 400 200 0
  • 38. Distribution of word count (N=1897) top six removed Std. Dev = 474.04 Mean = 368.0 N = 1897.00 Skewness = 3.49 Kurtosis = 16.40 Median = 223 4000.0 3600.0 3200.0 2800.0 2400.0 2000.0 1600.0 1200.0 800.0 400.0 0.0 800 600 400 200 0
  • 39.
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