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# Lies, Damn Lies And Anti Statistics

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### Lies, Damn Lies And Anti Statistics

1. 1. Lies, Damn Lies and Anti- Statistics Alan McSweeney
2. 2. Objective • Introduce the concept of distorting “anti-statistics”, illustrate how “anti-statistics” can be identified and define how statistics should be constructed to yield insight and meaning May 18, 2010 2
3. 3. Statistics • A statistic has two roles - primary and secondary − Primary - to summarise and describe the data while preserving information and reducing the volume of raw data − Secondary - to provide and enable insight • Where an alleged statistic does not perform these functions it is an “anti-statistic” − Distorting the underlying information (raw data), either deliberately or accidentally − Not providing insight or providing an inaccurate view of the underlying information • Most people are scared of large sets of numbers − The use of anti-statistics uses this fear May 18, 2010 3
4. 4. Statistics and Anti-Statistics • Statistics • Anti-Statistics • Descriptive • Distorting • Insightful • Promoting Misinterpretation • Informative • Misinformative • Enlightening • Concealing May 18, 2010 4
5. 5. Statistics - Primary Function • To describe the data while preserving information and reducing the volume of raw data • This means taking a large amount of raw data, producing descriptive summaries while not losing or distorting the underlying raw data • More important function of a statistic May 18, 2010 5
6. 6. Statistics - Secondary Function • To provide and enable insight • By reducing the volume of raw data, you can gain insight into what the data means − Enabling you to see the wood from the trees, know the amount and type of wood and make decisions about the use of the wood • Secondary function if primary function satisfied May 18, 2010 6
7. 7. Data, Information, Knowledge and Action Cycle • Good Knowledge statistics provide Action information that creates knowledge and enables correct actions Information Data May 18, 2010 7
8. 8. Information – Lots of It May 18, 2010 8
9. 9. Sample Information • 4,000 numbers representing the annual salaries of individuals − Sample data only • 100% of the information is available here • Very hard to see patterns, understand the situation, gain insight and make effective decisions and understand their consequences • The numbers do not lie but they are innocent creatures and can be made to lie • Need techniques that extract meaning and provide insight without losing the information the data represents May 18, 2010 9
10. 10. Statistics • I can take all this … • … And give you one derived number (average) − 107941.931 May 18, 2010 10
11. 11. Statistic • 4,000 numbers reduced to 1 • Reduced the amount of data by 99.975% (another “statistic”) • But I have lost information • Average value of 107941.931 is at best a simplistic view of the data and at worst a distortion that misrepresents the source data • If I use the average without looking to understand the raw data in more detail I am potentially creating a distortion May 18, 2010 11
12. 12. More Statistics Average Sum of all the values divided by the number of values 107941.93 Standard A measure of how widely values are dispersed from the average value 59904.19 Deviation Kurtosis Value that describes the relative peakedness or flatness of a distribution 0.112 where a positive value indicates a relatively peaked distribution and a negative value indicates a relatively flat distribution Skewness A measure of the asymmetry of a distribution around the average where a 0.731 positive value indicates a distribution with an asymmetric tail extending toward more positive values and a negative value indicates a distribution with an asymmetric tail extending toward more negative values Mode The most frequently occurring value 23958 Median This the number in the middle where, half the numbers have values that are 97909.5 greater than the median and half have values that are less – also called the 50th percentile • Be careful what statistics are used • Do not generate statistics just because you can • The use of statistics can give a false impression of certainty or meaning where there is none May 18, 2010 12
13. 13. Interpreting the Statistics Statistic Value Interpretation Average 107941.93 The average is higher than the median indicating that the data is dispersed unequally towards higher values Standard Deviation 59904.19 The high standard deviation indicates the underlying data is spread across a wide range of values Kurtosis 0.112 The positive value indicates that there is a peak in the data Skewness 0.731 The positive values indicates a distribution with an unequal and heavy tail extending toward more higher values Mode 23958 In a large set of data where only a small number of data values are the same, this is meaningless Median 97909.5 When the median is less than the average, it means the data is unequally distributed with a heavy tail extending toward more higher values • I now know that the data is skewed towards lower values and has a heavy tail indicating a small number of people earning large salaries May 18, 2010 13
14. 14. Number of People 0 10 20 30 40 50 0 60 May 18, 2010 20 00 0 40 00 0 60 00 0 80 00 0 10 00 00 12 00 00 14 00 00 16 00 00 Let’s Take a Look at the Data 18 00 Annual Salary 00 20 00 00 22 00 00 24 00 00 26 00 00 28 00 00 30 00 00 14
15. 15. Let’s Take a Look at the Data Clustered Increases around quickly Gradual drop lower values from peak • Characteristics from zero 60 − Increases quickly from zero 50 − Distribution skewed to the left 40 Number of People − Clustered around lower Heavy tail values 30 − Gradual drop from 20 peak − Heavy tail 10 • This type of data 0 distribution is very 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 common 20 40 60 80 10 12 14 16 18 20 22 24 26 28 30 Annual Salary Distribution skewed to the left May 18, 2010 15
16. 16. Statistics 0.4 • The usefulness of a statistic 0.35 depends on the underlying data 0.3 • Average really only makes sense when the data is 0.25 symmetrically/equally 0.2 distributed 0.15 − Otherwise, the average is distorted because of unequal distribution of 0.1 data 0.05 • Deviation also really only makes 0 sense when the data is -5 -4.5 -4.1 -3.6 -3.2 -2.7 -2.2 -1.8 -1.3 -0.9 -0.4 0.06 0.52 0.98 1.44 1.9 2.36 2.82 3.28 3.74 4.2 4.66 symmetrically distributed May 18, 2010 16
17. 17. Statistics • Be careful of obscure statistics such as Kurtosis and Skewness • They have a use but the meaning is quite specific and may not be appropriate May 18, 2010 17
18. 18. Descriptive Statistics • Look for statistics that contain − Measures of data location and clustering − Measures of dispersion and variability − Measures of association • Look at the underlying data, how it was collected, what it measures − If the data is of poor quality or measures the wrong values, any derived information will have very limited worth • There are lots of statistics that can be produced from the raw data − Produce only meaningful statistics − Do not throw statistics at the data May 18, 2010 18
19. 19. Some Common Descriptive and Summarising Statistics Statistic Type Statistic Description Data location and Clustering Average Simple average Weighted Average Average of values weighted according to a value such as their importance Truncated/Interpercentile Average Average of centralised subset of data Median The 50th percentile Mode The most commonly occurring value Dispersion, Variability and Shape Variance Measure of the amount of variation within the data Standard Deviation Square root of the Variance Range The spread of the data values Skewness Measure of the asymmetry of the distribution of the data Kurtosis Measure of the "peakedness” and the length of the tail of the distribution of the data Percentiles Value below which a certain percent of the data fall Association Correlation Correlation has a specific meaning that may not be relevant to the data May 18, 2010 19
20. 20. Another Look at the Sample Data 320000 300000 280000 260000 240000 Annual Salary 220000 200000 180000 160000 140000 120000 100000 80000 60000 40000 20000 0 0% 5% % % % % % % % % % % % % % % % % % % 0% 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 10 Percentage Earning Up to Salary Amount • This shows the salaries of cumulative percentages of the people surveyed May 18, 2010 20
21. 21. Another Look at the Sample Data 290000 - 300000 15 290000 - 300000 0.4% 280000 - 290000 17 280000 - 290000 0.4% 270000 - 280000 20 270000 - 280000 0.5% 260000 - 270000 22 260000 - 270000 0.6% 250000 - 260000 27 250000 - 260000 0.7% 240000 - 250000 32 240000 - 250000 0.8% 230000 - 240000 38 230000 - 240000 1.0% 220000 - 230000 47 220000 - 230000 1.2% 210000 - 220000 55 210000 - 220000 1.4% 200000 - 210000 67 200000 - 210000 1.7% 190000 - 200000 84 190000 - 200000 2.1% 180000 - 190000 96 180000 - 190000 2.4% 170000 - 180000 112 170000 - 180000 2.8% Salary Range Salary Range 160000 - 170000 128 160000 - 170000 3.2% 150000 - 160000 146 150000 - 160000 3.7% 140000 - 150000 166 140000 - 150000 4.2% 130000 - 140000 187 130000 - 140000 4.7% 120000 - 130000 209 120000 - 130000 5.2% 110000 - 120000 230 110000 - 120000 5.8% 100000 - 110000 249 100000 - 110000 6.2% 90000 - 100000 267 90000 - 100000 6.7% 80000 - 90000 280 80000 - 90000 7.0% 70000 - 80000 285 70000 - 80000 7.1% 60000 - 70000 283 60000 - 70000 7.1% 50000 - 60000 268 50000 - 60000 6.7% 40000 - 50000 237 40000 - 50000 5.9% 30000 - 40000 193 30000 - 40000 4.8% 20000 - 30000 133 20000 - 30000 3.3% 10000 - 20000 83 10000 - 20000 2.1% 0 - 10000 24 0 - 10000 0.6% 0 50 100 150 200 250 300 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% Number of People Percentage of People May 18, 2010 21
22. 22. Percentiles • Percentile of a set of data is the number or value below which that percent of data lies • Median = 50th percentile − Value below which 50% of data lies • Quartiles are percentiles for 25%, 50% and 75% • Percentiles are useful in summarising data May 18, 2010 22
23. 23. Percentiles for Sample Data • This … • … becomes this … • 4,000 numbers reduced to 10 numbers − 10% of people earn 38,332 or less − 20% of people earn 54,834 or less − 10% of people earn between 192,871 and 299,433 • Successfully reduced the volume of data while preserving more information May 18, 2010 23
24. 24. Anti-Statistics • Unfortunately everywhere • Take a number of general forms or types such as − Statement based on measurement of incorrect value − Statement without scale or reference − Statement based on grouping of categories (with possible distortion of categories) − Statements based on inaccurate on unspecified association or correlation May 18, 2010 24
25. 25. Sample Type 1 Anti-Statistic • Chimpanzee DNA is 99.7% the same as Human DNA • What does this statement mean? − Do chimpanzees make cars/houses/PCs/etc. that are 99.7% as good as those made by humans? • If the statement is true then what is being measured may be invalid, such as • 000000000000000000000000 and 000000000000000000000001 • These numbers are 99% the same based on the length of the lines in their characters − Or • A lot of DNA is not involved in the development process and this is being included in measurements − Or • A small change in DNA has a substantial impact on what is produced May 18, 2010 25
26. 26. Sample Type 2 Anti-Statistic • Statements of the form − X is the greatest cause of Y, such as • Car crashes are the greatest cause of deaths among males in their 20s and 30s • Meaningless because there is no scale or reference point • Statement creates an impression of scale and severity that is at best not justified or at worst incorrect • Take a look at the underlying life expectancy data May 18, 2010 26
27. 27. Type 2 Anti-Statistic • Probability of a person dying • Probability of a person dying within a year at each year of life within a year for first 35 years 0.6 0.0045 0.004 Probability of Dying Within One Year Probability of Dying Within One Year 0.5 0.0035 0.4 0.003 0.0025 0.3 0.002 0.2 0.0015 0.001 0.1 0.0005 0 0 20 Yea s 25 ea s 30 Yea s 35 Yea s Y rs 45 Yea s rs 55 Yea s 60 Yea s rs 70 Yea s rs 80 Yea s 85 Yea s rs 95 Yea s 10 Ye rs 10 Ye rs 5 ars s rs 5 0 0 5 10 15 20 25 30 35 r Y r r r r r r r r r r ar 15 Yea 40 ea 50 Yea 65 Yea 75 Yea 90 Yea 0 a 10 Yea Ye Years Years Years Years Years Years Years May 18, 2010 27
28. 28. Type 2 Anti-Statistic • The underlying life expectancy data shows that young people have very little chance of dying • Death rates are uniformly very low after the first year of life until about age 50 • So a statement such as − Car crashes are the greatest cause of deaths among males in their 20s and 30s • Will inevitably be true because nothing else really kills young males − Death due to illness is uncommon among this group so any other cause will dominate May 18, 2010 28
29. 29. Sample Type 3 Anti-Statistic • Statements of the form − N% of people do/have done X at least N times/with defined frequency − Typically arise as the results of tendentious surveys designed to create a false impression of severity • Such as − 75% of people admit to X up to N times a year • No indication of how the 75% is spread across the range of 1 to N times − 65% of people admit to having a negative experience up to N times due to X • No indication of the spread of negative experiences across the range of 1 to N • Generally a result of combining the responses to two or more questions or categories − Have often have you done/experienced X? • Once • Twice • Three times • … May 18, 2010 29
30. 30. Type 3 Anti-Statistic • Have often have you • Have often have you done/experienced X? done/experienced X? − Once − 45% − Twice − 10% − Three times − 8% − 4-8 times − 5% − 8-12 times − 2% • Total of these is 75% • Statement that 75% of people have done/experienced X up to 12 times a year distorts the distribution of the underlying data that is skewed towards lower rates of occurrence May 18, 2010 30
31. 31. Sample Type 4 Anti-Statistic • Statements of the form − Taking /doing A makes you N% more likely to be/experience B • Two issues − Causation – is there a real causal relationship − Degree of causation – how strong is the causal relationship • An association does not imply a causation − A might cause B − B might cause A − A might cause B and B might cause A − A might cause C that might cause B − A might cause C that might cause D … that might cause B − A might cause C that might cause B and A might cause D that might not cause B but A-C- D causation is greater than A-D-B negative causation − Measuring error − Random data that was skewed − Deliberate or malicious misrepresentation • Cause might be partial or contributory • Be careful of any statement of a relationship that does not demonstrate how causation happens May 18, 2010 31
32. 32. Association and Causation Scenarios Causes or Influences A B A B Causes or Influences Causes or Influences A B C D A Causes or Influences B D Negatively Causes or A B A Influences B Causes or Influences Causes or Influences C C May 18, 2010 32
33. 33. Association and Causation • Very common scenario where an association or causation is asserted Takes or Taking or Doing Does D D Affects or Causes B A B May 18, 2010 33
34. 34. Association and Causation • The real association or causation is actually along the lines of: Takes or Taking or Doing D Has Does D Little or No Effect or Influence on B or Even Members of Negatively Impacts B Group C Have a Greater Tendency to A Take or do D B Members of Group C Also Take or Do E Taking or Doing E Is a Affects or Causes Member of B a Group C E May 18, 2010 34
35. 35. Type 4 Anti-Statistic • Occurs very frequently • A percentage association can give a false sense of certainty − Just measures the looseness of association • Often misrepresents the degree of causation • Unless the precise nature of the causative relationship can be defined, take with a large dose of salt May 18, 2010 35
36. 36. Summary • Statistics are designed to provide insight without distorting the meaning of the underlying data or losing information • Anti-statistics are used to distort the underlying data to create false impressions • So there are Lies, Damn Lies and Anti-Statistics May 18, 2010 36