Descriptive stat
- 2. Descriptive Statistics Class A--IQs of 13 Students 102 115 128 109 131 89 98106 140 119 9397 110 Class B--IQs of 13 Students 127 162 131 103 96 111 80 109 93 87 120 105 109 An Illustration: Which Group is Smarter? Each individual may be different. If you try to understand a group by remembering the qualities of each member, you become overwhelmed and fail to understand the group.
- 3. Descriptive Statistics Which group is smarter now? Class A--Average IQ Class B--Average IQ 110.54 110.23 They’re roughly the same! With a summary descriptive statistic, it is much easier to answer our question.
- 4. Ordered sample characteristics • MMaaxxiimmuumm –– tthhee llaasstt nnuummbbeerr iinn oorrddeerreedd ssaammppllee >> mmaaxx((xx)) • MMiinniimmuumm –– tthhee ffiirrsstt nnuummbbeerr iinn oorrddeerreedd ssaammppllee >> mmiinn((xx)) • MMeeddiiaann –– tthhee nnuummbbeerr llooccaatteedd aatt tthhee cceenntteerr ooff oorrddeerreedd ssaammppllee >> mmeeddiiaann((xx)) • QQuuaannttiillee –– ssuucchh nnuummbbeerr xxpp,, tthhaatt tthhee mmaaxxiimmuumm vvaalluuee ooff pp--tthh ppaarrtt ooff ssaammppllee iiss lleessss oorr eeqquuaall xxpp >> qquuaannttiillee((xx,,pp)) • QQuuaarrttiilleess -- 00.2255--qquuaannttiillee iiss ccaalllleedd tthhee ffiirrsstt ((oorr lloowweerr)) qquuaarrttiillee;; 00.55--qquuaannttiillee iiss ccaalllleedd mmeeddiiaann oorr tthhee sseeccoonndd qquuaarrttiillee;; 00.7755--qquuaannttiillee iiss ccaalllleedd tthhee tthhiirrdd ((oorr uuppppeerr)) qquuaarrttiillee • IInntteerrqquuaarrttiillee rraannggee –– tthhee ddiiffffeerreennccee bbeettwweeeenn tthhee tthhiirrdd aanndd tthhee ffiirrsstt qquuaarrttiillee,, ii.ee. xx00.7755 −− xx00.2255 >> IIQQRR((xx))
- 5. Descriptive Statistics Types of descriptive statistics: • Organize Data –Tables –Graphs • Summarize Data –Central Tendency –Variation
- 6. Descriptive Statistics Summarizing Data: – Central Tendency (or Groups’ “Middle Values”) • Mean • Median • Mode – Variation (or Summary of Differences Within Groups) • Range • Interquartile Range • Variance • Standard Deviation
- 7. Mean The mean is the “balance point.” Each person’s score is like 1 pound placed at the score’s position on a see-saw. Below, on a 200 cm see-saw, the mean equals 110, the place on the see-saw where a fulcrum finds balance: 17 units below 4 units below 110 cm 21 units above 0 units The scale is balanced because… 17 + 4 on the left = 21 on the right 1 lb at 93 cm 1 lb at 106 cm 1 lb at 131 cm
- 8. Mean 1. Means can be badly affected by outliers (data points with extreme values unlike the rest) 2. Outliers can make the mean a bad measure of central tendency or common experience Income in the world All of Us Bill Gates Mean Outlier
- 9. Median The middle value when a variable’s values are ranked in order; the point that divides a distribution into two equal halves. When data are listed in order, the median is the point at which 50% of the cases are above and 50% below it. The 50th percentile.
- 10. Median Median = 109 (six cases above, six below) Class A--IQs of 13 Students 89 93 97 98 102 106 109 110 115 119 128 131 140
- 11. If the first student were to drop out of Class A, there would be a new median: 89 93 97 98 102 106 109 110 115 119 128 131 140 Median Median = 109.5 109 + 110 = 219/2 = 109.5 (six cases above, six below)
- 12. Median 1. The median is unaffected by outliers, making it a better measure of central tendency, better describing the “typical person” than the mean when data are skewed. All of Us Bill Gates outlier
- 13. Median 2. If the recorded values for a variable form a symmetric distribution, the median and mean are identical. 3. In skewed data, the mean lies further toward the skew than the median. Symmetric Skewed Mean Median Mean Median
- 14. Mode The most common data point is called the mode. The combined IQ scores for Classes A & B: 80 87 89 93 93 96 97 98 102 103 105 106 109 109 109 110 111 115 119 120 127 128 131 131 140 162 mode BTW, It is possible to have more than one mode!
- 15. Mode It may mot be at the center of a distribution. Data distribution on the 2.0 1.8 1.6 1.4 right is “bimodal” (even statistics can be open-minded) 82.00 89.00 87.00 96.00 93.00 98.00 97.00 103.00 102.00 106.00 105.00 109.00 107.00 115.00 111.00 120.00 119.00 128.00 127.00 140.00 131.00 162.00 IQ 1.2 1.0 Count
- 17. Descriptive Statistics Summarizing Data: Central Tendency (or Groups’ “Middle Values”) Mean Median Mode – Variation (or Summary of Differences Within Groups) • Range • Interquartile Range • Variance • Standard Deviation
- 18. Measures of Variability • Range (largest score – smallest score) • Variance (S2=Σ(x-M)2/N) • Standard deviation – Square root of the variance, so it’s in the same units as the mean – In a normal distribution, 68.26% of scores fall within +/- 1 sd of the mean; 95.44% fall within +/- 2 sd of the mean. • Coefficient of variation = the standard deviation divided by the sample mean
- 19. Range The spread, or the distance, between the lowest and highest values of a variable. To get the range for a variable, you subtract its lowest value from its highest value. Class A--IQs of 13 Students 102 115 128 109 131 89 98106 140 119 9397 110 Class A Range = 140 - 89 = 51 Class B--IQs of 13 Students 127 162 131 103 96111 80109 9387 120 105 109 Class B Range = 162 - 80 = 82
- 20. Interquartile Range A quartile is the value that marks one of the divisions that breaks a series of values into four equal parts. The median is a quartile and divides the cases in half. 25th percentile is a quartile that divides the first ¼ of cases from the latter ¾. 75th percentile is a quartile that divides the first ¾ of cases from the latter ¼. The interquartile range is the distance or range between the 25th percentile and the 75th percentile. Below, what is the interquartile range? 25% of cases 25% 25% 25% of cases 0 250 500 750 1000
- 21. Variance A measure of the spread of the recorded values on a variable. A measure of dispersion. The larger the variance, the further the individual cases are from the mean. Mean The smaller the variance, the closer the individual scores are to the mean. Mean
- 22. Variance Variance is a number that at first seems complex to calculate. Calculating variance starts with a “deviation.” A deviation is the distance away from the mean of a case’s score. Yi – Y-bar If the average person’s car costs $20,000, my deviation from the mean is - $14,000! 6K - 20K = -14K
- 23. Variance The deviation of 102 from 110.54 is? Deviation of 115? Class A--IQs of 13 Students 102 115 128 109 131 89 98106 140 119 9397 110 Y-barA = 110.54
- 24. Variance The deviation of 102 from 110.54 is? Deviation of 115? 102 - 110.54 = -8.54 115 - 110.54 = 4.46 Class A--IQs of 13 Students 102 115 128 109 131 89 98106 140 119 9397 110 Y-barA = 110.54
- 25. Variance • We want to add these to get total deviations, but if we were to do that, we would get zero every time. Why? • We need a way to eliminate negative signs. Squaring the deviations will eliminate negative signs... A Deviation Squared: (Yi – Y-bar)2 Back to the IQ example, A deviation squared for 102 is: of 115: (102 - 110.54)2 = (-8.54)2 = 72.93 (115 - 110.54)2 = (4.46)2 = 19.89
- 26. Variance If you were to add all the squared deviations together, you’d get what we call the “Sum of Squares.” Sum of Squares (SS) = Σ (Yi – Y-bar)2 SS = (Y1 – Y-bar)2 + (Y2 – Y-bar)2 + . . . + (Yn – Y-bar)2
- 27. Variance Class A, sum of squares: (102 – 110.54)2 + (115 – 110.54)2 + (126 – 110.54)2 + (109 – 110.54)2 + (131 – 110.54)2 + (89 – 110.54)2 + (98 – 110.54)2 + (106 – 110.54)2 + (140 – 110.54)2 + (119 – 110.54)2 + (93 – 110.54)2 + (97 – 110.54)2 + Class A--IQs of 13 Students 102 115 128 109 131 89 98106 140 119 9397 110 Y-bar = 110.54
- 28. Variance The last step… The approximate average sum of squares is the variance. SS/N = Variance for a population. SS/n-1 = Variance for a sample. Variance = Σ(Yi – Y-bar)2 / n – 1
- 29. Variance For Class A, Variance = 2825.39 / n - 1 = 2825.39 / 12 = 235.45 How helpful is that???
- 30. Standard Deviation To convert variance into something of meaning, let’s create standard deviation. The square root of the variance reveals the average deviation of the observations from the mean. s.d. = Σ(Yi – Y-bar)2 n - 1
- 31. Standard Deviation For Class A, the standard deviation is: 235.45 = 15.34 The average of persons’ deviation from the mean IQ of 110.54 is 15.34 IQ points. Review: 1. Deviation 2. Deviation squared 3. Sum of squares 4. Variance 5. Standard deviation
- 32. Standard Deviation 1. Larger s.d. = greater amounts of variation around the mean. For example: 19 25 31 13 25 37 Y = 25 Y = 25 s.d. = 3 s.d. = 6 2. s.d. = 0 only when all values are the same (only when you have a constant and not a “variable”) 3. If you were to “rescale” a variable, the s.d. would change by the same magnitude—if we changed units above so the mean equaled 250, the s.d. on the left would be 30, and on the right, 60 4. Like the mean, the s.d. will be inflated by an outlier case value.
- 33. Practical Application for Understanding Variance and Standard Deviation Even though we live in a world where we pay real dollars for goods and services (not percentages of income), most employers issue raises based on percent of salary. Why do supervisors think the most fair raise is a percentage raise? Answer: 1) Because higher paid persons win the most money. 2) The easiest thing to do is raise everyone’s salary by a fixed percent. If your budget went up by 5%, salaries can go up by 5%. The problem is that the flat percent raise gives unequal increased rewards. . .
- 34. Practical Application for Understanding Variance and Standard Deviation Acme Toilet Cleaning Services Salary Pool: $200,000 Incomes: President: $100K; Manager: 50K; Secretary: 40K; and Toilet Cleaner: 10K Mean: $50K Range: $90K Variance: $1,050,000,000 These can be considered “measures of inequality” Standard Deviation: $32.4K Now, let’s apply a 5% raise.
- 35. Practical Application for Understanding Variance and Standard Deviation After a 5% raise, the pool of money increases by $10K to $210,000 Incomes: President: $105K; Manager: 52.5K; Secretary: 42K; and Toilet Cleaner: 10.5K Mean: $52.5K – went up by 5% Range: $94.5K – went up by 5% Variance: $1,157,625,000 Measures of Inequality Standard Deviation: $34K –went up by 5% The flat percentage raise increased inequality. The top earner got 50% of the new money. The bottom earner got 5% of the new money. Measures of inequality went up by 5%. Last year’s statistics: Acme Toilet Cleaning Services annual payroll of $200K Incomes: $100K, 50K, 40K, and 10K Mean: $50K Range: $90K; Variance: $1,050,000,000; Standard Deviation: $32.4K
- 36. Practical Application for Understanding Variance and Standard Deviation The flat percentage raise increased inequality. The top earner got 50% of the new money. The bottom earner got 5% of the new money. Inequality increased by 5%. Since we pay for goods and services in real dollars, not in percentages, there are substantially more new things the top earners can purchase compared with the bottom earner for the rest of their employment years. Acme Toilet Cleaning Services is giving the earners $5,000, $2,500, $2,000, and $500 more respectively each and every year forever. What does this mean in terms of compounding raises? Acme is essentially saying: “Each year we’ll buy you a new TV, in addition to everything else you buy, here’s what you’ll get:”
- 37. Practical Application for Understanding Variance and Standard Deviation Toilet Cleaner Secretary Manager President The gap between the rich and poor expands. This is why some progressive organizations give a percentage raise with a flat increase for lowest wage earners. For example, 5% or $1,000, whichever is greater.
- 38. Descriptive Statistics Summarizing Data: Central Tendency (or Groups’ “Middle Values”) Mean Median Mode Variation (or Summary of Differences Within Groups) Range Interquartile Range Variance Standard Deviation – …Wait! There’s more
- 39. Box-Plots A way to graphically portray almost all the descriptive statistics at once is the box-plot. A box-plot shows: Upper and lower quartiles Mean Median Range Outliers (1.5 IQR)
- 40. Box-Plots IQ 180.00 160.00 140.00 120.00 100.00 80.00 123.5 106.5 96.5 162 82 M=110.5 IQR = 27; There is no outlier.
- 42. Confidence Intervals • Confidence intervals express the range in which the true value of a population parameter (as estimated by the sample statistic) falls, with a high degree of confidence (usually 95% or 99%).
- 43. Standard Deviation Versus Standard Error • The mean of the sampling distribution equals the population mean. • The standard deviation of the sampling distribution (also called the standard error of the mean) equals the population standard deviation / the square root of the sample size. • The standard error is an index of sampling error —an estimate of how much any sample can be expected to vary from the actual population value.