Introduction to Statistics

1. Introduction to Statistics • Concentration on applied statistics • Statistics appropriate for measurement • Today’s lecture will cover basic concepts • You should already be familiar with these

2. Independent Events • Occurrence of one event doesn’t affect probability of other • Examples: • Coin flips • Inputs from separate users • “Unrelated” traffic accidents • What about second basketball free throw after the player misses the first?

3. Random Variable • Variable that takes values with a specified probability • Variable usually denoted by capital letters, particular values by lowercase • Examples: • Number shown on dice • Network delay • What about disk seek time?

4. Cumulative Distribution Function (CDF) • Maps a value a to probability that the outcome is less than or equal to a: • Valid for discrete and continuous variables • Monotonically increasing • Easy to specify, calculate, measure

5. CDF Examples • Coin flip (T = 1, H = 2): • Exponential packet interarrival times:

6. Probability Density Function (pdf) • Derivative of (continuous) CDF: • Usable to find probability of a range:

7. Examples of pdf • Exponential interarrival times: • Gaussian (normal) distribution:

8. Probability Mass Function (pmf) • CDF not differentiable for discrete random variables • pmf serves as replacement: f(xi) = pi where piis the probability that x will take on the value xi

9. Examples of pmf • Coin flip: • Typical CS grad class size:

10. Expected Value (Mean) • Mean • Summation if discrete • Integration if continuous

11. Variance & Standard Deviation • Var(x) = • Usually denoted 2 • Square root is called standard deviation

12. Coefficient of Variation (C.O.V. or C.V.) • Ratio of standard deviation to mean: • Indicates how well mean represents the variable

13. Covariance • Given x, y with means x and y, their covariance is: • typos on p.181 of book • High covariance implies y departs from mean whenever x does

14. Covariance (cont’d) • For independent variables,E(xy)= E(x)E(y)so Cov(x,y)= 0 • Reverse isn’t true: Cov(x,y) = 0 doesn’t imply independence • If y = x, covariance reduces to variance

15. Correlation Coefficient • Normalized covariance: • Always lies between -1 and 1 • Correlation of 1 x ~ y, -1 

16. Quantile • x value at which CDF takes a value is called a-quantile or 100-percentile, denoted by x. • If 90th-percentile score on GRE was 1500, then 90% of population got 1500 or less

17. Quantile Example 0.5-quantile -quantile

18. Median • 50-percentile (0.5-quantile) of a random variable • Alternative to mean • By definition, 50% of population is sub-median, 50% super-median • Lots of bad (good) drivers • Lots of smart (stupid) people

19. Mode • Most likely value, i.e., xi with highest probability pi, or x at which pdf/pmf is maximum • Not necessarily defined (e.g., tie) • Some distributions are bi-modal (e.g., human height has one mode for males and one for females)

20. Examples of Mode Mode • Dice throws: • Adult human weight: Mode Sub-mode

21. Normal (Gaussian) Distribution • Most common distribution in data analysis • pdf is: • -x+ • Mean is  , standard deviation 

22. Notationfor Gaussian Distributions • Often denoted N(,) • Unit normal is N(0,1) • If x has N(,), has N(0,1) • The -quantile of unit normal z ~ N(0,1) is denoted z so that

23. Why Is GaussianSo Popular? • If xi ~ N(,) and all xi independent, thenixi is normal with mean ii and variance i2i2 • Sum of large no. of independent observations from any distribution is itself normal (Central Limit Theorem) • Experimental errors can be modeled as normal distribution.

24. Central Limit Theorem • Sum of 2 coin flips (H=1, T=0): • Sum of 8 coin flips:

25. Measured Data Measured Data But, we don’t know F(x) – all we have is a bunch of observed values – a sample. What is a sample? • Example: How tall is a human? • Could measure every person in the world (actually even that’s a sample) • Or could measure every person in this room • Population has parameters • Sample has statistics • Drawn from population • Inherently erroneous

26. Central Tendency • Sample mean – x (arithmetic mean) • Take sum of all observations and divide by the number of observations • Sample median • Sort the observations in increasing order and take the observation in the middle of the series • Sample mode • Plot a histogram of the observations and choose the midpoint of the bucket where the histogram peaks

27. Indices of Dispersion • Measures of how much a data set varies • Range • Sample variance • And derived from sample variance: • Square root -- standard deviation, S • Ratio of sample mean and standard deviation – COVs / x • Percentiles • Specification of how observations fall into buckets

28. Interquartile Range • Yet another measure of dispersion • The difference between Q3 and Q1 • Semi-interquartile range - • Often interesting measure of what’s going on in the middle of the range

29. Which Index of Dispersion to Use? Yes Range Bounded? No Unimodal symmetrical? Yes C.O.V No Percentiles or SIQR But always remember what you’re looking for

30. Determining a Distribution for a Data Set • If a data set has a common distribution, that’s the best way to summarize it • Saying a data set is uniformly distributed is more informative than just giving its sample mean and standard deviation • So how do you determine if your data set fits a distribution? • Plot a histogram • Quantile-quantile plot • Statistical methods

31. Quantile-Quantile Plots • Most suitable for small data sets • Basically -- guess a distribution • Plot where quantiles of data should fall in that distribution • Against where they actually fall in the sample • If plot is close to linear, data closely matches that distribution

32. ObtainingTheoretical Quantiles • We need to determine where the quantiles should fall for a particular distribution • Requires inverting the CDF for that distribution qi = F(xi)t xi = F-1(qi) • Then determining quantiles for observed points • Then plugging in quantiles to inverted CDF

33. Inverting a Distribution • Many common distributions have already been inverted (how convenient…) • For others that are hard to invert, tables and approximations are often available (nearly as convenient)

34. Is Our Example Data Set Normally Distributed? • Our example data set was -17, -10, -4.8, 2, 5.4, 27, 84.3, 92, 445, 2056 • Does this match the normal distribution? • The normal distribution doesn’t invert nicely • But there is an approximation for N(0,1): • Or invert numerically

35. Data For Example Normal Quantile-Quantile Plot i qi yi xi 1 0.05 -17 -1.64684 2 0.15 -10 -1.03481 3 0.25 -4.8 -0.67234 4 0.35 2 -0.38375 5 0.45 5.4 -0.1251 6 0.55 27 0.1251 7 0.65 84.3 0.383753 8 0.75 92 0.672345 9 0.85 445 1.034812 10 0.95 2056 1.646839

36. Example NormalQuantile-Quantile Plot • Definitely not normal • Because it isn’t linear • Tail at high end is too long for normal

37. Estimating Populationfrom Samples • How tall is a human? • Measure everybody in this room • Calculate sample mean • Assume population mean equals • What is the error in our estimate?

38. Estimating Error • Sample mean is a random variable • Mean has some distribution • Multiple sample means have “mean of means” • Knowing distribution of means can estimate error

39. Confidence Intervals • Sample mean value is only an estimate of the true population mean • Bounds c1 and c2 such that there is a high probability, 1-a, that the population mean is in the interval (c1,c2): Prob{ c1 < m < c2} =1-awhere a is the significance level and100(1-a) is the confidence level • Overlapping confidence intervals is interpreted as “not statistically different”

40. Confidence Intervals • How tall is Fred? • Suppose average human height is 170 cm • Fred is 170 cm tall Yeah, right • Suppose 90% of humans are between 155 and 190 cm • Fred is between 155 and 190 cm • We are 90% confident that Fred is between 155 and 190 cm

41. Confidence Intervalof Sample Mean • Knowing where 90% of sample means fall, we can state a 90% confidence interval • Key is Central Limit Theorem: • Sample means are normally distributed • Only if independent • Mean of sample means is population mean  • Standard deviation (standard error) is

42. EstimatingConfidence Intervals • Two formulas for confidence intervals • Over 30 samples from any distribution: z-distribution • Small sample from normally distributed population: t-distribution • Common error: using t-distribution for non-normal population • Central Limit Theorem often saves us

43. The z Distribution • Interval on either side of mean: • Significance level is small for large confidence levels • Tables of z are tricky: be careful!

44. Example of z Distribution • 35 samples: 10 16 47 48 74 30 81 42 57 67 7 13 56 44 54 17 60 32 45 28 33 60 36 59 73 46 10 40 35 65 34 25 18 48 63 • Sample mean = 42.1. Standard deviation s = 20.1. n = 35 • 90% confidence interval is

45. The t Distribution • Formula is almost the same: • Usable only for normally distributed populations! • But works with small samples

46. Example of t Distribution • 10 height samples: 148 166 170 191 187 114 168 180 177 204 • Sample mean = 170.5. Standard deviation s = 25.1, n = 10 • 90% confidence interval is • 99% interval is (144.7, 196.3)

47. Getting More Confidence • Asking for a higher confidence level widens the confidence interval • Counterintuitive? • How tall is Fred? • 90% sure he’s between 155 and 190 cm • We want to be 99% sure we’re right • So we need more room: 99% sure he’s between 145 and 200 cm

48. For Discussion Next Tuesday Project Proposal • Statement of hypothesis • Workload decisions • Metrics to be used • Method Reading for Next Time Elson, Girod, Estrin, “Fine-grained Network Time Synchronization using Reference Broadcasts, OSDI 2002 – see readings.html

49. For Discussion Today • Bring in one either notoriously bad or exceptionally good example of data presentation from your proceedings. The bad ones are more fun. Or if you find something just really different, please show it.