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Statistics

Statistics. Introduction to. Statistic [ st uh- tis -tik ] noun . A numerical fact or datum, especially one computed from a sample. How long does the ball take to fall?. Measured values: See Board. How do we decide which of these measured values is correct ?.

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Statistics

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  1. Statistics Introduction to Statistic [stuh-tis-tik] noun . A numerical fact or datum, especially one computed from a sample

  2. How long does the ball take to fall? • Measured values: • See Board • How do we decide which of these measured values is correct? • How do we discuss the variation in our measurements?

  3. Mean • Also known as “Average” • Add all results, and divide by the number of measurements. • Equation form:

  4. Propagation of Uncertainty Accuracy • Sources of Inaccuracy: • Broken measurement device • Parallax • Random error • ? Low bias, high variability Precision • Sources of Imprecision: • Multiple measurement methods • Systematic error? High bias, low variability

  5. Variance and Standard Deviation • Squared deviation: How much variation is there from the mean? • Variance: measures the absolute distance observations are from the mean

  6. Error • Error is the difference between the measured and expected value • Error is how we make sense of differences between two measurements that should be the same • Error is NOT mistakes! If you made a mistake, do it again.

  7. Types of Error Descriptions For a true mean, µ, and standard deviation, σ, the sample mean has an uncertainty of the mean over the square root of the number of samples. Gives a measure of reliability of the mean. Sample standard error tells you how close your sample mean should be to the true mean.

  8. Using the Standard Error This is the simplest way of using data to confirm or refute a hypothesis. confirmed inside not confirmed outside This is also what is used to create the error bars.

  9. Example with data • Set of values: 2, 4, 4, 4, 5, 5, 7, 9 • Mean: • Standard Deviation:

  10. Error Types • False positive: Say things are different when they are the same • False negative: Say things are the same when they are different.

  11. Group Discussion: • What happens to Standard Deviation as sample size increases? • What does that imply about sample error? • Define standard deviation and sample error in your own words?

  12. Summary Mean Standard Deviation Variance Sample Standard Error

  13. Summary: Often used alternative Mean Variance Standard Deviation Sample Standard Error

  14. Types of Graphs: Continuous vs. Catagorical • Examples? • Options: • Times of ball rolling down ramp with increasing steepness • Sales of coffee, tea and soft drinks at a restaurant • Time it takes students to commute to UCI • SAT scores of varying ethnic groups

  15. Density Curve Low values indicate a small spread (all values close to the mean) high values indicate a large spread (all values far from the mean)

  16. Normal Distribution • Particularly important class of density curve • Symmetric, unimodal, • bell-shaped • Mean, μ, is at the center of the curve • Probabilities are the area under the curve • Total area = 1

  17. The Empirical Rule • In a normal distribution with mean μ and standard deviation of σ: • 68% of observations fall within 1 σ of the mean • 95% of observation fall within 2 σ of the mean • 99.7% observations fall within 3 σ of the mean F D B A C

  18. Example with data • Set of values: 2, 4, 4, 4, 5, 5, 7, 9 • Mean: • Standard Deviation:

  19. Data Distribution 5-6 5-4 5 5+6 5-2 5+2 5+4

  20. Confidence Interval 5-6 5-4 5 5+6 5-2 5+2 5+4

  21. Central Limit Theorem • If X follows a normal distribution with mean μ and standard deviation σ, then x̄ is also normally distributed with mean • What if X is not normally distributed? • When sampling from any population with mean μ and standard deviation σ, when n is large, the sampling distribution of x̄ is approximately normal: • As the number of measurements increase, they will approach a normal distribution (Gaussian). Visit This webpage to play with the numbers http://www.intuitor.com/statistics/CLAppClasses/CentLimApplet.htm

  22. Applications • Simulated examples: Dice rolling, coin flipping ect… Exit polling

  23. Non-normal Distributions

  24. Central Limit Theorem Summary • For large N of sample, the distribution of those mean values will be: which is a normal distribution. • Normal distribution of CLT is independent of the type of distribution of data.

  25. Where else would this become problematic? Where can it still be used, but issues should be considered?

  26. Questions?

  27. Effective Statistics You might have strong association, but how do you prove causation? (that x causes y?) Good evidence for causation: a well designed experiment where all other variables that cause changes in the response variable are controlled

  28. The Scientific/Statistic Process • Formulating a scientific question • Decide on the population you are interested in • Select a sample • Observational study or experiment? • Collect data • Analyze data • State your conclusion

  29. Ways to collect information from sample • Anecdotal evidence • Available data • Observational study • Experiment

  30. Sampling and Inference population sample sampling inference s x̄ σ μ

  31. Some Cautions • Statistics can not account for poor experimental design • There is no sharp border between “significant” and “non-significant” correlation, only increasing and decreasing evidence • Lack of significance may be due to poorly designed experiment

  32. Fit Tests t-test, z-test, and χ2 test

  33. z-Test

  34. z-test • All normal distributions are the same if we standardize our data: • Units of size σ • Mean μ as center • If x is an observation from a normal distribution, the standardized value of x is called the z-score • Z-scores tell how many standard deviations away from the mean an observation is

  35. z- test procedure • To use: find the mean, standard deviation, and standard error • Use these statistics along with the observed value to find Z value • Consult the z-score table to find P(Z) the determined z Equation for hypothesis testing:

  36. Example • Jacob scores 16 on the ACT. Emily scores 670 on the SAT. Assuming that both tests measure scholastic aptitude, who has the higher score? The SAT scores for 1.4 million students in a recent graduating class were roughly normal with a mean of 1026 and standard deviation of 209. The ACT scores for more than 1 million students in the same class were roughly normal with mean of 20.8 and standard deviation of 4.8.

  37. Example Continued

  38. Interpreting Results

  39. “Backwards” z-test • What if we are given a probability (P(Z)) and we are interested in finding the observed value corresponding to the probability.? • Find the Z-score • Set up the probability (could be 2 sided) P(-z0<Z<zo) = • Convert the score to x by

  40. t Tests

  41. Necessary assumptions for t-Test 1. Population is normally distributed. 2. Sample is randomly selected from the unknown population. 3. Standard deviation of the unknown population is the same as the known population. So, we can take the sample standard deviation as an estimate of the known population.

  42. This is typical of the kind of data many of you may generate. Let’s take a quick Look at how this T Test calculated from the data, using Excel.

  43. z versus t procedures • Use z procedures if you know the population standard deviation • Use t procedure if you don’t know the population standard deviation • Usually we don’t know the population standard deviation, unless told otherwise • Central Limit Theorem

  44. χ2-test (kai)

  45. χ2-test (Goodness-of-fit) Users Guide • χ2-test tells us whether distributions of categorical variables differ from one another • Can use to determine if your data conforms to a functional fit. • Compares multiple means to multiple expected values. • Can only use when you have multiple data sets that cannot be combined into one mean. • Use when comparing means to expected values.

  46. χ2-test • Xi is each individual mean • µi is each expected value • ΔXi = uncertainty in Xi • d = # of mean values • χ2/d table gives probability that data matches expected values. • In χ2/d , d is count of independent measurements.

  47. χ2- (Goodness-of-fit) Test Procedure • Find averages and uncertainty for each average. • Calculate χ2 using averages, uncertainties, and expected values. • Count number of independent variables. • Use table to find probability of fit accuracy based on χ2/d and number of independent variables (d).

  48. Example • Launch a bottle rocket with several different volumes of water. • Measure height of flight multiple times for each volume. • You decide you have a fit of: • Plot of fit with data on left.

  49. Example • 7 degrees offreedom • Probability of fit ≈50% • 50% of the time, chance alone could produce a larger χ2 value. • No reason to reject fit. • This does not mean that other fits might not match the data better, so try other fits and see which one is closest.

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