Statistics

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

3. How Long is a Meter Stick? • Expected value: 1m • Measured values: 1.01m, .98m, 1.03m • How do we decide which of these measured values is correct? • How do we discuss the variation in our measurements?

4. Mean • The arithmetic average

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

6. Variance and Standard Deviation • Variance: • Standard Deviation: measures how far observations are from the mean

7. 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.

8. 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.

9. 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.

10. 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)

11. 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

12. 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

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

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

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

16. 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

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

18. Non-normal Distributions

19. 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.

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

21. Questions?

22. 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

23. 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

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

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

26. 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

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

28. z-Test

29. 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

30. 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:

31. 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.

32. Example Continued

33. Interpreting Results

34. “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

35. t Tests

36. 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.

38. 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.

39. 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

40. χ2-test (kai)

41. 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.

42. 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.

43. 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).

44. 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.

45. 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.

46. Interpreting Results • Probability is how similar data is to expected value. • Large P means data is similar to expected value. • Small P means data is different than expected value.

47. Summary • Propagation of uncertainty • Mean • Accuracy vs. Precision • Error • Standard deviation • Central Limit Theorem • Fit Tests • z-test • t-test • χ2-test