Paul Cornwell March 31,

 Let X 1,…,X n be independent, identically distributed random variables with positive variance. Averages of these variables will be approximately normally distributed with mean μ and standard deviation σ/√n when n is large. 2

 How large of a sample size is required for the Central Limit Theorem (CLT) approximation to be good?  What is a ‘good’ approximation? 3

 Permits analysis of random variables even when underlying distribution is unknown  Estimating parameters  Hypothesis Testing  Polling 4

 Performing a hypothesis test to determine if set of data came from normal  Considerations ◦ Power: probability that a test will reject the null hypothesis when it is false ◦ Ease of Use 5

 Problems ◦ No test is desirable in every situation (no universally most powerful test) ◦ Some lack ability to verify for composite hypothesis of normality (i.e. nonstandard normal) ◦ The reliability of tests is sensitive to sample size; with enough data, null hypothesis will be rejected 6

 Symmetric  Unimodal  Bell-shaped  Continuous 7

 Skewness: Measures the asymmetry of a distribution. ◦ Defined as the third standardized moment ◦ Skew of normal distribution is 0 8

 Kurtosis: Measures peakedness or heaviness of the tails. ◦ Defined as the fourth standardized moment ◦ Kurtosis of normal distribution is 3 9

 Cumulative distribution function: 10

11 parametersKurtosisSkewness% outside 1.96*sd K-S distance Mean Std Dev n = 20 p = (.25).3325 (1.5) n = 25 p = n = 30 p = n = 50 p = n = 100 p = *from R

 Cumulative distribution function: 12

13 parametersKurtosisSkewness% outside 1.96*sd K-S distance Mean Std Dev n = 5 (a,b) = (0,1) (-1.2).004 (0) (.129) n = 5 (a,b) = (0,50) (6.455) n = 5 (a,b) = (0,.1) (.0129) n = 3 (a,b) = (0,50) (8.333) *from R

 Cumulative distribution function: 14

15 parametersKurtosisSkewness% outside 1.96*sd K-S distance Mean Std Dev n = 5 λ = (6).904 (2) (.4472) n = (.316) n = (.2581) *from R

 Find n values for more distributions  Refine criteria for quality of approximation  Explore meanless distributions  Classify distributions in order to have more general guidelines for minimum sample size 16

Paul Cornwell May 2,

 Central Limit Theorem: Averages of i.i.d. variables become normally distributed as sample size increases  Rate of converge depends on underlying distribution  What sample size is needed to produce a good approximation from the CLT? 18

 Real-life applications of the Central Limit Theorem  What does kurtosis tell us about a distribution?  What is the rationale for requiring np ≥ 5?  What about distributions with no mean? 19

 Probability for total distance covered in a random walk tends towards normal  Hypothesis testing  Confidence intervals (polling)  Signal processing, noise cancellation 20

 Measures the “peakedness” of a distribution  Higher peaks means fatter tails 21

 Traditional assumption for normality with binomial is np > 5 or 10  Skewness of binomial distribution increases as p moves away from.5  Larger n is required for convergence for skewed distributions 22

 Has no moments (including mean, variance)  Distribution of averages looks like regular distribution  CLT does not apply 23

 α = β = 1/3  Distribution is symmetric and bimodal  Convergence to normal is fast in averages 24

 Heavier-tailed, bell-shaped curve  Approaches normal distribution as degrees of freedom increase 25

 4 statistics: K-S distance, tail probabilities, skewness and kurtosis  Different thresholds for “adequate” and “superior” approximations  Both are fairly conservative 26

27 Distribution∣Kurtosis∣ <.5 ∣Skewness∣ <.25 Tail Prob..04<x<.06 K-S Distance <.05 max Uniform31223 Beta (α=β=1/3)41334 Exponential Binomial (p=.1) Binomial (p=.5) Student’s t with 2.5 df NA 1320 Student’s t with 4.1 df

28 Distribution∣Kurtosis∣ <.3 ∣Skewness∣ <.15 Tail Prob..04<x<.06 K-S Distance <.02 max Uniform41224 Beta (α=β=1/3)61346 Exponential Binomial (p=.1) Binomial (p=.5) Student’s t with 2.5 df NA Student’s t with 4.1 df

 Skewness is difficult to shake  Tail probabilities are fairly accurate for small sample sizes  Traditional recommendation is small for many common distributions 29