B AD 6243: Applied Univariate Statistics Data Distributions and Sampling Professor Laku Chidambaram Price College of Business University of Oklahoma

BAD 6243: Applied Univariate Statistics 2 Distributions A distribution is a way of representing the frequency of occurrence of values for a variable A distribution can be discrete (e.g., Bernoulli, Binomial, Poisson) or continuous (e.g., Normal, Exponential, Uniform) A histogram, representing a probability density function, depicts a distribution Distributions are defined by the functional form and the values of parameters Our focus is on the shape of such distributions and their implications for statistical inference

BAD 6243: Applied Univariate Statistics 3 Normal Distribution Refers to a family of distributions (a.k.a Gaussian distributions) that are bell shaped and: –Represent a continuous probability distribution –Are symmetric (with scores concentrated in the middle) –Can be specified mathematically in terms of two parameters: the mean (μ) and the standard deviation (σ) –Have one mode –Are asymptotic

BAD 6243: Applied Univariate Statistics 4 An Example

BAD 6243: Applied Univariate Statistics 5 Standard Normal Distribution The area P under the standard normal probability curve, with the respective z-statistic

BAD 6243: Applied Univariate Statistics 6 The z Distribution The standard normal distribution, sometimes called the z distribution (as indicated by the formula below), is a normal distribution with a mean of 0 and a standard deviation of 1 Normal distributions can be transformed to a standard normal distribution using the formula: where X is a score from the original normal distribution, μ is its mean and σ is the standard deviation A z-score represents the number of standard deviations above or below the mean Note that the z distribution will only be a normal distribution if the original distribution (X) is normal

BAD 6243: Applied Univariate Statistics 7 Areas Under the Curve The Empirical Rule: Chebyshev’s Theorem: The portion of any set of data within K standard deviations of the mean is always at least 1 - 1/K 2, where K > 1 (For K=2, at least 75% of data must be within 2 SDs of mean, for K=3, 89% …)

BAD 6243: Applied Univariate Statistics 8 An Example If IQ scores are normally distributed, with a mean of 100 and a standard deviation of 15, –what proportion of scores would be greater than 125? –what proportion of scores would fall between 90 and 120? –what proportion of scores would be less than 85?

BAD 6243: Applied Univariate Statistics 9 Distribution and Sampling Concepts Central Limit Theorem –As sample size increases, the sampling distribution of the mean for simple random samples of n cases, taken from a population with a mean equal to  and a finite variance equal to  2, approximates a normal distribution Sampling Distribution of the Mean* Standard Deviation vs. Standard Error of the Mean Sample Size vs. Number of Samples

BAD 6243: Applied Univariate Statistics 10 t Distributions t-distributions refer to a family of distributions, which like normal distributions, are bell-shaped, but whose shape changes with the sample size; smaller sample sizes have flatter distributions, while larger sizes approximate normal distributions

BAD 6243: Applied Univariate Statistics 11 Chi-squared Distributions The chi-square (  2 ) distribution refers to a family of distributions (derived from the normal distribution) with one parameter, k, the degrees of freedom (=n-1) with mean=k and variance = 2k

BAD 6243: Applied Univariate Statistics 12 F Distributions The F distributions are a family of right-skewed distributions (often used in the Analysis of Variance) that represent the ratio of two chi-square distributions. Thus, an F distribution for a given significance level is denoted by the degrees of freedom for the numerator chi-square (df1) and the degrees of freedom for the denominator chi-square (df2).