The Central Limit Theorem can be demonstrated by taking sample mean distributions from a population that is not normally distributed. Graph the probability distribution for rolling a six-sided die as a relative frequency histogram. Determine the mean and standard deviation for this distribution. Construct a relative frequency distribution for a sample mean distribution with sample size of 4 for rolling a six-sided die using 100 trials. Sketch this as a relative frequency histogram using the same horizontal and vertical scales as the relative frequency histogram for the population. This means using classes of ̅̅  1  ≤  x ̅  < 2,  2  ≤ x ̅  <3,  etc. This can be done using real dice rolls or simulations using random number generators. Determine the mean and standard deviation for the sample mean distribution with n = 4. Repeat the process for the sample mean when n = 9. Graph the relative frequency histogram using the same scale as the previous wo distributions. Enter the mean and standard deviations for all three distributions in the table below.   Distribution Mean Standard Deviation Population     Sample Mean, n=4     Sample Mean, n=9

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...