1. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-1 Supplement 2: Comparing the two estimators of population variance by simulations

2. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-2 Average of squared deviations of values from the mean Population variance: Population Variance Where = population mean N = population size x i = i th value of the variable x

3. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-3 Sample analogue of the population variance Average of squared deviations of values from the mean Sample variance #1: Where = arithmetic mean n = sample size X i = i th value of the variable X

4. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-4 Average (approximately) of squared deviations of values from the mean Sample variance #2: Sample Variance Where = arithmetic mean n = sample size X i = i th value of the variable X

5. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-5 Sample variance #1 vs. #2 Claim: Sample variance #1 is a biased estimator of population variance. Sample variance #2 is a unbiased estimator of population variance. An estimator is unbiased of the population variance only if on average, the estimator correctly estimates the population variance.

6. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-6 Using Monte Carlo simulation to compare the properties of the two estimators Create a population of 100 population values from 1 to 100 with equal chance of occurrence in a worksheet called “population.” Generate 1000 sample of size 10 in a worksheet called “samples” based on random sample with replacement from the population according to the probability assigned. [Tools > Data Analysis > Random Number Generation > Discrete ….] For each of the sample compute the variance #1 and variance #2. Produce average of the 1000 sample variance #1 and variance #2 in a worksheet called “summary”. Result: variance.xlsvariance.xls

7. Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson0-7 - END - Supplement 2: Comparing the two estimators of population variance by simulations