Calculating Measures of Centrality and Variation for a Set of Data in Economics, Study notes of Economics

Download Calculating Measures of Centrality and Variation for a Set of Data in Economics and more Study notes Economics in PDF only on Docsity! Economics 3640-001 Instructor: Sanghoon Lee Lecture 3 * Reading Assignment: Ch.2 Methods for Describing Sets of Data (2.6 ~ 2.7) Ex. Active nuclear power plants. (p.66 & p.71) The U.S. Energy Information Administration monitors all nuclear power plants operating in the U.S. The table below lists the number of active nuclear power plants operating in each of a sample of 20 states. a. Find the mean, median, and mode of this data set. b. Eliminate the largest value from the data set and repeat part a. What effect does dropping this measurement have on the measures of central tendency found in part a? c. Arrange the 20 values in the table from lowest to highest. Next, eliminate the lowest two values and the highest two values from the data set and find the mean of the remaining data values. The result is called a 10% trimmed mean, since it is calculated after removing the highest 10% and the lowest 10% of the data values. What advantages does a trimmed mean have over the regular arithmetic mean? d. Find the variance and the standard deviation of the data set. e. Eliminate the largest value from the data set and repeat part d. What effect does dropping this measurement have on the measures of variation found in part d? f. Eliminate the smallest and largest value from the data set and repeat part d. What effect does dropping both of these measurements have on the measures of variation found in part d? You now know that the s measures the variability of a set of data and how to calculate it. The larger the s, the more variable the data are. The smaller the s, the less variation in the data. 2.6 Interpreting the Standard Deviation In Table 2.2, x─ = 36.99 and s = 2.42. x─ – s = 34.57 x─ + s = 39.41 If we examine the data, we find that 68 of the 100 measurements are in the interval 34.57 (x─ – s) to 39.41 (x─ + s). 1 x─ – 2s = 32.15, x─ + 2s = 41.83: 96% Table 2.7 The Empirical Rule applies to mound-shaped and symmetric distributions of data. a. Approximately 68% of the measurements will fall within 1 standard deviation of the mean, i.e., within the interval (x─ – s, x─ + s) for samples and (μ – σ, μ + σ) for populations. b. Approximately 95% of the measurements will fall within 2 standard deviations of the mean, i.e., within the interval (x─ – 2s, x─ + 2s) for samples and (μ – 2σ, μ + 2σ) for populations. c. Approximately 99.7% (essentially all) of the measurements will fall within 3 standard deviations of the mean, i.e., within the interval (x─ – 3s, x─ + 3s) for samples and (μ – 3σ, μ + 3σ) for populations. Table 2.6 Chebyshev's Rule applies to any data set. d. Generally, for any number k greater than 1, at least (1 – 1/k2) of the measurements will fall within the interval (x─ – ks, x─ + ks) for samples and (μ – kσ, μ + kσ) for populations. Ex. 2.84 The following is a sample of 25 measurements: 7 6 6 11 8 9 11 9 10 8 7 7 5 9 10 7 7 7 7 9 12 10 10 8 6 a. Compute x─, s2, and s for this sample. b. Count the number of measurements in the intervals x─ ± s, x─ ± 2s, and x─ ± 3s. Express each count as a percentage of the total number of measurements. c. Compare the percentages found in part b to the percentages given by the Empirical Rule and Chebyshev's Rule. Ex. 2.85 Given a data set with a largest value of 760 and a smallest value of 135, what would you estimate the standard deviation to be? Explain the logic behind the procedure you used to estimate the standard deviation. Suppose the standard deviation is reported to be 25. Is this feasible? Explain. Ex. 2.88 Recommendation letters for professors. Recall that n = 148, x─ = 2.28 and s = 1.48. a. Sketch the relative frequency distribution for the number of recommendation letters included in each application for the experimental psychology position. Assume the distribution is mound-shaped and relatively symmetric. b. Locate an interval on the distribution, part a, that captures approximately 95% of the measurements in the sample. c. Locate an interval on the distribution, part a, that captures almost all the sample measurements. 2.7 Numerical Measures of Relative Standing Def. 2.13) The sample z-score for a measurement x is z = (x – x─) / s The population z-score for a measurement x is z = (x – μ) / σ Eg., sample z = – 1.0: x is 1.0 standard deviation below the sample mean. 2