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Perform Descriptive Measures Statistical Procedures FINANCE SCHOOL 805A-44A-1033-VG-1 Terminal Learning Objective Action: Perform descriptive measures statistical procedures by computing the measures of central tendency and dispersion; constructing a frequency distribution (absolute, relative, and cumulative); plotting a histogram, frequency polygon, and cumulative frequency distribution; and identify the skewness of the direction. Conditions: Given the Formulas provided in the Summary Sheet and a data set (population or sample). Standard: IAW the Descriptive Measures Programmed Text and Accepted Statistical Techniques. FINANCE SCHOOL 805A-44A-1033-VG-2 What is (or are) Statistics? “There are Three Types of Lies: LIES, DAMNED LIES, and STATISTICS.” -- DISRAELI. Other quotes are: There was a Man who Drowned in a River only 2 Feet Deep --- On Average, That Is. People use Statistics as a Drunk uses a Lamppost --- More for Support than for Illumination. Statistics are like a Razor --- Vital, Variable, And Vicious! Figures Don’t Lie, But Liars Figure! FINANCE SCHOOL 805A-44A-1033-VG-3 Descriptive Terms • Population (N): is the whole thing. • Sample (n): is a part of the whole. • Parameter. • Statistic. FINANCE SCHOOL 805A-44A-1033-VG-4 Sampling Procedures Select a Sample Population Sample Parameter Statistic Make an Inference FINANCE SCHOOL 805A-44A-1033-VG-5 Measures of Central Tendency Mean. Median. Mode. FINANCE SCHOOL 805A-44A-1033-VG-6 MEAN • Population Mean (): = X Where: X = observations = arithmetic population mean N = # observations Where: • Sample Mean (X): X = X X = observations n X = arithmetic population mean n = # observations FINANCE SCHOOL 805A-44A-1033-VG-7 Population = X = 7 + 9 + 10 + 10 + 11 + 13 = 60 = 10 N 6 6 FINANCE SCHOOL 805A-44A-1033-VG-8 Extreme Values = 7 + 9 + 10 + 10 + 11 + 13 + 31 = 91 = 13 7 7 FINANCE SCHOOL 805A-44A-1033-VG-9 Characteristics of the Mean • Familiar to most people. • Always exists. • Always unique. • Lends itself to statistical manipulation. • Accounts for each individual item. • Reliable estimator. FINANCE SCHOOL 805A-44A-1033-VG-10 MEDIAN The middle item (or mean of the two middle items) found within a set of data, when the data is arranged in order of magnitude. FINANCE SCHOOL 805A-44A-1033-VG-11 Characteristics of the Median • Always exists. • Always unique. • Ordered array required. • Less reliable than the mean as an estimator. FINANCE SCHOOL 805A-44A-1033-VG-12 MODE The most frequently occurring observation or observations in the data. X X X 7 7 7 9 9 9 10 10 9 ] 13 10 ] 10 16 13 13 17 15 13 ] No Mode Unimodal Bimodal FINANCE SCHOOL 805A-44A-1033-VG-13 Characteristics of the Mode • May not always exist. • Requires no calculation. • Are not unique. FINANCE SCHOOL 805A-44A-1033-VG-14 Arithmetic Mean Population Mean = Sample Mean = X FINANCE SCHOOL 805A-44A-1033-VG-15 Measures of Dispersion Range. Variance. Standard Deviation. FINANCE SCHOOL 805A-44A-1033-VG-16 RANGE The difference between the two extreme values of a set of a data. RANGE = LARGEST VALUE - SMALLEST VALUE Rg = Xmax - Xmin FINANCE SCHOOL 805A-44A-1033-VG-17 Average Deviation Each data point of the data set differs from the mean by a measurable amount. We can measure this in a general sense by the formula: di = Xi - where di is the difference between any specific data point, Xi , and the mean. FINANCE SCHOOL 805A-44A-1033-VG-18 Original Data Set Let’s consider our original data set: 7, 9, 10, 10, 11, 13 the mean is: = 7 + 9 + 10 + 10 + 11 + 13 = 60 = 10 6 6 FINANCE SCHOOL 805A-44A-1033-VG-19 Data Point Now we find the deviation of each data point from the mean: X X- = d 7 7 - 10 = -3 9 9 - 10 = -1 10 10 - 10 = 0 10 10 - 10 = 0 11 11 - 10 = +1 13 13 - 10 = +3 (X - ) = 0 N For this and any data set, the average of the deviations about the mean is Zero. FINANCE SCHOOL 805A-44A-1033-VG-20 VARIANCE The mean of the squares of the variations from the mean of a frequency distribution. 2 x) 2 x - N 2 Population Variance: N 2 x) 2 x - n 2 Sample Variance: S n-1 FINANCE SCHOOL 805A-44A-1033-VG-21 Standard Deviation The square root of the mean of the standard deviations. 2 Population Std Deviation: x) x - N 2 2 N Sample Std Deviation: 2 x) 2 x - n 2 S S n-1 FINANCE SCHOOL 805A-44A-1033-VG-22 Squared Deviations X X- = d (X - )2 7 7 - 10 = -3 9 9 9 - 10 = -1 1 10 10 - 10 = 0 0 10 10 - 10 = 0 0 11 11 - 10 = +1 1 13 13 - 10 = +3 9 X = 60 (X - ) = 0 20 N FINANCE SCHOOL 805A-44A-1033-VG-23 Mean Squared Deviation The average or mean squared deviation is found by dividing the sum of the squared deviations by the number of observations: (X - )2 = 20 = 3 1/3 N 6 The mean squared deviation is also called the variance, and is denoted by the lower case Greek letter (sigma) squared: 2 FINANCE SCHOOL 805A-44A-1033-VG-24 Alternative Formulation An alternative formulation for both the variance and standard deviation simplifies the work involved. These are: (X)2 Variance: 2 = (X - )2 = X2 - N N N (X)2 Standard Deviation = = X2 - N N FINANCE SCHOOL 805A-44A-1033-VG-25 Worksheet A Worksheet A for 2 = (X - )2 N X X- = d (X - )2 7 7 - 10 = -3 9 9 9 - 10 = -1 1 10 10 - 10 = 0 0 10 10 - 10 = 0 0 11 11 - 10 = +1 1 13 13 - 10 = +3 9 X = 60 (X - ) = 0 20 N = 60 = 10 2 = 20 = 3.3333 = 3.3333 = 1.8257 6 6 FINANCE SCHOOL 805A-44A-1033-VG-26 Worksheet B (X)2 Worksheet B for 2 = X2 - N N X X2 7 49 9 81 10 100 10 100 11 121 13 169 X = 60 X2 = 620 (60)2 2 = 620 - 6 = 620 - 600 = 20 = 3.3333 6 6 6 = 3.3333 = 1.8257 FINANCE SCHOOL 805A-44A-1033-VG-27 Coefficient of Variation C = 100(/) - standard deviation is a measure of absolute variability. - coefficient of variation is a measure of relative variability. Group A Group B = $6.61 $5.82 = .57 .51 C = 8.62% 8.76% FINANCE SCHOOL 805A-44A-1033-VG-28 Federalist Papers 0.4 Hamilton 0.35 Madison 0.3 Proportion of Papers Disputed 0.25 0.2 0.15 0.1 0.05 0 0 1 3 5 7 9 11 13 15 17 19 Rate of Usage FINANCE SCHOOL 805A-44A-1033-VG-29 Pure Chemical Company Output of 10 curing VATs 19 Jan **. VAT Gallons Produced A 65 B 67 C 66 D 68 E 66 F 67 G 66 H 65 I 64 J 68 FINANCE SCHOOL 805A-44A-1033-VG-30 Frequency Distribution (Pure Chemical Company) VAT output 19 Jan **. Frequency Output Gallons Absolute Relative 64 1 1/10 = 0.1 65 2 2.10 = 0.2 66 3 3.10 = 0.3 67 2 2/10 = 0.2 68 2 2/10 = 0.2 TOTALS 10 1.0 FINANCE SCHOOL 805A-44A-1033-VG-31 Histogram (Pure Chemical Company) Curing VAT output 19 Jan **. 3 0.3 Relative Frequency Absolute Frequency 2 0.2 1 0.1 0 0 0 64 65 66 67 68 0 Output (Gallons) FINANCE SCHOOL 805A-44A-1033-VG-32 Frequency Polygon (Pure Chemical Company) Curing VAT output 19 Jan **. 3 0.3 Relative Frequency Absolute Frequency 2 0.2 0.1 1 0 0 0 64 65 66 67 68 0 Output (Gallons) FINANCE SCHOOL 805A-44A-1033-VG-33 Population Test N = 34 ROSTER ROSTER NUMBER GRADE NUMBER GRADE 1 98 18 99 2 84 19 78 3 86 20 99 4 100 21 86 5 94 22 95 6 97 23 100 7 81 24 93 8 86 25 97 9 99 26 97 FINANCE SCHOOL 805A-44A-1033-VG-34 Population Test N = 34 (Cont) ROSTER ROSTER NUMBER GRADE NUMBER GRADE 10 87 27 79 11 86 28 91 12 90 29 88 13 89 30 93 14 96 31 94 15 94 32 84 16 98 33 89 17 90 34 91 FINANCE SCHOOL 805A-44A-1033-VG-35 Frequency Distribution for Test Scores FREQUENCY . MIDPOINT CLASS INTERVAL TALLY ABSOLUTE RELATIVE CUMULATIVE 80 78-81 3 .088 .088 84 82-85 2 .059 .147 88 86-89 8 .235 .382 92 90-93 6 .176 .558 96 94-97 8 .235 .793 100 98-100 7 .206 .999 34 .999* * NOT EXACT DUE TO ROUNDING. FINANCE SCHOOL 805A-44A-1033-VG-36 Test Score Histogram 8 Absolute Frequency 6 4 2 0 78 82 86 90 94 98 102 Grade FINANCE SCHOOL 805A-44A-1033-VG-37 Test Score Frequency Polygon Cumulative Frequency Distribution 1 Cumulative Frequency 0.8 0.6 0.4 0.2 0 76 80 84 88 92 96 100 Grade (Midpoint) FINANCE SCHOOL 805A-44A-1033-VG-38 Symmetrical Frequency Distribution Frequency 50% 50% Mean X Median Mode FINANCE SCHOOL 805A-44A-1033-VG-39 Skewed Frequency Distributions MODE MEDIAN F MEAN X (1) SKEWED RIGHT MODE F MEDIAN MEAN X (2) SKEWED LEFT FINANCE SCHOOL 805A-44A-1033-VG-40 Descriptive Measures SUMMARY • Compute and explain the uses of the following measures of central tendency and dispersion: - Mean - Range. - Median - Variance. - Mode - Standard Deviation. • Construct and plot a frequency distribution and identify the skewness of the distribution. FINANCE SCHOOL 805A-44A-1033-VG-41 Terminal Learning Objective Action: Perform descriptive measures statistical procedures by computing the measures of central tendency and dispersion; constructing a frequency distribution (absolute, relative, and cumulative); plotting a histogram, frequency polygon, and cumulative frequency distribution; and identify the skewness of the direction. Conditions: Given the Formulas provided in the Summary Sheet and a data set (population or sample). Standard: IAW the Descriptive Measures Programmed Text and Accepted Statistical Techniques FINANCE SCHOOL 805A-44A-1033-VG-42

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