Population Parameter and Sample Statistics Describing Data: Numerical Measures z 統計上用來描述資料特性的摘要性數值，稱之 為統計表徵數 (Statistical Characteristic ) 或統計 測量數 (Statistical Measurement)。 z 依描述母體與樣本的不同，分別稱之為： Chapter 3 – – McGraw-Hill/Irwin ©The McGraw-Hill Companies, Inc. 2008 2 樣本統計量 (Sample Statistic) 母體參數 (Population Parameter) z 母體參數是描述母體資料特性的統計測量數，一般簡 稱為參數或母數。參數是統計想要獲得的數值，是統 計的核心。 z 母體參數是個固定的常數。 z 但因常常無法直接求得，因此必須使用統計技巧，由 樣本資料估計。 z 3 母體參數 (Population Parameter) ------ 用來描述母體特質的測量數。 樣本統計量 (Sample Statistic) ------ 用來描述樣本特質的測量數。 z 樣本統計量是描述樣本資料特性的統計測量數，一般 簡稱為統計量。 z 樣本統計量可由樣本觀察值計算而得。 z 樣本統計量常用來推論母體參數。 z 例如：樣本平均數、樣本變異數、樣本標準差。 例如：母體平均數、變異數、標準差、、、、 4 1 常用統計表徵數 z Center of Location (中心位置): 測量母體或樣本中心位置 的指標。 z – – – – – Dispersion (分散程度)： 測量母體或樣本元素離散程度的 指標。 – – – – – Range（全距） Mean Deviation（平均離差） Variance（變異數） Standard Deviation（標準差） 5 – Mean Median Mode Geometric Mean 6 Population Mean Characteristics of the Mean For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: The arithmetic mean is the most widely used measure of location. It requires the interval scale. Its major characteristics are: – – – – 7 Center of Location (中心位置):測量母體 或樣本中心位置的指標。 Mean（算數平均數） Median（中位數） Mode（眾數） Geometric Mean（幾何平均數） – z Center of Location All values are used. It is unique. The sum of the deviations from the mean is 0. It is calculated by summing the values and dividing by the number of values. 8 2 EXAMPLE – Population Mean Sample Mean z 9 10 EXAMPLE – Sample Mean Properties of the Arithmetic Mean z z z z z 11 For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: Every set of interval-level and ratio-level data has a mean. All the values are included in computing the mean. A set of data has a unique mean. The mean is affected by unusually large or small data values. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero. 12 3 Weighted Mean z EXAMPLE – Weighted Mean The Carter Construction Company pays its hourly employees $16.50, $19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the $25.00 rate. What is the mean hourly rate paid the 26 employees? The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula: 13 14 The Median z The Median is the midpoint of the values after they have been ordered from the smallest to the largest. – – 15 Properties of the Median z There is a unique median for each data set. z It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. There are as many values above the median as below it in the data array. z It can be computed for ratio-level, interval-level, and ordinal-level data. For an even set of values, the median will be the arithmetic average of the two middle numbers. z It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. 16 4 EXAMPLES - Median The ages for a sample of five college students are: 21, 25, 19, 20, 22 Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. 17 The Mode z The heights of four basketball players, in inches, are: 76, 73, 80, 75 The mode is the value of the observation that appears most frequently. Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5 18 Example - Mode Mean vs. Median z z 當分配是對稱時 (Symmetric)，Mean 與 Median 是相 同的。 當分配是傾斜的 (Skewed)，Mean 與 Median 有較大 差別。 – – 19 可以把 Mean 當成重心，當分配向右傾斜時，重心會向右， 而 Median 對每個元素所給的重量（權重）是相同的，所以 不受極端值影響。 X={1,2,3,4,5} Y={1,2,3,4,90} Mean(X) = 3 Median(X)=3 Mean(Y) = 20 Median(Y) =3 20 5 The Relative Positions of the Mean, Median and the Mode The Geometric Mean z z z 21 Useful in finding the average change of percentages, ratios, indexes, or growth rates over time. It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other. The geometric mean will always be less than or equal to the arithmetic mean. 22 The Geometric Mean z z EXAMPLE – Geometric Mean Suppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. We begin by calculating the geometric mean. The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values. The formula for the geometric mean is written: GM = ( 1.05 )( 1.15 ) = 1.09886 23 24 6 EXAMPLE – Geometric Mean (2) EXAMPLE – Geometric Mean (3) The return on investment earned by Atkins construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment? The prices of a stock for three successive days were: 100, 60 , 90. What is the geometric mean rate of return of this investment? GM = 4 ( 1.3 )( 1.2 )( 0.6 )( 3.0 ) = 4 2.808 = 1.294 GM = 2 ( 0.6 )( 1.5 ) − 1 = 2 0.9 − 1 = −5.13% If you compute the arithmetic mean [30+20-40+200]/4 = 52.5, you would overstate the mean rate of return. 25 If you compute the arithmetic mean [-40%+50%]/2 = 5%, you would overstate the mean rate of return. 26 Why Study Dispersion? Dispersion or Variability （分散程度） z 27 Dispersion：測量母體或樣本元素離散程度的 指標。 – Range – Mean Deviation – Variance – Standard Deviation z A measure of location, such as the mean or the median, only describes the center of the data. It is valuable from that standpoint, but it does not tell us anything about the spread of the data. z For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth. 28 7 Why Study Dispersion? Why Study Dispersion? • A measure of dispersion can be used to evaluate the reliability of two or more measures of location. • Mean = 4.9, but the spread of the data is from 6m to 16.8y. • The average is not representative because of the large spread. 29 30 Mean Deviation Range Mean Deviation: The arithmetic mean of the absolute values of the deviations from the arithmetic mean. Advantage: 1. All data are used in the computation. 2. Easy to understand : average of deviations form the mean. Drawback: --- Absolute values are difficult to work with. Example: The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the range for the number of cappuccinos sold. Range = Largest – Smallest value = 80 – 20 = 60 31 32 8 EXAMPLE – Mean Deviation Variance and Standard Deviation The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the mean deviation for the number of cappuccinos sold. z z 33 34 EXAMPLE – Population Variance and Standard Deviation Population Variance and Standard Deviation z z 35 Variance --- The arithmetic mean of the squared deviations from the mean. Standard Deviation --- The square root of the variance. The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What is the population variance? Population Variance Population standard Deviation 36 9 EXAMPLE – Sample Variance Sample Variance and Standard Deviation z z The hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. What is the sample variance? Sample Variance Sample standard Deviation Square root of sample variance. 37 38 Computational Form Chebyshev’s Theorem n Sum of Square (SS) = n ∑ ( X -X ) = ∑ X 2 i i=1 2 i - n (X) 2 i=1 SS Population Variance = n SS Sample Variance = n-1 ≥ 1Mean -kth sd 39 1 k2 Mean Mean + kth sd 40 10 Chebyshev’s Theorem The Empirical Rule The arithmetic mean biweekly amount contributed by the Dupree Paint employees to the company’s profitsharing plan is $51.54, and the standard deviation is $7.51. At least what percent of the contributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean? 41 42 The Arithmetic Mean of Grouped Data Example The Arithmetic Mean of Grouped Data Recall in Chapter 2, we constructed a frequency distribution for the vehicle selling prices. The information is repeated below. Determine the arithmetic mean vehicle selling price. 43 44 11 The Arithmetic Mean of Grouped Data Example 45 Standard Deviation of Grouped Data 46 Standard Deviation of Grouped Data Example Exercises z 47 1,3,7,9,13,19, 25, 27, 29, 31, 35, 39, 41, 47, 51, 53, 55, 59,61, 65, 79, 81, 83 48 12 End of Chapter 3 49 13

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