Lecture-4-Summer2010

Lecture-4-Summer2010 - LECTURE 4 Descriptive Statistics:...

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LECTURE 4 Descriptive Statistics: Numerical Methods (Part 2) This lecture covers material on z-scores, Chebyshev's theorem, Empirical rule, and five number summary, covariance and correlation. Microsoft Excel is used for some applications. Read: Chapter 3, Sections 3.3, 3.4 and 3.5. Applications of the Mean and Standard Deviation z-scores: The z-score is also known as the standardized value. For a data set having a number of items, we can compute the mean and the standard deviations. The z-score for a data value x i shows the number of standard deviations that value is from the mean . Note: A positive z-score for a data value indicates it is greater than the mean; conversely a negative z-score indicates that the data value is less than the mean. --------------------------------- EXAMPLE 1 For the monthly payments data, we have calculated the mean = 1619.94; and the standard deviation s = 1326. Let us compute the z- score for two of the data values: x 1 = 542 and x 2 = 2345. For x 1 = 542, z = (x 1 - )/s or z = (542 - 1619.94)/1326 or z = -0.813 This means that the data item 542 is -0.813 standard deviations from the mean. 1
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For x 2 = 2345, z = (x 2 - )/s or z = (2345 - 1619.94)/1326 or z = 0.546 This means that the data item 2345 is 0.546 standard deviations from the mean. Chebyshev's theorem: For any distribution, and for z > 1, at least (1-1/ z 2 ) data values must be within z standard deviations of the mean. Essentially the theorem tells us that for any data set , a minimum number of points will be within z standard deviations from the mean. This is useful in understanding data sets since it gives us information on how much the data is dispersed around the mean. --------------------------------- EXAMPLE 2 For the monthly payments data, we have calculated the mean = 1619.94; and the standard deviation s = 1326. Let z = 2, then (1-1/ z 2 ) = (1-1/2 2 ) = 0.75 or 75%. Then at least 75% of the data will be within ± 2s of the mean . + 2s = 1619.94 + 2(1326) = 4271.94 - 2s = 1619.94 - 2(1326) = -1032.06 We can see from the monthly payments data that 15 of the 16 data points or 93.75% of the data are within the range -1032.06 to 4271.94. ----------------------------------- The Empirical Rule: For data having a bell-shaped distribution: Approximately 68% of data items are within ± 1 standard deviation from the mean. Approximately 95% of data items are within ± 2 standard deviation from the mean. Approximately all of data items are within ± 3 standard deviation from the mean. 2
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Note: This is an important result because many data sets exhibit a bell shaped distribution which can be approximated by the normal distribution (about which we will discuss later). The empirical rule therefore provides information about the distribution of data for this commonly found distribution. Since almost all data values should lie within ± 3s from the mean, those data
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Lecture-4-Summer2010 - LECTURE 4 Descriptive Statistics:...

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