chapter2 - 9/1/2011 Introduction to Probability and...

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9/1/2011 1 Introduction to Probability and Statistics Thirteenth Edition Chapter 2 Describing Data with Numerical Measures Describing Data with Numerical Measures • Graphical methods may not always be sufficient for describing data. Numerical measures can be created for both populations and samples. –A parameter is a numerical descriptive measure calculated for a population . statistic is a numerical descriptive measure calculated for a sample . Measures of Center • A measure along the horizontal axis of the data distribution that locates the center of the distribution. Arithmetic Mean or Average • The mean of a set of measurements is the sum of the measurements divided by the total number of measurements. i x x n where n = number of measurements sum of all the measurements x i 
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9/1/2011 2 Example •The set: 2, 9, 1, 5, 6 n x x i 6 . 6 5 33 5 6 5 11 9 2 If we were able to enumerate the whole population, the population mean would be called m (the Greek letter ― mu ‖). • The median of a set of measurements is the middle measurement when the measurements are ranked from smallest to largest. • The position of the median is Median .5( n + 1) once the measurements have been ordered. Example • The set: 2, 4, 9, 8, 6, 5, 3 n = 7 • Sort: 2, 3, 4, 5, 6, 8, 9 • Position: .5( n + 1) = .5(7 + 1) = 4 th Median = 4 th largest measurement • The set: 2, 4, 9, 8, 6, 5 n = 6 • Sort: 2, 4, 5, 6, 8, 9 • Position: .5( n + 1) = .5(6 + 1) = 3.5 th Median = (5 + 6)/2 = 5.5 — average of the 3 rd and 4 th measurements Mode • The mode is the measurement which occurs most frequently. • The set: 2, 4, 9, 8, 8, 5, 3 –The mode is 8 , which occurs twice • The set: 2, 2, 9, 8, 8, 5, 3 –There are two modes— 8 and 2 ( bimodal ) • The set: 2, 4, 9, 8, 5, 3 –There is no mode (each value is unique).
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9/1/2011 3 Example Mean? Median? Mode? (Highest peak) The number of quarts of milk purchased by 25 households: 0 0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 2 . 2 25 55 n x x i 2 m 2 mode Quarts Relative frequency 5 4 3 2 1 0 10/25 8/25 6/25 4/25 2/25 0 • The mean is more easily affected by extremely large or small values than the median. Extreme Values •The median is often used as a measure of center when the distribution is skewed. Extreme Values Skewed left: Mean < Median Skewed right: Mean > Median Symmetric: Mean = Median Measures of Variability • A measure along the horizontal axis of the data distribution that describes the spread of the distribution from the center.
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9/1/2011 4 The Range • The range, R, of a set of n measurements is the difference between the largest and smallest measurements. Example: A botanist records the number of petals on 5 flowers: 5, 12, 6, 8, 14 • The range is R = 14 – 5 = 9. •Quick and easy, but only uses 2 of the 5 measurements.
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chapter2 - 9/1/2011 Introduction to Probability and...

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