Chap 3 - Chapter 3 Descriptive Statistics / Describing...

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Chapter 3 Descriptive Statistics / Describing Distributions with Numbers A. Common Measurements of Location -these measurements give you a sense of where a data point falls in line relative to other data points. Note: If you calculate a stat from a sample it is called a sample statistic. If you calculate it from a population statistic then it is called a population statistic. 1. Mean – this is the most common measurement. It is simply the average. It is one of three measures of centrality. a. sample mean - _ x = ∑x i / n = x 1 + x 2 + …. + x n b. population mean – μ = ∑x i / N note: n = total number in sample N = total number in population 2. Median –M d – The middle value when the data is arranged in ascending order; second measure of center. - if the data has an odd number of points, then there is a true middle -if the data has an even number of points, then take the average of the two middle points. Example: If you are given 3, 6, 7, 8, 8, 10 Since there are an even number of points we take (x 3 + x 4 ) / 2 = (7 + 8) /2 = 7.5 3. Mode –M o - this is the value that occurs with the greatest frequency. If there is more than on value that takes on the greatest frequency then we say that the value is bi, tri, or multi-modal; last measure or centrality. 4. Percentiles/Quartiles – tells us how data are spread over a 100 percent interval from smallest to largest. How to calculate percentage: (a) arrange data in ascending order (b) Compute the index of the percent Index – i = (p/100)n p = percentile of interest n = number of operations (c) If it is not an integer round up. If it is an integer then the pth percentile is average of the i and i+1 value a. For Lower Quartile (Q1 or 25 th percentile): i. Sort all observations in ascending order 1
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ii. Compute the position L1 = 0.25 * N, where N is the total number of observations. iii. If L1 is a whole number, the lower quartile is midway between the L1-th value and the next one. iv. If L1 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the lower quartile. b. For Upper Quartile (Q3 or 75 th percentile): i. Sort all observations in ascending order ii. Compute the position L3 = 0.75 * N, where N is the total number of observations. iii. If L3 is a whole number, the lower quartile is midway between the L3-th value and the next one. iv. If L3 is not a whole number, change it by rounding up to the nearest integer. The value at that position is the lower quartile. Example: 61, 61, 61, 67, 73, 73, 74, 79, 81, 81, 87, 89, 89, 92, 97, 100 Given our data from test scores again if we wanted to know the 40 th percentile we obtain it as follows. i = (40/100) 16 = .4*16 = 6.4 = 7 So our 7 th value is our 40 th percentile. This corresponds to 74. 5. Quartiles
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Chap 3 - Chapter 3 Descriptive Statistics / Describing...

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