2-Chapter 4 - Chapter 4 Numerical Descriptive Descriptive...

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1 Numerical Descriptive Chapter 4 Descriptive Techniques Measures we will look at ± Usually, we focus our attention on three types of measures when describing population characteristics: ² Central location (e.g. average) Variability or spread (e g standard deviation ² Variability or spread (e.g. standard deviation) ² Association (correlation) The measure of central location reflects the locations of all the actual data points. 4.1 Measures of Central Location ± The measure of central location reflects the locations of all the actual data points. ± How? With two data points, the central location With one data point clearly the central location is at the point itself. should fall in the middle between them (in order to reflect the location of both of them).
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2 Measures of Central Location ± There are 4 main measures of central location ² The Arithmetic Mean ² The Median ² The Mode ² The Geometric Mean ± This is the most popular and useful measure of central location The Arithmetic Mean Sum of the observations Number of observations Mean = n Sample mean Population mean N The Arithmetic Mean – pg. 91 n x x i 1 i = = N x i 1 i = = μ Sample size Population size n x x i n 1 i = =
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3 = + + + = = = 10 ... 10 2 10 1 x x i i Example 4.1 (pg. 98) The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours. Find the mean time on the Internet. 22 11.0 The Arithmetic Mean Example 4.2 (pg. 98) Suppose the telephone bills represent the population of measurements. The population mean is = + + = = μ = 200 ... 200 x i 200 1 i 42.19 38.45 45.77 43.59 Example 4.3 (pg99) ± The Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude. The Median Comment Odd number of observations 9, 12, 14, 22 , 12, 14, 22, 33 0, 0, 5, 7, 8, 9, 12, 14, 22, 33 Even number of observations Find the median of the time on the internet for the 10 adults of example 4.1 Suppose only 9 adults were sampled (exclude, say, the longest time (33)) 8.5, 8 ± The Mode of a set of observations is the value that occurs most frequently. ± Set of data may have one mode (or modal class), or two or more modes. The Mode The modal class For large data sets the modal class is much more relevant than a single-value mode.
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4 ± Example 4.5 (pg. 101) ± Find the mode for the data in Example 4.1 (pg99). Here are the data again: 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 Solution The Mode • All observation except “0” occur once. There are two “0”. Thus, the mode is zero. • Is this a good measure of central location? • The value “0” does not reside at the center of this set (compare with the mean = 11.0 and the mode = 8.5). Relationship among Mean, Median, and Mode ± If a distribution is symmetrical, the mean, median and mode coincide ± If a distribution is asymmetrical, and skewed to the left or to the right, the three measures differ.
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2-Chapter 4 - Chapter 4 Numerical Descriptive Descriptive...

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