Measuring CenterMeasuring the CenterSlide 3Slide 4Mean, Median, and ModeSlide 6Slide 7Slide 8Slide 9The Median vs. The MeanThe MeanSlide 12Slide 13Slide 14Slide 15The MedianSlide 17Slide 18Slide 19Slide 20Slide 21TI-83 – The MeanCase Study 8Slide 24Slide 25Slide 26Slide 27Slide 28TI-83 – The MedianSlide 30The ModeModeWeighted MeansSlide 34Slide 35Lecture 15Sections 5.1 – 5.2Mon, Feb 11, 2008Measuring CenterMeasuring the CenterOften, we would like to have one number that that is “representative” of a population or sample.It seems reasonable to choose a number that is near the “center” of the distribution rather than in the left or right extremes.But there is no single “correct” way to do this.Measuring the CenterMean – the simple average of a set of numbers.Median – the value that divides the set of numbers into a lower half and an upper half.Mode – the most frequently occurring value in the set of numbers.Measuring the CenterIn a unimodal, symmetric distribution, these values will all be near the center.In skewed distributions, they will be spread out.Mean, Median, and ModeIf a distribution is symmetric, then the mean, median, and mode are all the same and are all at the center of the distribution.Mean, Median, and ModeHowever, if the distribution is skewed, then the mean, median, and mode are all different.Mean, Median, and ModeHowever, if the distribution is skewed, then the mean, median, and mode are all different.The mode is at the peak.ModeMean, Median, and ModeHowever, if the distribution is skewed, then the mean, median, and mode are all different.The mean is shifted in the direction of skewing.ModeMeanMean, Median, and ModeModeMedianMeanHowever, if the distribution is skewed, then the mean, median, and mode are all different.The median is (typically) between the mode and the mean.The Median vs. The MeanIf the data are strongly skewed, then the median is generally to give a more representative value.If the data are not skewed, then the mean is usually preferred.The MeanWhy is the average usually a good measure of the center?If we have only two numbers, the average is half way between them.What if we have more than two numbers?The mean balances the “deviations” on the left with the “deviations” on the right.The Mean1 2 3 4 5 6 7 8 109The Mean1 2 3 4 5 6 7 8 109AverageThe Mean1 2 3 4 5 6 7 8 109Average-2-5The Mean1 2 3 4 5 6 7 8 109Average-2-5+1+2+4The Median1 2 3 4 5 6 7 8 109The Median1 2 3 4 5 6 7 8 109MedianThe Median1 2 3 4 5 6 7 8 109Median-3-6The Median1 2 3 4 5 6 7 8 109Median+1+3-3-6The MeanWe use the letter x to denote a value from the sample or population.The symbol means “add them all up.”So, x means add up all the values in the population or sample (depending on the context).Then the sample mean isnxThe MeanWe denote the mean of a sample by the symbolx, pronounced “x bar”.We denote the mean of a population by , pronounced “mu” (myoo).Therefore,NxnxxTI-83 – The MeanEnter the data into a list, say L1.Press STAT > CALC > 1-Var Stats.Press ENTER. “1-Var-Stats” appears.Type L1 and press ENTER.A list of statistics appears. The first one is the mean.Case Study 82007 Small Arms Study (p. 47)Find the average number of guns per country for India, China, Germany, France, and Pakistan.Is the value representative of the group?Then include the U.S. and compute the average for the six countries.Is the value representative of the group?The Median1 2 3 4 5 6 7 8 109The Median1 2 3 4 5 6 7 8 109MedianThe MedianMedian – The middle value, or the average of the middle two values, of a sample or population, when the values are arranged from smallest to largest.The median, by definition, is at the 50th percentile.It separates the lower 50% of the sample from the upper 50%.The MedianWhen n is odd, the median is the middle number, which is in position (n + 1)/2.When n is even, the median is the average of the middle two numbers, which are in positions n/2 and n/2 + 1.Case Study 82007 Small Arms Study (p. 47)Find the median number of guns per country for India, China, Germany, France, and Pakistan.Is the value representative of the group?Then include the U.S. and compute the median for the six countries.Is the value representative of the group?TI-83 – The MedianFollow the same procedure that was used to find the mean.When the list of statistics appears, scroll down to the one labeled “Med.” It is the median.TI-83 – The MedianUse the TI-83 to find the median number of guns.46, 40, 25, 19, 18.46, 40, 25, 19, 18, 270.The ModeMode – The value in the sample or population that occurs most frequently.The mode is a good indicator of the distribution’s central peak, if it has one.ModeThe problem is that many distributions do not have a peak or they have several peaks.In other words, the mode does not necessarily exist or there may be several modes.Weighted MeansFor the countries India, China, Germany, France, and Pakistan, the average number of guns per country is 29.6 million. For Mexico and Brazil, the average is 15.4 million.What is the average for all seven countries?Weighted MeansThe averages areIndia, et al: x1 = 29.6.Mexico & Brazil: x2 = 15.4.How could we combine the two averages to get the average for all seven countries?Weighted MeansCompute the weighted average:7)4.15(2)6.29(5
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