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Measuring Center Lecture 15 Sections 5.1 – 5.2 Mon, Feb 11, 2008 Measuring the Center Often, 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 Center Mean – 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 Center In a unimodal, symmetric distribution, these values will all be near the center. In skewed distributions, they will be spread out. Mean, Median, and Mode If 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 Mode However, if the distribution is skewed, then the mean, median, and mode are all different. Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. The mode is at the peak. Mode Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. The mean is shifted in the direction of skewing. Mode Mean Mean, Median, and Mode However, if the distribution is skewed, then the mean, median, and mode are all different. The median is (typically) between the mode and the mean. Mode Median Mean The Median vs. The Mean If 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 Mean Why 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 Mean 1 2 3 4 5 6 7 8 9 10 The Mean Average 1 2 3 4 5 6 7 8 9 10 The Mean Average -5 -2 1 2 3 4 5 6 7 8 9 10 The Mean Average +4 -5 +2 -2 1 2 3 4 5 +1 6 7 8 9 10 The Median 1 2 3 4 5 6 7 8 9 10 The Median Median 1 2 3 4 5 6 7 8 9 10 The Median Median -6 -3 1 2 3 4 5 6 7 8 9 10 The Median Median -6 +3 -3 1 2 3 4 5 6 +1 7 8 9 10 The Mean We 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). x Then the sample mean is n The Mean We denote the mean of a sample by the symbolx, pronounced “x bar”. We denote the mean of a population by , pronounced “mu” (myoo). x Therefore, x n x N TI-83 – The Mean Enter 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 8 2007 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 Median 1 2 3 4 5 6 7 8 9 10 The Median Median 1 2 3 4 5 6 7 8 9 10 The Median Median – 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 Median When 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 8 2007 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 Median Follow 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 Median Use the TI-83 to find the median number of guns. 46, 40, 25, 19, 18. 46, 40, 25, 19, 18, 270. The Mode Mode – 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. Mode The 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 Means For 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 Means The averages are India, et al: Mexico & Brazil: x1 = 29.6. x2 = 15.4. How could we combine the two averages to get the average for all seven countries? Weighted Means Compute the weighted average: 5(29.6) 2(15.4) x 7