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 ModeMedianMean 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 symbol x 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 L 1 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 50 th 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
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