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Chapter 2

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Sociology and Anthropology
SOAN 3120
Michelle Dumas

Chapter 2: Describing Distributions with Numbers Measuring Center: The Mean  The most common measure of spread if the average, or the mean  To find the mean, add their values and divide by the number of observations  The capital Greek sigma is short for add them all up o The subscripts on the observations are just a way of keeping the n observations distinct o The bar over the x indicates the mean of all the x-values  An important fact about the mean as a measure of center: it is sensitive to the influence of a few extreme observations o Because the mean cannot resist the influence of extreme observations, we say that is not a resistant measure of center  The median is the formal version of the midpoint o The median M is the midpoint of a distribution, the number such that half the observations are smaller and the other half are larger o The find the median: 1. Arrange all observations in order of size, from smallest to largest 2. If the number of observations n is odd, the median M is the center observation in the ordered list. If the number of observations n is even, the median M is midway between the two center observations in the ordered list. 3. You can always locate the median in the ordered list of observations by counting up (n+1)/2 observations from the start of the list.  Note the formula (n+1)/2 does not give the median just the location of the median in the ordered list Comparing the Mean and the Median  The median unlike the mean is resistant  The outlier just counts as one observation above the center, no matter how far above the center it lies  The mean uses the actual value of each observation and so will chase a single large observation upward  The mean and the median of a roughly symmetric distribution are close together  If the distribution is exactly symmetric, the mean and median are exactly the same  In a skewed distribution, the mean is usually farther out in the long tail than is the median  Many economic variables have distributions that are skewed to the right  Reports about incomes and other strongly skewed distributions usually give the median rather than the mean  The mean and the median measure the center in different ways, and both are useful Measuring Spread: The Quartiles  The mean and the median do not tell the whole story  We are interested in the spread or variability  The simplest useful numerical description of a distribution requires both a measure of center and a measure of spread  We can improve our descriptions of spread by also looking at the spread of the middle half of the data  The quartiles mark out the middle half  To calculate the quartiles: 1. Arrange observations in increasing order and locate the median M in the ordered list of observations 2. The first quartile Q1 is the median of the observations who position in the order list is to the left of the location of the overall median 3. The third quartile Q3 is the median of the observations who position in the order list is to the right of the location of the overall median  When there is an odd number of observations leave out the overall median when you locate the quartiles in the ordered list  The quartiles are restraint because they are affected by a few extreme observations  When the number of observations is even include all the observations when you locate the quartiles The Five-Number Summary and BoxPlots  The smallest and largest observations tell us little about the distributions as a whole, but they give us information about the tails of the distribution that is missing if we know only the median and the quartiles  To get a quick summary of both center and spread combine all five numbers  The five number summary of a distribution consists of the smallest observation, the first quartile, the median, and the third quartile, and the largest observation, written in order from smallest to largest o Minimum Q1 m Q3 Maximum  The five number summary of a distribution leads to a new graph, the boxplot  The boxplot is a graph of the five number summary o A central box spans the quartiles Q1 and Q3 o A line in the box that marks the median M o Lines that extend from the box to the smallest and largest observations  Boxplots show less detail than histograms or stemplots so they are best used for side by side comparison of more than one distribution  When you first look at a boxplot locate the median, then look at the sp
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