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week2class

# week2class - Quick Review Describing Distributions...

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Quick Review

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Describing Distributions Numerically Measures of Center: The mean or arithmetic average To calculate the average, or mean, add all values, Heights of 25 Women then divide by the number of observations. Sum of heights is 1598.3 Divided by 25 women = 63.9 inches Your numerical summary must be meaningful The distribution of the women’s height appears coherent and symmetrical. The mean is a good numerical summary. Height of 25 women in a class 58.2 64.0 59.5 64.5 60.7 64.1 60.9 64.8 61.9 65.2 61.9 65.7 62.2 66.2 62.2 66.7 62.4 67.1 62.9 67.8 63.9 68.9 63.1 69.6 63.9 9 . 63 25 1598 1 = = = = n x x n i i
Here, reporting a single numerical summary would not make sense in discribing the overall pattern of the data because there are really three seperated distribution. Measures of Center: The median The median is the midpoint of a distribution—the number such that half of the observations are smaller and half are larger. Here the shape of the distribution is wildly irregular. Why? Multimodal 6 . 69 = x Height of plants by color 0 1 2 3 4 5 Height in centimeters Number of plants red pink blue

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Comparing the mean and median: The mean and the median are roughly equal only if the distribution is symmetric. The median is a measure of the center that is resistant to skew and outliers. The mean is not resistant to such deviant values. mean= meadian 1 1 0.6 2 2 1.2 3 3 1.6 4 4 1.9 5 5 1.5 6 6 2.1 7 7 2.3 8 8 2.3 9 9 2.5 10 10 2.8 11 11 2.9 12 12 3.3 13 3.4 14 1 3.6 15 2 3.7 16 3 3.8 17 4 3.9 18 5 4.1 19 6 4.2 20 7 4.5 21 8 4.7 22 9 4.9 23 10 5.3 24 11 5.6 25 12 6.1 n=25 (n+1)/2=26/2=13 Median =3.4 2. If n is odd, the median is observation ( n +1)/2 down the list 1. Sort observations from smallest to largest. n = number of observations ______________________________ 1 1 0.6 2 2 1.2 3 3 1.6 4 4 1.9 5 5 1.5 6 6 2.1 7 7 2.3 8 8 2.3 9 9 2.5 10 10 2.8 11 11 2.9 12 3.3 13 3.4 14 1 3.6 15 2 3.7 16 3 3.8 17 4 3.9 18 5 4.1 19 6 4.2 20 7 4.5 21 8 4.7 22 9 4.9 23 10 5.3 24 11 5.6 N=24 n/2=12 Median=(3.3+3.4)/2=3.35 3. If n is even, the median is the mean of the two center observations Mean and median for a symmetric distribution
mean < median median < mean Mean and Median of a distribution with outliers Mean and median for a distribution that is skewed left or negatively skewed Mean and median for a distribution that is skewed right or positively skewed

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Impact of skewed data Percent of people dying 4 . 3 = x Without the outlier outliers 2 . 4 = x With the outlier outliers The median, on the other hand is only (pulled to the right a little). The mean is pulled to the right a lot by the outliers. (From 3.4 to 4.2)
Mean vs. Median Mean: - easy to calulate - easy to work with algebraically - highly affected by outliers - not a resistant measure Median: - can be time consuming to calculate - more resistant to a few extreme observations Measures of Center: The mode The most frequent value in the data Important for categorical data Possible to have more than one mode Mean and median of a symmetric distribution 4 . 3 4 . 3 = = M x Mean and median of a skewed distribution 5 . 2 4 . 3 = = M x

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Mean, Median, Mode If the distribution is exactly symmetric (normal), the mean, the median and the mode are exactly the same.
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