Chapter_2 - STAT 2053 Elementary Statistics Chapter 2...

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STAT 2053 – Elementary Statistics Chapter 2 – Describing Distributions with Numbers 1
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2 Numerical Summaries Center of the data mean median mode Variation range quartiles (interquartile range) variance standard deviation
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Notation When we have a set of n observations, we identify the value of each observation with x 1 , x 2 , … , x n The subscripts do not order the observations or give any special info. They just keep the values separate. Nothing is special about the letter x either. Most letters within reason can be used. 3
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4 x = x 1 + x 2 + ... + x n n To calculate the mean of a set of n observations, we simply take the average of all the values: Sum the values and divide by the number of values Or, using more compact notation: x = 1 n x i i = 1 n Mean or Average
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5 Mean or Average ( 29 x n x x x n x n i i n = + + + = = 1 1 1 2 1 The Σ (capital Greek sigma) in the formula for the mean is short for “add them all up”. The bar over the x indicates the mean of all the x -values. Pronounce the mean as “ x -bar”. When writers who are discussing data x x y
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Mean or Average Example : Calculate the mean of the following values: 10 15 13 19 18 14 17 12 96 Now take out the 96, and calculate the mean of the remaining values: 6 ( 29 = = + + + = n i i n x n n x x x x 1 1 2 1
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Measuring Center The mean can be heavily influenced or altered by extreme values (outliers) or even heavy skewness. A skewed distribution that has no outliers will pull the mean toward its long tail. Definition : A resistant measure is relatively unaffected by changes in/removal of a small number of values within a data set. Because the mean cannot resist the influence of extreme observations, we say that it is not a 7
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Measuring Center The median, however, is a good example of a resistant measure of center. Definition : The median of a distribution (denoted M ), is the number such that half of the ordered observations are less than M and the other half of the ordered observations are greater than M. This is the formal definition of the midpoint of our distribution (the “typical” value of a 8
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Median ( M ) To calculate the median for a set of n observations: Order your data from smallest value to largest value. If n is odd, then M is the value in the center of your ordered list. If n is even, there will be two observations in the center of your list. M is the average of these two. 9
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10 Median ( M ) Location of the median: If n is odd or even L(M) = (n+1)/2 , where n = sample size. Example: If 25 data values are recorded, the Median would be the (25+1)/2 = 13 th ordered value. If 24 data values are recorded, the Median would be the (24+1)/2 = 12.5 so we know it would be the average of the 12 th and 13 th ordered values.
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11 Median ( M ) Example 1 data: 2 4 6 Median ( M ) = 4 Example 2 data: 2 4 6 8 Median = 5 (ave. of 4 and 6) Example 3 data: 6 2 4 Median 2 ( order the values: 2 4 6 , so Median = 4)
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This note was uploaded on 09/23/2011 for the course STAT 2053 taught by Professor Staff during the Fall '08 term at Oklahoma State.

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Chapter_2 - STAT 2053 Elementary Statistics Chapter 2...

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