Describing Distributions Numerically

# Describing Distributions Numerically - 2.3 Describing...

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2.3 Describing Distributions Numerically Numerical and More Graphical Methods to Describe Univariate Data

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2 characteristics of a data set to measure center measures where the “middle” of the data is located variability measures how “spread out” the data is
The median: a measure of center Given a set of n measurements arranged in order of magnitude, Median= middle value n odd mean of 2 middle values, n even Ex. 2, 4, 6, 8, 10; n =5; median=6 Ex. 2, 4, 6, 8; n =4; median=(4+6)/2=5

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The median splits the histogram into 2 halves of equal area
Examples Example: n = 7 17.5 2.8 3.2 13.9 14.1 25.3 45.8 Example n = 7 (ordered): 2.8 3.2 13.9 14.1 17.5 25.3 45.8 Example: n = 8 17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8 Example n =8 (ordered) 2.8 3.2 13.9 14.1 17.5 25.3 35.7 45.8 m = 14.1 m = (14.1+17.5)/2 = 15.8

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Measures of Spread The range and interquartile range
Ways to measure variability range=largest-smallest OK sometimes; in general, too crude; sensitive to one large or small data value The range measures spread by examining the ends of the data A better way to measure spread is to examine the middle portion of the data

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m = median = 3.4 Q 1 = first quartile = 2.3 Q 3 = third quartile = 4.2 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 6 2.3 9 5 2.5 10 4 2.8 11 3 2.9 12 2 3.3 13 1 3.4 14 2 3.6 15 3 3.7 16 4 3.8 17 5 3.9 18 6 4.1 19 7 4.2 20 6 4.5 21 5 4.7 22 4 4.9 23 3 5.3 24 2 5.6 25 1 6.1 Quartiles:  Measuring spread by  examining the middle The first quartile, Q 1 , is the value in the sample that has 25% of the data at or below it (Q 1 is the median of the lower half of the sorted data) .
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