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Measures of Dispersion How far the data is spread out

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Range Difference between the largest and smallest value in a data set. 3 5 8 1 -7 2 5 Smallest value = -7 Largest value = 8 Range = 15

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Variance

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Calculating the Variance Example: Consider the following data set 4, 5, 6, 6, 9The sample mean = 6 x 46-24 561 6600 6600 0639

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Calculating the Variance Here’s the 4th column from the previous slide. The sum of the square of the residuals = 14 Divide the sum of the square of the residuals by the number in the sample minus one. 14/4 = 3.5

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Standard Deviation Important value to study of statistics Measures average absolute distance of values to the mean Population Sample

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Quantiles Dividing data into equal groups ▫Percentiles ▫Quartiles First quartile – Q 1 = Second quartile – Q 2 = Third quartile – Q 3 = Interquartile range ▫Q 3 – Q 1

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Percentiles Commonly used measure of relative position. Remember the median? For any data set, the p th percentile is a value (x) such that p percent of the data is less than x and 1-p percent of the data is greater than x.

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Boxplots (Box and whisker plots) Visual display of data 5-number summary ▫Minimum ▫Q1 ▫Q2 ▫Q3 ▫Maximum IQR Outliers

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Boxplots

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z-scores The z-score tells us how far a data value is from the mean in terms of the number of standard deviations This is another one of the fundamental values in statistics that we will use again, later.

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Summary Mean, median, and mode ▫What information does each convey? ▫Which is the most resistant to outliers?

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Summary Range Variance/Standard deviation Z-scores (more later) Quantiles ▫IQR ▫Boxplots

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Which to use? Data is symmetric and unimodal ▫Use the mean and s.d. Data is skewed ▫Use the median and 5-number summary Mode?

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Practice Exercise Consider the following data: 4, -8, 4, -12, 8, 4, -14 What is the range?

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Practice Exercise Consider the following data: 4, -8, 4, -12, 8, 4, -14 What is the variance and standard deviation?

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Practice Exercise Consider the following data: 4, -8, 4, -12, 8, 4, -14 What are the values of Q1, Q2, and Q3? What is the value of the IQR?

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