# Measures of Dispersion How far the data is spread out.

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Measures of Dispersion How far the data is spread out

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

Variance

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

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

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

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

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.

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

Boxplots

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.

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

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

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?

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

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

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?