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BBA182 Applied Statistics Week 4 (1)Measures of variation Dr Susanne Hansen Saral Email: susanne.saral@okan.edu.tr

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Numerical measures to describe data

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Interquatile range, IQR Alternative way to calculate the IQR Khan Academy

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Five-Number Summary of a data set

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Five-Number Summary: Example

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Exercise Consider the data given below: 110 125 99 115 119 95 110 132 85 a. Compute the mean. b. Compute the median. c. What is the mode? d. What is the shape of the distribution? e. What is the lower quartile, Q1? f. What is the upper quartile, Q3? g. Indicate the five number summary

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Exercise Consider the data given below. 85 95 99 110 110 115 119 125 132 a. Compute the mean. 110 b. Compute the median. 110 c. What is the mode? 110 d. What is the shape of the distribution? Symmetric, because mean = median=mode e. What is the lower quartile, Q1? 97 f. What is the upper quartile, Q3? 122 g. Indicate the five number summary 85 < 97 < 110 < 122 < 132

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Five number summary and Boxplots Boxplot is created from the five-number summary A boxplot is a graph for numerical data that describes the shape of a distribution, in terms of the 5 number summary. It visualizes the spread of the data in the data set.

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Five number summary and Boxplots Boxplot is created from the five-number summary The central box shows the middle half of the data from Q1 to Q3, (middle 50% of the data) with a line drawn at the median Two lines extend from the box. One line is the line from Q1 to the minimum value, the other is the line from Q3 to the maximum value A boxplot is a graph for numerical data that describes the shape of a distribution, like the histogram

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Five number summary and boxplot 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 Minimum number = 1 Maximum number = 5 = 1 2.5 Median = 2 Five number summary: 1 = 1 < 2 < 2.5 < 5 (plot a dot chart, then boxplot)

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Five number summary and boxplot 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Minimum number = 1 Maximum number = 120 = 1 2.5 Median = 2 Five number summary: 1 = 1 < 2 < 2.5 < 120

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Boxplot

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Gilotti’s Pizza Sales in $100s

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Gilotti’s Pizza Sales What are the shapes of the distribution of the four data set?

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Gilotti’s Pizza Sales - boxplot

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Gilotti’s Pizza Sales in $100s

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Measuring variation in a data set that follows a normal distribution

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Measuring variation in a data set Data set 1 : 23 19 21 18 24 21 23 Mean: 21.3 Data set 2 : 23 35 19 7 21 24 22 Mean: 21.6 Which of these two data sets has the highest spread/variation? Why?

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Average distance to the mean: Standard deviation Most commonly used measure of variability Measures the standard (average) distance of each individual data point from the mean.

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Calculating the average distance to the mean Our goal is to measure the standard distance of each single data in the data set from the mean. 1st step: Calculate the mean of the data set = 2nd step: Calculate the standard distance from the mean is to determine distance from the mean for each individual score: deviation score = X - μ Where x is the value of each individual score and μ the population mean.

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Calculating the average distance to the mean Step 3: Once we have calculated the distance between each single score and the mean, we add up the those deviation scores. Our mean in this example is = 3. Example: We have a set of 4 scores (): 8, 1, 3, 0,

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Calculating the average distance to the mean Notice that the deviation score adds up to zero! This is not surprising because the mean serves as balance point (middle point) for the distribution. (!Remember: In a symmetric distribution the mean and the median are identical) The distances of the single score above the mean equal the distances of the single scores below the mean. Therefore the deviation score always adds up to zero.

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Calculating the average distance to the mean Step 3: The solution is to get rid of the + and – which causes the cancelling out effect. We square each deviation score and sum them up

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Population Variance, Average of squared deviations from the mean Population variance:

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Sample Variance, Average of squared deviations from the mean Sample variance:

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Population Standard Deviation, Most commonly used measure of variation in a population Shows variation about the mean in a symmetric data set Has the same units as the original data, Example: If original data is in meters than the standard deviation will also be in meters. Population standard deviation:

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Sample Standard Deviation, s Most commonly used measure of variation in a sample Shows variation about the mean Has the same units as the original data Sample standard deviation:

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Calculation Example: Sample Standard Deviation, s

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Class example Calculating sample variance and standard deviation Compute the variance, and standard deviation, s, of the following sample data: 6 8 7 10 3 5 9 8

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Class example (continued) When we analyze the variance formula we, see that we need to calculate the sample mean, first: = = 7

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Class example (continued) The mean = 7

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C Class example (continued) Calculating the sample variance: 6 8 7 10 3 5 9 8 = = = 5.14 Sample standard deviation, s = 2.27 (average distance to the mean of 7)