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The Practice of Statistics Third Edition Chapter 2: Describing Location in a Distribution Section 2.1 Measures of Relative Standing and Density Curves Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

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Objectives What are measures of relative standing? What is a Standardize Value? How do you compute a z-score of an observation given the mean and standard deviation of a distribution? What does the z-score measure? How do you find the pth percentile of an observation in a data set? What is a Mathematical Model? What is a Density Curve?

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Measures of Relative Standing Suppose we have a data set of grades for Algebra 2AB Chapter Test : ( 94, 61, 40, 72, 73, 88, 68, 62, 73, 57, 35, 82, 48, 66, 65, 79, 45, 91, 66, 71, 63, 11, 69, 64, 38, 59, 70, 70, 79, 77, 39, 55) We can discuss a particular students grade relative standing in the class in two ways: – Relative to the median (percentile) – Relative to the mean (how far way from the mean)

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p th Percentile Pth percentile of a distribution – the value with p percent of the observations less than or equal to the observation in question. For example we are interested in the percentile for a test grade of 45. Data sorted: (11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71, 72, 73, 73, 77, 79, 79, 82, 88, 91, 94 ) 6/32 x 100 = 18.75%

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Your turn! Data sorted: (11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71, 72, 73, 73, 77, 79, 79, 82, 88, 91, 94 ) What is the percentile for a grade of 66? 17/32 x 100 = 53.1% What is the 50% percentile? Grade of 66.

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Relative to the Means of the Data Set We standardize each data by: The standard value (z-score) is a measure of how many standard deviations a data value is from the means of the data set.

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Back to Our Example Data sorted: (11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71, 72, 73, 73, 77, 79, 79, 82, 88, 91, 94 ) What is the standard value for a test grade of 45? First we need to find the mean and standard deviation. Mean = and S x = What is the z-score for a grade of 66? z = 0.144

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Calculator Exercise We will convert all the grades to z-score. Data sorted: (11,35, 38, 39, 40, 45, 48, 55, 57, 59, 61, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 71, 72, 73, 73, 77, 79, 79, 82, 88, 91, 94 )

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Impact on the Distribution When we Standardize We need to look at the mean and standard deviation, to find out what is the impact. Recall: Linear Transformation: x new = a + bx When we add (or subtract) a constant from each data we move the distribution by that amount but we do not change the spread. When we multiply (or divide) each data by a constant we change the spread. We can quickly compute the new standard deviation by multiplying the old standard deviation by dividing it by the constant.

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Impact Continued So looking at the formula to convert data to a standard value we can see we are moving the distribution by a constant and by dividing the standard deviation we are changing the spread. If x = mean, then mean – mean = 0. The new mean is 0. By dividing by the standard deviation we are changing the standard deviation to 1 since

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Data Analysis Toolbox (p123) When describing a distribution – 1.Always plot the data. 2.Look for overall pattern (shape, center, spread) and striking deviations such as outliers. 3.Calculate a numerical summary to describe center and spread. 4.For large data sets, can we fit a smooth curve to the distribution.

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For Example The smooth curve are an idealized description (mathematical model) for the distribution. Smooth curves are easier to work with than histograms

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Density Curves When we adjust the scale so that the area underneath the curve is one we have density curve. Definition :

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Density Curves Density Curves come in different shapes, but they all have the same area of 1.

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Density Curve Parameters Since density curves are idealized descriptions of the data distribution we use different symbols to represent: Sample Density curve Mean µ Standard Deviation SxSx σ

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Algebra 2 Grades

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The Median and Mean of a Density Curve

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Problem 2.12 page 128 (a) Mean C, Median B (b) Mean A, Median A (c) Mean A, Median B

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