The Practice of StatisticsThird 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

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?

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 student’s grade relative standing in the class in two ways:– Relative to the median (percentile)– Relative to the mean (how far way from the mean)

pth 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%

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.

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.

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 = 63.4375 and Sx = 17.8234

034.18234.17

4375.6345

deviation standard

meanxz

• What is the z-score for a grade of 66?

• z = 0.144

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 )

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: xnew = 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.

Impact Continued

deviation standard

meanxz

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

1deviation standard

deviation standard

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.

For Example

The smooth curve are an idealized description (mathematical model) for the distribution.

Smooth curves are easier to work with than histograms

Density Curves

When we adjust the scale so that the area underneath the curve is one we have density curve.

Definition :

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

Density Curve Parameters

Since density curves are idealized descriptions of the data distribution we use different symbols to represent:

Sample Density curve

xMean µ

Standard Deviation Sxσ

Algebra 2 Grades

The Median and Mean of a Density Curve

Problem 2.12 page 128

(a) Mean C, Median B

(b) Mean A, Median A

(c) Mean A, Median B