Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Numerically Summarizing Data 3.

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.

ChapterNumerically Summarizing Data

3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.

SectionMeasures of Central Tendency and Dispersion from Grouped Data

3.3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.3-3

Objectives

1. Approximate the mean of a variable from grouped data

2. Compute the weighted mean3. Approximate the standard deviation of a

variable from grouped data


Objective 1• Approximate the Mean of a Variable from

Grouped Data


We have discussed how to compute descriptive statistics from raw data, but often the only available data have already been summarized in frequency distributions (grouped data). Although we cannot find exact values of the mean or standard deviation without raw data, we can approximate these measures using the techniques discussed in this section.


Approximate the Mean of a Variable from a Frequency Distribution

xi fifi

x1 f1 x2 f2 ... xn fn

f1 f2 ... fn

Sample MeanPopulation Mean

where xi is the midpoint or value of the ith classfi is the frequency of the ith classn is the number of classes

x xi fifi

x1 f1 x2 f2 ... xn fn

f1 f2 ... fn


Hours 0 1-5 6-10 11-15 16-20 21-25 26-30 31-35 Frequency 0 130 250 230 180 100 60 50

The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the mean number of hours spent preparing for class each week.

Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf

EXAMPLE Approximating the Mean from a Relative Frequency Distribution


Time Frequency xi xi fi

0 0 0 01 - 5 130 3.5 4556 - 10 250 8.5 2125

11 - 15 230 13.5 310516 - 20 180 18.5 333021 - 25 100 23.5 235026 – 30 60 28.5 171031 – 35 50 33.5 1675

14,750i ix f fi 1000

14,7501000

14.75

i i

i

x fx

f


Objective 2• Compute the Weighted Mean


The weighted mean, , of a variable is found by multiplying each value of the variable by its corresponding weight, adding these products, and dividing this sum by the sum of the weights. It can be expressed using the formula

xw wi xiwi

w1x1 w2 x2 ... wn xn

w1 w2 ... wn

where w is the weight of the ith observationxi is the value of the ith observation

xw


EXAMPLE Computed a Weighted Mean

Bob goes to the “Buy the Weigh” Nut store and creates his own bridge mix. He combines 1 pound of raisins, 2 pounds of chocolate covered peanuts, and 1.5 pounds of cashews. The raisins cost $1.25 per pound, the chocolate covered peanuts cost $3.25 per pound, and the cashews cost $5.40 per pound. What is the cost per pound of this mix?

xw 1($1.25) 2($3.25)1.5($5.40)

1 2 1.5

$15.85

4.5$3.52


Objective 3• Approximate the Standard Deviation of a

Variable from Grouped Data


Approximate the Standard Deviation of a Variable from a Frequency Distribution

xi 2 fi

fi

SampleStandard Deviation

PopulationStandard Deviation

where xi is the midpoint or value of the ith classfi is the frequency of the ith class

s xi x 2 fi

fi 1


xi2 fi

xi f 2

fifi

An algebraically equivalent formula for the population standard deviation is


Hours 0 1-5 6-10 11-15 16-20 21-25 26-30 31-35 Frequency 0 130 250 230 180 100 60 50

The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the standard deviation number of hours spent preparing for class each week.

Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf

EXAMPLE Approximating the Standard Deviation from a Relative Frequency Distribution


TimeFrequency xi

0 0 0 0 01 - 5 130 3.5 –11.25 16,453.125

6 - 10 250 8.5 –6.25 9765.62511 - 15 230 13.5 –1.25 359.37516 - 20 180 18.5 3.75 2531.2521 - 25 100 23.5 8.75 7656.2526 – 30 60 28.5 13.75 11,343.7531 – 35 50 33.5 18.75 17,578.125

xi x 2

fi 65,687.5 fi 1000

s2 xi x 2

fi

fi 1

65,687.51000 1

65.8

s s2 65.88.1 hours

xi x 2fi xi x

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Numerically Summarizing Data 3.

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