Visual Description of Data

Visual description is an art

http://www.xach.com/moviecharts/2010.htmlhttp://simplecomplexity.net/visualization-galle

ry/http://cohort11.americanobserver.net/latoyae

gwuekwe/multimediafinal.html

Chapter 2 - Learning Objectives

• Convert raw data into a data array.• Construct:

a frequency distribution.a relative frequency distribution.a cumulative relative frequency distribution.

• Construct a stem-and-leaf diagram.• Construct a histogram.

Hospital Cost ProblemThe average daily cost to community hospitals for patient stays for each of the 50 U.S. states is given in the next table.

a) Arrange these into a data array.

b) Construct a frequency distribution, a relative frequency distribution and a cumulative relative frequency distribution.

c) Construct a stem-and-leaf display.

d) Construct a histogram.

The DataAL $775 HI 823 MA 1,036 NM 1,046 SD

506AK 1,136 ID 659 MI 902 NY 784 TN

859AZ 1,091 IL 917 MN 652 NC 763 TX

1,010AR 678 IN 898 MS 555 ND 507 UT

1,081CA 1,221 IA 612 MO 863 OH 940 VT

676CO 961 KS 666 MT 482 OK 797 VA

830CT 1,058 KY 703 NE 626 OR 1,052 WA

1,143DE 1,024 LA 875 NV 900 PA 861 WV

701FL 960 ME 738 NH 976 RI 885 WI

744GA 775 MD 889 NJ 829 SC 838 WY

537

Data ArrayCA 1,221 TX 1,010 RI 885 NY 784 KS

666WA 1,143 NH 976 LA 875 AL 775 ID

659AK 1,136 CO 961 MO 863 GA 775 MN

652AZ 1,091 FL 960 PA 861 NC 763 NE

626UT 1,081 CH 940 TN 859 WI 744 IA

612CT 1,058 IL 917 SC 838 ME 738 MS

555OR 1,052 MI 902 VA 830 KY 703 WY

537NM 1,046 NV 900 NJ 829 WV 701 ND

507MA 1,036 IN 898 HI 823 AR 678 SD

506DE 1,024 MD 889 OK 797 VT 676 MT

482

The Stem-and-Leaf Display

Stem-and-Leaf Display N = 50Leaf Unit: 100

1 12 21 2 11 43, 36 8 10 91, 81, 58, 52, 46, 36, 24, 10 7 9 76, 61, 60, 40, 17, 02, 00(11) 8 98, 89, 85, 75, 63, 61, 59, 38, 30, 29, 23 9 7 97, 84, 75, 75, 63, 44, 38, 03, 01 7 6 78, 76, 66, 59, 52, 26, 12 4 5 55, 37, 07, 06 1 4 82

Range: $482 - $1,221

Constructing the Frequency Distribution

• Step 1. Number of classes• The range is: $1,221 – $482 = $739

$739/7 $106 and $739/8 $92

• Step 2. The class interval So, if we use 8 classes, we can make each class

$100 wide.

• Step 3. The lower class limitIf we start at $450, we can cover the range in 8 classes,

each class $100 in width.The first class : $450 up to $550

• Step 4. Setting class limits$450 up to $550 $850 up to $950$550 up to $650 $950 up to $1,050$650 up to $750 $1,050 up to $1,150$750 up to $850 $1,150 up to $1,250

Constructing the Frequency Distribution

Average daily cost NumberMark

$450 – under $550 4 $500 $550 – under $650 3 $600 $650 – under $750 9 $700 $750 – under $850 9 $800 $850 – under $950 11 $900 $950 – under $1,050 7 $1,000$1,050 – under $1,150 6 $1,100$1,150 – under $1,250 1 $1,200Interval width: $100

Frequency Distribution

Cumulative Frequency Distribution

Average daily cost Number Cum. Freq.

$450 – under $550 4 4

$550 – under $650 3 7

$650 – under $750 9 16

$750 – under $850 9 25

$850 – under $950 11 36

$950 – under $1,050 7 43

$1,050 – under $1,150 6 49

$1,150 – under $1,250 1 50

Converting to a Relative Frequency Distribution

1. Retain the same classes defined in the frequency distribution.

2. Sum the total number of observations across all classes of the frequency distribution.

3. Divide the frequency for each class by the total number of observations, forming the percentage of data values in each class.

The Relative Frequency Distribution

Average daily cost Number Rel. Freq.

$450 – under $550 4 4/50 = .08

$550 – under $650 3 3/50 = .06

$650 – under $750 9 9/50 = .18

$750 – under $850 9 9/50 = .18

$850 – under $950 11 11/50 = .22

$950 – under $1,050 7 7/50 = .14

$1,050 – under $1,150 6 6/50 = .12

$1,150 – under $1,250 1 1/50 = .02

Forming a Cumulative Relative Frequency Distribution

1. List the number of observations in the lowest class.

2. Add the frequency of the lowest class to the frequency of the second class. Record that cumulative sum for the second class.

3. Continue to add the prior cumulative sum to the frequency for that class, so that the cumulative sum for the final class is the total number of observations in the data set.

Forming a Cumulative Relative Frequency Distribution4. Divide the accumulated frequencies for each class by

the total number of observations -- giving you the percent of all observations that occurred up to and including that class.

• An Alternative: Accrue the relative frequencies for each class instead of the raw frequencies. Then you don’t have to divide by the total to get percentages.

The Cumulative Relative Frequency Distribution

Average daily cost Cum.Freq. Cum.Rel.Freq.

$450 – under $550 4 4/50 = .02

$550 – under $650 7 7/50 = .14

$650 – under $750 16 16/50 = .32

$750 – under $850 25 25/50 = .50

$850 – under $950 36 36/50 = .72

$950 – under $1,050 43 43/50 = .86

$1,050 – under $1,150 49 49/50 = .98

$1,150 – under $1,250 50 50/50 = 1.00

The Histogram

0

2

4

6

8

10

12

500 600 700 800 900 1000 1100 1200

The Stem-and-Leaf Display

Stem-and-Leaf Display N = 50Leaf Unit: 100

1 12 21 2 11 43, 36 8 10 91, 81, 58, 52, 46, 36, 24, 10 7 9 76, 61, 60, 40, 17, 02, 00(11) 8 98, 89, 85, 75, 63, 61, 59, 38, 30, 29, 23 9 7 97, 84, 75, 75, 63, 44, 38, 03, 01 7 6 78, 76, 66, 59, 52, 26, 12 4 5 55, 37, 07, 06 1 4 82

Range: $482 - $1,221

Summary

Frequency distributionsStem and leaf plotHistogram