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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 Problem The 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 Data AL $775HI 823MA 1,036NM 1,046SD 506 AK 1,136ID 659MI 902NY 784TN 859 AZ 1,091IL 917MN 652NC 763TX 1,010 AR 678IN 898MS 555ND 507UT 1,081 CA 1,221IA 612MO 863OH 940VT 676 CO 961KS 666MT 482OK 797VA 830 CT 1,058KY 703NE 626OR 1,052WA 1,143 DE 1,024LA 875NV 900PA 861WV 701 FL 960ME 738NH 976RI 885WI 744 GA 775MD 889NJ 829SC 838WY 537

Data Array CA 1,221TX 1,010RI 885NY 784KS 666 WA 1,143NH 976LA 875AL 775ID 659 AK 1,136CO 961MO 863GA 775MN 652 AZ 1,091FL 960PA 861NC 763NE 626 UT 1,081CH 940TN 859WI 744IA 612 CT 1,058IL 917SC 838ME 738MS 555 OR 1,052MI 902VA 830KY 703WY 537 NM 1,046NV 900NJ 829WV 701ND 507 MA 1,036IN 898HI 823AR 678SD 506 DE 1,024MD 889OK 797VT 676MT 482

The Stem-and-Leaf Display Stem-and-Leaf DisplayN = 50 Leaf Unit: , , 81, 58, 52, 46, 36, 24, , 61, 60, 40, 17, 02, 00 (11) 898, 89, 85, 75, 63, 61, 59, 38, 30, 29, , 84, 75, 75, 63, 44, 38, 03, , 76, 66, 59, 52, 26, , 37, 07, 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 limit If 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 $5504$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 $1,200 Interval width: $100 Frequency Distribution

Cumulative Frequency Distribution Average daily cost Number Cum. Freq. $450 – under $ $550 – under $ $650 – under $ $750 – under $ $850 – under $ $950 – under $1, $1,050 – under $1, $1,150 – under $1,

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 $55044/50 =.08 $550 – under $650 33/50 =.06 $650 – under $750 99/50 =.18 $750 – under $850 99/50 =.18 $850 – under $ /50 =.22 $950 – under $1,050 77/50 =.14 $1,050 – under $1,150 66/50 =.12 $1,150 – under $1,250 11/50 =.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 Distribution 4. 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 44/50 =.02 $550 – under $650 77/50 =.14 $650 – under $ /50 =.32 $750 – under $ /50 =.50 $850 – under $ /50 =.72 $950 – under $1, /50 =.86 $1,050 – under $1, /50 =.98 $1,150 – under $1, /50 = 1.00

The Histogram

The Stem-and-Leaf Display Stem-and-Leaf DisplayN = 50 Leaf Unit: , , 81, 58, 52, 46, 36, 24, , 61, 60, 40, 17, 02, 00 (11) 898, 89, 85, 75, 63, 61, 59, 38, 30, 29, , 84, 75, 75, 63, 44, 38, 03, , 76, 66, 59, 52, 26, , 37, 07, Range: $482 - $1,221

Summary Frequency distributions Stem and leaf plot Histogram