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Displaying Categorical Variables Frequency Table. Variable. Categories of the Variable. Count of elements from sample in each category. Total = 498. Displaying Categorical Variables Relative Frequency Circle Graph. Relative Frequency or Proportion. 74/498 = 15%.

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Displaying Categorical VariablesFrequency Table

Variable

Categories of the Variable

Count of elements from sample in each category. Total = 498

Section 2.1, Page 24

Displaying Categorical VariablesRelative Frequency Circle Graph

Relative Frequency or Proportion.74/498 = 15%

Section 2.1, Page 25

Displaying Categorical VariablesVertical Bar Graph

Section 2.1, Page 25

Displaying Categorical VariablesPareto Chart for Hate Crimes USA 1993

Cumulative count/relative frequency

Frequency, Relative Frequency, and Cumulative Relative Frequency Table

Section 2.1, Page 25

Section 2.2, Page 26

Displaying Quantitative DataStem-and-Leaf Displays

To make stem-and-leaf display, first find the minimum and maximum number, 52 and 96. We then graph the tens digits in the left column, 5 – 9. We then plot each number opposite its tens digit. The plot point is the ones digit.

Section 2.2, Page 27

Stem-and Leaf-DisplaysProblems

Section 2.2, Page 50

Displaying Quantitative DataUngrouped Frequency Distribution

Values of the Variable in the data set

Frequency or number of times each value occurs in the data set

Section 2.1, Page 29

Displaying Quantitative DataGrouped Frequency Distribution

Classes or bins, usually 5 to 12 of equal width

95 or more to less than 105

Section 2.2, Page 30

Displaying Quantitative DataHistograms

Histogram: A bar graph that represents a frequency distribution of a quantitative variable.

15

Count

10

5

5 to 12 equal sized classes or bins

Section 2.2, Page 32

Histograms Shapes of Distributions

Section 2.2, Page 33

Constructing HistogramTI-83 Calculator

(50 States)

Enter Data:STAT-1:Edit-ENTER Type all the data in L1

Set up Plot:

2nd Stat Plot Enter --Turn plot ON, select Histogram Icon,

enter XList: as L1 and Freq: as 1

Set the Viewing WindowZoom 9: ZoomStat –Hit Trace key then arrows to view

axes values.

Change category size to 7Window –Make Xscl= 7. Then hit Graph Key

Display class width and frequency.

Trace

Section 2.2 WS #21

# of States

% College Students Enrolled in Public Institutions

The leftmost class or bin shows the number of states between 44 and <51. There are 2 states in this bin. To see the next bin, hit the right arrow button.

Section 2.2 WS #21

2.4 Heights of NBA players selected in the June 2004 Draft.

• Construct a histogram. Be sure to show the scale and the label for the x and y axes.

• Describe the shape of the distribution.

Section 2.2, Page 50

Cumulative Frequency DistributionFinal Exam Scores for 50 Students

Cumulative Relative Frequency

2/50 = .04

4/50 = .08

11/50 =.22

24/50 = .48

35/50 = .70

46/50 = .92

50/50 = 1.0

Cumulative Relative Frequency

For classes <65, 11/50 = .22

Section 2.2, Page 34

Find the sample mean for the set {6, 3, 8, 6, 4}

Section 2.3, Page 35

The median is the value of the middle number when the date are ranked according to size.

Find the median for the data the following set with an odd n: {3, 3, 5, 6, 8}, n=5. The data values are in ascending order. Depth of median = (n+1)/2. For this set: (5+1)/2 = 3The median is the 3rd number, 5.

Find the median for the following data values that are in ascending order with even n: {6, 7, 8, 9, 9, 10}, n=6.

Dept of median = (n+1)/2 = 3.5

The median is then the average of the 3rd and 4th number. The median is (8+9)/2 = 8.5

Section 2.3, Page 36

Measures of Central TendencyMode and Midrange

(L+H)/2 = (3+8)/2 = 5.5

Section 2.3, Page 37

Measures of Central TendencySummary

The most useful measure is the mean. However, when a set of numbers has outliers, the mean gets distorted and may not be representative of the central tendency. When this happens, the median is a better measure of central tendency because it is not affected by outliers.

Section 2.3, Page 37

Secton 2.4, Page 39

Measures of DispersionVariance and Standard Deviation

{6, 3, 8, 5, 3 }

Section 2.4, Page 41

Measures of PositionPercentiles

Percentiles: Values of the variable that divide a set of ranked data into 100 equal subsets: each set of data has 99 percentiles.

A specific number fromwithin the range of values

In the set

Section 2.5, Page 42

Finding PercentilesExample

Sample data set of 20 numbers in ascending rank order:

{6, 12, 14, 17, 23, 27, 29, 33, 42, 51, 59, 65, 69, 74, 79, 82, 84, 88, 92, 97}

Find the 21st Percentile. Sample size n=20.

Calculate the depth: percentile*n/100 = 21*20/100 = 4.2.

(If the depth is an integer, Pk is the average of the number and the next number. If the depth contains a decimal, Pk is the next number.)

Since the depth contains a decimal, Pk is the next number, the 5th number, Pk = 23.

Find the 75th Percentile:Depth = 75*20/100 = 15. Since the depth is an integer, the 75th percentile is the average of the 15th and 16th numbers, (79+82)/2=80.5.

Section 2.5, Page 43

Find the 21st and the 75th percentile of the following data set.

{6, 12, 14, 17, 23, 27, 29, 33, 42, 51, 59, 65, 69, 74, 79, 82, 84, 88, 92, 97}

STAT-EDIT: Enter the data in L1

PRGM: down arrow to PRCNTILE

ENTER: (Displays Program Input Page)

2nd L1: (Enters the List name)

21.0: (Enters the desired percentile)

ENTER: (Displays the 21st percentile)

ENTER-2ND L1-75: (Displays the 75th percentile)

CLEAR: (Clears the home screen)

Section 2.5, Page 43

5-Number SummaryBox and Whisker Display

Q1

Q3

H

L

Med

Interquartile Range = Q3-Q1

Range of middle 50% of values

Measure of dispersion resistant to outliers.

Section 2.5, Page 44

• Find the mean and the standard deviation for the above sample data.

• STAT – EDIT: Enter the data is L1

• STAT – CALC-1: 1-VAR Stats - ENTER

• 2ND L1 - ENTER

• DISPLAY:

Sample Mean

Sample Standard Deviation Sx

• Find the variance of the sample

• (2.7999)2 =7.8394

Problems, Page 50

• Find the 5 number summary

• PRESS THE DOWN ARROW FIVE TIMES

• DISPLAY:

• Find the interquartile range

• 32 - 28 = 4

• e. Find the range.

• 34 – 25 = 9

Problems, Page 50

• Make a box and whisker display of the data.

• 2ND STAT PLOT-ENTER

• ENTER: Sets plot to ON

• DOWN ARROW

• RIGHT ARROW 5 TIMES: Select box Plot

• DOWN ARROW – 2nd L1: Select List

• Display:

ZOOM – 9TRACE: Display:

RIGHT-LEFT ARROW: Display 5-number summary

Problems, Page 50

The mean and median are measures of the center of a distribution. Outliers will distort the mean, so when outliers are present the mean is not a good measure of the center. The median is not distorted by outliers.

The standard deviation, variance, range, and Interquartile range (IQR) are measures of the spread or variability of a distribution. Outliers will distort the standard deviation, variance, and range, so when outliers are present, these are not good measures of the spread or variability. The Interquartile range is not distorted by outliers.

When outliers are present, then use the median and IQR as measures of the center and spread.

When no significant outliers are present, use the mean and standard deviation as measures of center and spread. These measures allow use of the maximum number statistical tools using the distribution.

Section 2.4

• Find the mean, variance, and standard deviation.

• Find the 5-number summary.

• Make a box and whisker display and label the numbers.

• Calculate the Interquartile range and the range

• Describe the shape of the distribution

• Find the 33 percentile.

Problems, Page 50

• Find the mean, variance, and standard deviation.

• Find the 5-number summary.

• Make a box and whisker display and label the numbers.

• Calculate the Interquartile range and the range

• Describe the shape of the distribution.

• Find the 90th percentile.

Problems, Page 50