Displaying Categorical VariablesFrequency Table

1Section 2.1, Page 24

Variable

Categories of the Variable

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

Displaying Categorical VariablesRelative Frequency Circle Graph

2Section 2.1, Page 25

Relative Frequency or Proportion.74/498 = 15%

Displaying Categorical VariablesVertical Bar Graph

3Section 2.1, Page 25

Displaying Categorical VariablesPareto Chart for Hate Crimes USA 1993

4Section 2.1, Page 25

Cumulative count/relative frequency

Frequency, Relative Frequency, and Cumulative Relative Frequency Table

Displaying Quantitative DataDot Plots

5Section 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.

6Section 2.2, Page 27

Stem-and Leaf-DisplaysProblems

7Problems, Page 50

Displaying Quantitative DataUngrouped Frequency Distribution

8Section 2.1, Page 29

Values of the Variable in the data set

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

Displaying Quantitative DataGrouped Frequency Distribution

9Section 2.2, Page 30

Classes or bins, usually 5 to 12 of equal width

95 or more to less than 105

Displaying Quantitative DataHistograms

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

10Section 2.2, Page 32

5 to 12 equal sized classes or bins

5

10

15

Histograms Shapes of Distributions

11Section 2.2, Page 33

Calculator Fundamentals

Clear the Home screen: “Clear” key

Setting the calculator decimal places: “Mode” key: Down arrow key to “FLOAT”, right arrow key to desired number, then “ENTER”

Entering data into a List: “STAT-Edit-Enter”: Type in each number followed by “ENTER” or down arrow.

Deleting number for a list: Position the cursor over the number and press “Del” key.

Clear a List: Position the cursor over the list title, press “Clear-Enter”. Caution if you press “DEL” you will eliminate the entire list position.Return to Home Screen: “2nd – Quit

Scientific Notation: When an answer will be less than 1 with more than three zeros after the decimal point, the calculator will return the answer in Scientific Notation. For example, the number 5.3E-5 is converted to normal notation by moving the decimal point 5 places left: 0.000053.

12Section 2.2

Constructing HistogramTI-83 Calculator

Enter Data:STAT-1:Edit-ENTER Type all the data in L1Set up Plot:2nd Stat Plot Enter --Turn plot ON, select Histogram Icon, enter XList: as L1 and Freq: as 1Set the Viewing WindowZoom 9: ZoomStat – Hit Trace key then arrows to viewaxes values.Change category size to 7Window –Make Xscl= 7. Then hit Graph KeyDisplay class width and frequency.Trace

(50 States)

13Section 2.2

TI-83 Histogram Display

# 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.

14Section 2.2 WS #21

Histogram Problem2.4 Heights of NBA players selected in the June 2004 Draft.

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

b. Describe the shape of the distribution.

15Section 2.2, Page 50

Cumulative Frequency DistributionFinal Exam Scores for 50 Students

For classes <65, 11/50 = .22

16Section 2.2, Page 34

Cumulative Relative Frequency

2/50 = .044/50 = .08

11/50 =.22

24/50 = .48

35/50 = .70

46/50 = .92

50/50 = 1.0

Cumulative Relative Frequency

Measures of Central TendencyMean

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

17Section 2.3, Page 35

Measures of Central TendencyMedian

The median is the value of the middle number when the data 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 = 3.5The median is then the average of the 3rd and 4th number. The median is (8+9)/2 = 8.5

18Section 2.3, Page 36

Measures of Central TendencyMode and Midrange

19Section 2.3, Page 37

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

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.

20Section 2.3, Page 37

Measures of DispersionRange

21Secton 2.4, Page 39

Measures of DispersionVariance and Standard Deviation

22Section 2.4, Page 41

{6, 3, 8, 5, 3 }

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

23Section 2.5, Page 42

Finding PercentilesExample

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.

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}

24Section 2.5, Page 43

Using the TI-83 to Find Percentiles

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 L1PRGM: down arrow to PRCNTILE ENTER: (Copies program to home

screen)

ENTER: (Displays Program Input Page)

2nd L1: (Enters the List name)

ENTER: (Asks for Percentile)

21.0: (Enters the desired percentile)

ENTER: (Displays the 21st percentile)

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

CLEAR: (Clears the home screen)

25Section 2.5, Page 43

5-Number SummaryBox and Whisker Display

L

Q1 Q3

H

Med

26Section 2.5, Page 44

Interquartile Range = Q3-Q1

Range of middle 50% of valuesMeasure of dispersion resistant to outliers.

TI – 83 Problem (1)

a. Find the mean, standard deviation sample data.

STAT – EDIT: Enter the data is L1PRGM – SAMPSTAT - ENTER2ND L1 - ENTERDISPLAY:

Sample Mean

27Problems, Page 52

Standard Deviation

Variance

TI-83 Problem (2)

b. Find the Interquartile range (IQR)Q3 – Q1 = 32 – 28 = 4

c. Find the range.Max – Min = 34 – 25 = 9

28Problems, Page 50

TI-83 Problem (3)d. Make a box and whisker display of the data.

2ND STAT PLOT-ENTERENTER: Sets plot to ONDOWN ARROWRIGHT ARROW 5 TIMES: Select box PlotDOWN ARROW – 2nd L1: Select ListDisplay:

ZOOM – 9TRACE: Display:

RIGHT-LEFT ARROW: Display 5-number summary

29Problems, Page 50

Summary: Measures of Center and Spread

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.

30Section 2.4

Problem

a. Find the mean, variance, and standard deviation.

b. Find the 5-number summary.

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

d. Calculate the Interquartile range and the range

e. Describe the shape of the distribution

f. Find the 33rd percentile.

31Problems, Page 50

Problem

a. Find the mean, variance, and standard deviation.

b. Find the 5-number summary.

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

d. Calculate the Interquartile range and the range

e. Describe the shape of the distribution.

f. Find the 90th percentile.

32Problems, Page 50