021 describing data - full of my life with mathematics only and lower class limits. class frequency:...

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DESCRIBING DATA

Frequency Tables, FrequencyDistributions, and Graphic Presentation

A raw data is the data obtained before it is being processed or arranged.

Raw Data

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A raw score is the score obtained by a particular student in a particular test before it is being processed or arranged.

Example: Raw Score

3

78, 74, 65, 74, 74, 67, 63, 67, 80, 58 74, 50, 65, 74,

86, 78, 63, 65, 80, 89

The raw scores for 20 students in a test

The Raw Score is a Variable

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� Data in raw form are usually not easy to use for decision making� Some type of organization is needed

� Table� Graph

� Techniques reviewed here:� Ordered Array� Stem-and-Leaf Display� Frequency Distributions and Histograms� Bar charts and pie charts� Contingency tables

Organizing and Presenting Data Graphically

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Interval Data

Ordered Array

Stem-and-LeafDisplay Histogram Polygon Ogive

Frequency Distributions and

Cumulative Distributions

Tables and Charts for Interval Data

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2

A sorted list of data:

� Shows range (minimum to maximum)

� Provides some signals about variability within the range

� May help identify outliers (unusual observations)

� If the data set is large, the ordered array is less useful

The Ordered Array

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� Data in raw form (as collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38

� Data in ordered array from smallest to largest:

21, 24, 24, 26, 27, 27, 30, 32, 38, 41

(continued)

The Ordered Array

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� A simple way to see distribution details in a data set

METHOD: Separate the sorted data series

into leading digits (the stem ) and

the trailing digits (the leaves )

Stem-and-Leaf Diagram

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Stem-and-Leaf

� The major advantage to organizing the data into stem-and-leaf display is that we get a quick visual picture of the shape of the distribution.

� Stem-and-leaf display is a statistical technique to present a set of data. Each numerical value is divided into two parts. The leading digit(s) becomes the stem and the trailing digit the leaf. The stems are located along the vertical axis, and the leaf values are stacked against each other along the horizontal axis.

� Advantage of the stem-and-leaf display over a frequency distribution - the identity of each observation is not lost.

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� Here, use the 10’s digit for the stem unit:

Data in ordered array:21, 24, 24, 26, 27, 27, 30, 32, 38, 41

� 21 is shown as

� 38 is shown as

Stem Leaf

2 1

3 8

Example

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Example

Suppose the seven observations in the 90 up to 100 class are: 96, 94, 93, 94, 95, 96, and 97.

The stem value is the leading digit or digits, in this case 9. The leaves are the trailing digits. The stem is placed to the left of a vertical line and the leaf values to the right. The values in the 90 up to 100 class would appear as

Then, we sort the values within each stem from smallest to largest. Thus, the second row of the stem-and-leaf display would appear as follows:

12

3

� Completed stem-and-leaf diagram:

Stem Leaves

2 1 4 4 6 7 7

3 0 2 8

4 1

(continued)

Data in ordered array:21, 24, 24, 26, 27, 27, 30, 32, 38, 41

Example

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� Using the 100’s digit as the stem:

� Round off the 10’s digit to form the

leaves

� 613 would become 6 1

� 776 would become 7 8

� . . .

� 1224 becomes 12 2

Stem Leaf

Using other stem units

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�Using the 100’s digit as the stem:

� The completed stem-and-leaf

display: Stem Leaves

(continued)

6 1 3 6

7 2 2 5 8

8 3 4 6 6 9 9

9 1 3 3 6 8

10 3 5 6

11 4 7

12 2

Data:

613, 632, 658, 717,722, 750, 776, 827,841, 859, 863, 891,894, 906, 928, 933,955, 982, 1034, 1047,1056, 1140, 1169, 1224

Using other stem units

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Stem-and-leaf: Another Example

Listed in Table 4–1 is the number of 30-second radio advertising spots purchased by each of the 45 members of the Greater Buffalo Automobile Dealers Association last year. Organize the data into a stem-and-leaf display. Around what values do the number of advertising spots tend to cluster? What is the fewest number of spots purchased by a dealer? The largest number purchased?

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Stem-and-leaf: Another Example

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What is a Frequency Distribution?

� A frequency distribution is a list or a table…

� containing class groupings (categories or ranges within which the data fall) ...

� and the corresponding frequencies with which data fall within each grouping or category

Tabulating Numerical Data: Frequency Distributions

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� A frequency distribution is a way to summarize data

� The distribution condenses the raw data into a more useful form...

� and allows for a quick visual interpretation of the data

Why Use Frequency Distributions?

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Score (X)

Frequency(f)

50586365677478808689

1123252211

Total 20

Frequency distribution table for ungrouped data

Frequency Distribution (ungrouped data)

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A Frequency Distribution is a grouping of data into mutually exclusive categories showing the number of observations in each class.

Frequency Distribution (grouped data)

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EXAMPLE – Creating a Frequency Distribution Table

Ms. Kathryn Ball of AutoUSA wants to develop tables, charts, and graphs to show the typical selling price on various dealer lots. The table on the right reports only the price of the 80 vehicles sold last month at Whitner Autoplex.

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Constructing a Frequency Table -Example

� Step 1: Decide on the number of classes. A useful recipe to determine the number of classes (k) is the “2 to the k rule.” such that 2k > n.There were 80 vehicles sold. So n = 80. If we try k = 6, which means we would use 6 classes, then 26 = 64, somewhat less than 80. Hence, 6 is not enough classes. If we let k = 7, then 27 128, which is greater than 80. So the recommended number of classes is 7.

� Step 2: Determine the class interval or width. The formula is: i ≥ (H-L)/k where i is the class interval, H is the highest observed value, L is the lowest observed value, and k is the number of classes.($35,925 - $15,546)/7 = $2,911Round up to some convenient number, such as a multiple of

10 or 100. Use a class width of $3,000

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Largest observation

Collect data

Bills

42.19

38.45

29.23

89.35

118.04

110.46

0.00

72.88

83.05..

(There are 200 data points

Prepare a frequency distributionHow many classes to use?

Number of observations Number of classes

Less then 50 5-7

50 - 200 7-9

200 - 500 9-10

500 - 1,000 10-11

1,000 – 5,000 11-13

5,000- 50,000 13-17

More than 50,000 17-20

Class width = [Range] / [# of classes]

[119.63 -0] / [8] = 14.95 15

Largest

observationLargest

observation

Smallest observationSmallest

observationSmallest

observationSmallest observation

Largest observation

NO of Class= 1 +3.3 log (n)

n: No of data/observation

Guide

line

Or Use No. of Class = 1 + 3.3log(n) Guide line

24

5

� Step 3: Set the individual class limits


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� Step 4: Tally the vehicle selling prices into the classes.

� Step 5: Count the number of items in each class.


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Frequency Distribution –Characteristics

Class midpoint: A point that divides a class into two equal parts. This is the average of the upper and lower class limits.

Class frequency: The number of observations in each class.

Class interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class.

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Relative Frequencies

� Class frequencies can be converted to relative class frequencies to show the fraction of the total number of observations in each class.

� A relative frequency captures the relationship between a class total and the total number of observations.

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Relative Frequency Distribution

To convert a frequency distribution to a relative frequency distribution, each of the class frequencies is divided by the total number of observations.

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Score (X)

Frequency(f)

Relative Frequency

Percentage(%)

50 1 0.05 5

58 1 0.05 5

63 2 0.10 10

65 3 0.15 15

67 2 0.10 10

74 5 0.25 25

78 2 0.10 10

80 2 0.10 10

86 1 0.05 5

89 1 0.05 5

Σf = 20

Example

30

6

Relative frequency =

Example: At score 65Relative frequency =

Percentage = Relative frequency x 100

Example: At score 65Percentage = 0.15 x 100 = 15%

f

f

∑=

students ofnumber Total

score particular aat students ofNumber

15.020

3 =

Relative Frequencies – Definition

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The three commonly used graphic forms are:

� Histograms� Frequency polygons� Cumulative frequency distributions� Ogive

Graphic Presentation of a Frequency Distribution

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Histogram for a frequency distribution based on quantitative data is very similar to the bar chart showing the distribution of qualitative data. The classes are marked on the horizontal axis and the class frequencies on the vertical axis. The class frequencies are represented by the heights of the bars.

Histogram

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Histogram Using Excel

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Frequency Polygon

� A frequency polygon also shows the shape of a distribution and is similar to a histogram.

� It consists of line segments connecting the points formed by the intersections of the class midpoints and the class frequencies.

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Frequency Polygon: Daily High Temperature

0

1

2

3

4

5

6

7

5 15 25 35 45 55 More

Fre

quen

cy

Class Midpoints

Class

10 but less than 20 15 320 but less than 30 25 630 but less than 40 35 540 but less than 50 45 450 but less than 60 55 2

FrequencyClass

Midpoint

(In a percentage polygon the vertical axis would be defined to show the percentage of observations per class)

Example: Frequency Polygon

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7

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

Skor

Kek

erap

an

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

Score

Fre

quen

cy

50 60 70 80 90

Example: Frequency Polygon

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0.00

0.05

0.10

0.15

0.20

0.25

0.30

1 2 3 4 5 6 7 8 9 10

Score

Rel

ativ

e F

requ

ency

50 60 70 80 90

Example: Relative Frequency Polygon

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0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11

Score

Per

cent

(%

)

50 60 70 80 90

Example: Percent Graph

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Frequency Polygon and Histogram

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Cumulative Frequency Distribution

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42

8

Class

10 but less than 20 3 15 3 15

20 but less than 30 6 30 9 45

30 but less than 40 5 25 14 70

40 but less than 50 4 20 18 90

50 but less than 60 2 10 20 100

Total 20 100

Percentage Cumulative Percentage

Data in ordered array:

12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58

FrequencyCumulative Frequency


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Ogive: Daily High Temperature

0

20

40

60

80

100

10 20 30 40 50 60

Cum

ulat

ive

Per

cent

age

Class Boundaries (Not Midpoints)

Class

Less than 10 10 010 but less than 20 20 1520 but less than 30 30 4530 but less than 40 40 7040 but less than 50 50 9050 but less than 60 60 100

Cumulative Percentage

Lower class

boundary

The Ogive (Cumulative % Polygon)

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Score(X)

Frequency (f)

Cumulative Frequency (cf)

Cumulative Relative Frequency (crf)

Cumulative Percent (cp)

50 1 1 0.05 5

58 1 2 0.10 10

63 2 4 0.20 20

65 3 7 0.35 35

67 2 9 0.45 45

74 5 14 0.70 70

78 2 16 0.80 80

80 2 18 0.90 90

86 1 19 0.95 95

89 1 20 1.00 100

Σf = 20

Cumulative Relative Frequency Distribution

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0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

Score

Cum

ulat

ive

Freq

uenc

y

18 students obtain score 85 or less

50 60 70 80 90

Cumulative Frequency Curve

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Grouped Data – Cumulative Frequency

Distribution and Cumulative Percent

Class Interval

(CI)

(score X)

Class Limit (CL)

(score X)

Class Mid Point

(m)

Frequency (less than

Upper Class Limit

(UCL))(f)

Relative Frequency (less than

UCL)

(cf)

Cumulative Relative

Frequency (less than

UCL) (crf)

Cumulative Percent (less than UCL)

(cp)

50 – 5455 – 5960 – 6465 – 6970 – 7475 – 7980 – 8485 – 89

49.5 – 54.554.5 – 59.559.5 – 64.564.5 – 69.569.5 – 74.574.5 – 79.579.5 – 84.584.5 – 89.5

5257626772778287

11255222

124914161820

0.050.100.200.450.700.800.901.00

5102045708090100

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0

5

10

15

20

1 2 3 4 5 6 7 8 9

Score

Cum

ulat

ive

Freq

uenc

y

100%

75%

50%

25%

0

Cumulative Percent

49.5 54.5 59.5 64.5 69.5 74.5 79.5 84.5 89.5

Cumulative Frequency Curve and Cumulative Percent for Grouped Data

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9

Ogive

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� Orgive is a smooth cumulative frequency curve.

� The curve moves from the left and increasessmoothly to the right.

� The smooth increase is called monotonic.

OGIVE

Ogive

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Categorical Data

Graphing Data

Pie Charts

Pareto Diagram

Bar Charts

Tabulating Data

Summary Table

Tables and Charts for Categorical Data

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Investment Amount PercentageType (in thousands $) (%)

Stocks 46.5 42.27Bonds 32.0 29.09CD 15.5 14.09Savings 16.0 14.55

Total 110.0 100.0(Variables are Categorical)

Summarize data by category

The Summary Table

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� Bar charts and Pie charts are often used for qualitative (category/nominal) data

� Height of bar or size of pie slice shows the frequency or percentage for each category

Bar and Pie Charts

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Bar Charts

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10

Bar Chart: Example

Investor's Portfolio

0 10 20 30 40 50

Stocks

Bonds

CD

Savings

Amount in $1000's



Total 110.0 100.0

Current Investment Portfolio

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Pie Charts

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Percentages are rounded to the nearest percent


Savings 15%

CD 14%

Bonds 29%

Stocks42%



Total 110.0 100.0

Pie Charts: Example

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PIE CHART USING EXCEL

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Pareto Diagram

� Used to portray categorical data

� A bar chart, where categories are shown in

descending order of frequency

� A cumulative polygon is often shown in the

same graph

� Used to separate the “vital few” from the “trivial many”

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cumulative %

invested (line graph)

% in

vest

ed in

eac

h ca

tego

ry

(bar

gra

ph)

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

Stocks Bonds Savings CD

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%


Pareto Diagram: Example

60

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Contingency Tables

� A scatter diagram requires that both of the variables be at least interval scale.

� What if we wish to study the relationship between two variables when one or both are nominal or ordinal scale? In this case we tally the results in a contingency table .

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Contingency Tables – Example

A manufacturer of preassembled windows produced 50 windows yesterday. This morning the quality assurance inspector reviewed each window for all quality aspects. Each was classified as acceptable or unacceptable and by the shift on which it was produced. Thus we reported two variables on a single item. The two variables are shift and quality. The results are reported in the following table.

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� Contingency Table for Investment Choices ($1000’s)

Investment Investor A Investor B Investor C Total Category

Stocks 46.5 55 27.5 129

Bonds 32.0 44 19.0 95CD 15.5 20 13.5 49Savings 16.0 28 7.0 51

Total 110.0 147 67.0 324

(Individual values could also be expressed as percentages of the overall total, percentages of the row totals, or percentages of the column totals)

Contingency Table

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� Example � To conduct an efficient advertisement

campaign the relationship between occupation and newspapers readership is studied. The following table was created

Blue Collar White collar ProfessionalG&M 27 29 33Post 18 43 51Star 38 15 24Sun 37 21 18

Blue Collar White collar ProfessionalG&M 27 29 33Post 18 43 51Star 38 15 24Sun 37 21 18

Contingency Table: Example

64

� SolutionIf there is no relationship between occupation and newspaper read, the bar charts describing the frequency of readership of newspapers should look similar across occupations.


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Blue

0

10

20

30

40

1 2 3 4

Blue-collar workers prefer

the “Star” and the “Sun”.

White-collar workers and professionals mostly read the“Post” and the “Globe

and Mail”

White

0

10

20

30

40

50

1 2 3 4

Prof

0

10

20

30

40

50

60

1 2 3 4


66

12

� We create a contingency table.� This table lists the frequency for each

combination of values of the two variables.

� We can create a bar chart that represent the frequency of occurrence of each combination of values.

Graphing the Relationship Between Two Nominal Variables

67

� Data can be classified according to the time it is collected.� Cross-sectional data are all collected at

the same time.� Time-series data are collected at

successive points in time.

� Time-series data is often depicted on a line chart (a plot of the variable over time).

Describing Time-Series Data

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� Example � The total amount of income tax paid by

individuals in 1987 through 1999 are listed below.

� Draw a graph of this data and describe the information produced.

Line Chart

69

Line Chart

0200,000400,000600,000800,000

1,000,0001,200,000

87 88 89 90 91 92 93 94 95 96 97 98 99

For the first five years – total tax was relatively flatFrom 1993 there was a rapid increase in tax revenues.

Line charts can be used to describe nominal data time series.

Line Chart

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� Present data in a way that provides substance, statistics and design

� Communicate complex ideas with clarity, precision and efficiency

� Give the largest number of ideas in the most efficient manner

� Excellence almost always involves several dimensions

� Tell the truth about the data

Principles of Graphical Excellence

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Providing information concerning the monthly bills of new subscribers in the first month after signing on with a telephone company. (Refer to file)� Collect data� Prepare a frequency distribution� Draw a histogram

APPLICATION EXAMPLE

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13

42.19 103.15 39.21 89.5 75.71 2.42 8.37 77.21 1.62 109.08 28.77 104.4 35.32 115.78 13.9 6.95

38.45 94.52 48.54 13.36 88.62 1.08 7.18 72.47 91.1 2.45 9.12 2.88 117.69 0.98 9.22 6.48

29.23 26.84 93.31 44.16 99.5 76.69 11.07 0 10.88 21.97 118.75 65.9 106.84 19.45 109.94 11.64

89.35 93.93 104.88 92.97 85 13.62 1.47 5.64 30.62 17.12 0 20.55 8.4 0 10.7 83.26

118.04 90.26 30.61 99.56 0 88.51 26.4 6.48 100.05 19.7 13.95 3.43 90.04 27.21 0 15.42

110.46 72.78 22.57 92.62 8.41 55.99 13.26 6.95 26.97 6.93 14.34 10.44 3.85 89.27 11.27 24.49

0 101.36 63.7 78.89 70.48 12.24 21.13 19.6 15.43 10.05 79.52 21.36 91.56 14.49 72.02 89.13

72.88 104.8 104.84 87.71 92.88 119.63 95.03 8.11 29.25 99.03 2.72 24.42 10.13 92.17 7.74 111.14

83.05 74.01 6.45 93.57 3.2 23.31 29.04 9.01 1.88 29.24 9.63 95.52 5.72 21 5.04 92.64

95.73 56.01 16.47 0 115.5 11.05 5.42 84.77 16.44 15.21 21.34 6.72 33.69 106.59 33.4 53.9

114.67 19.34 15.3 112.94

27.57 13.54 75.49 20.12

64.78 18.89 68.69 53.21

45.81 1.57 35 15.3

56.04 0 9.12 49.24

20.39 5.2 18.49 9.44

31.77 2.8 84.12 2.67

94.67 5.1 13.68 4.69

44.32 3.03 20.84 41.38

3.69 9.16 100.04 45.77

Data of Bill

73

Largest observation

Collect data

Bills

42.19

38.45

29.23

89.35

118.04

110.46

0.00

72.88

83.05..

(There are 200 data points

Prepare a frequency distributionHow many classes to use?

Number of observations Number of classes

Less then 50 5-7

50 - 200 7-9

200 - 500 9-10

500 - 1,000 10-11

1,000 – 5,000 11-13

5,000- 50,000 13-17

More than 50,000 17-20

Class width = [Range] / [# of classes]

[119.63 -0] / [8] = 14.95 15

Largest

observationLargest

observation

Smallest observationSmallest

observationSmallest

observationSmallest observation

Largest observation

NO of Class= 1 +3.3 log (n)

n: No of data/observation

Guide

line

Preparing Frequency Distribution

74

Draw a HistogramBill Frequency

15 7130 3745 1360 975 1090 18

105 28120 14

0

20

40

60

80

15 30 45 60 75 90 105 120

Bills

Fre

quen

cy

Draw Histogram

75

0

20

40

60

80

15 30 45 60 75 90 105

120

Bills

Fre

quen

cy

What information can we extract from this histogramAbout half of all the bills are small

71+37=108 13+9+10=32

A few bills are in the middle range

Relatively,large numberof large bills

18+28+14=60

Extracting Information

76

� There are four typical shape characteristics

Shapes of Histograms

77

Positively skewed Negatively skewed

•One with the long tail

extending to either right or

left side

Shapes of Histograms

78

14

A modal class is the one with the largest number of observations.

A unimodal histogram

The modal class

Modal classes

79

A bimodal histogram

A modal class A modal class

Modal classes

80

• A special type of symmetric unimodal histogram is bell shaped• Many statistical techniques require that the population be bell

shaped.• Drawing the histogram helps verify the shape of the population in

question

Bell shaped histograms

81

� Example 2 : Comparing students’ performance� Students’ performance in two statistics classes

were compared.� The two classes differed in their teaching

emphasis� Class A – mathematical analysis and

development of theory.� Class B – applications and computer based

analysis.� The final mark for each student in each course

was recorded.� Draw histograms and interpret the results.

Interpreting Histograms

82

Marks (Manual) Marks (Computer)

77 59 75 60 65 81 72 59

74 83 71 50 71 53 85 66

75 77 75 52 66 70 72 71

75 74 74 47 79 76 77 68

67 78 53 46 65 73 64 72

72 67 49 50 82 73 77 75

81 82 56 51 80 85 89 74

76 55 61 44 86 83 87 77

79 73 61 52 67 80 78 69

73 92 54 53 64 67 79 60

75 71 44 56 62 78 59 92

52 53 54 53 74 68 63 69

72 75 78 76 67 67 84 69

72 70 73 82 72 62 74 73

83 59 81 82 68 83 74 65

Data

83

Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Freq

uenc

y

Histogram

02040

50 60 70 80 90 100

Marks(Manual)

Freq

uenc

y

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Freq

uenc

y

Histogram

02040

50 60 70 80 90 100

Marks(Computer)

Freq

uenc

y

The mathematical emphasis

creates two groups, and a

larger spread.

Interpreting Histograms

84

021 describing data - full of my life with mathematics only and lower class limits. class frequency:...

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