understandable statistics seventh edition by brase and brase prepared by: lynn smith gloucester...

Understandable StatisticsSeventh Edition

By Brase and BrasePrepared by: Lynn Smith

Gloucester County College

Chapter Two

Organizing Data

A graphical display should:

• Show the data

• Induce the viewer to think about the substance of the graphic

• Avoid distorting the message

Bar Graph• bars of uniform width• uniformly spaced• may be vertical or

horizontal• lengths represent

quantities being compared

• provide title, labels for each bar and lengths of bars

Reasons for Returns

0

10

20

30

40

50

60

Col

or

Size

Did

n't

like

Qua

lity

Pareto Chart

• tool of quality control• start with a bar chart• arrange bars in

decreasing order of frequency

• frequently used to investigate causes of problems

Reasons for returns

50

20

105

0

10

20

30

40

50

60

Circle Graph (Pie Chart)

• shows division of whole into component parts

• label parts with appropriate percentages of the whole

Conventions held

49%

27%

19%5%

FloridaCaliforniaVirginiaTexas

Time Plot

• Shows data values in chronological order

• time on horizontal scale

• variable being measured on vertical scale

• connect data points with line segments

Sales (in thousands of dollars)

050

100150200250300

Time Series

• Time series are data sets composed of similar measurements taken at regular intervals over time.

• A time plot can reveal some features of a time series.

Histogram

Differences from a bar chart:

• bars have equal width and always touch

• width of bars represents quantity

• heights of bars represent frequency

f

Measured quantity

To construct a histogram from raw data:

• Decide on the number of classes (5 to 15 is customary).

• Find a convenient class width.

• Organize the data into a frequency table.

• Find the class midpoints and the class boundaries.

• Sketch the histogram.

Finding class width

1. Compute:

classesofnumberdesiredvaluedatasmallestvaluedataestargl

2. Increase the value computed to the next highest whole number

Class Width

Raw Data:

10.2 18.7 22.3 20.0

6.3 17.8 17.1 5.0

2.4 7.9 0.3 2.5

8.5 12.5 21.4 16.5

0.4 5.2 4.1 14.3

19.5 22.5 0.0 24.7

11.4

Use 5 classes.

24.7 – 0.0

5

= 4.94

Round class width up to 5.

Frequency Table

• Determine class width.

• Create the classes. May use smallest data value as lower limit of first class and add width to get lower limit of next class.

• Tally data into classes.

• Compute midpoints for each class.

• Determine class boundaries.

Tallying the Data# of miles tally frequency

0.0 - 4.9 |||| | 6

5.0 - 9.9 |||| 5

10.0 - 14.9 |||| 4

15.0 - 19.9 |||| 5

20.0 - 24.9 |||| 5

Grouped Frequency Table

# of miles f

0.0 - 4.9 6

5.0 - 9.9 5

10.0 - 14.9 4

15.0 - 19.9 5

20.0 - 24.9 5

Class limits:

lower - upper

Computing Class Width

difference between the lower class limit of one class and the lower class

limit of the next class

# of miles f class widths

0.0 - 4.9 6 5

5.0 - 9.9 6 5

10.0 - 14.9 4 5

15.0 - 19.9 5 5

20.0 - 24.9 5 5

Finding Class Widths

Computing Class Midpoints

lower class limit + upper class limit

2

# of miles f class midpoints

0.0 - 4.9 6 2.45

5.0 - 9.9 5

10.0 - 14.9 4

15.0 - 19.9 5

20.0 - 24.9 5

Finding Class Midpoints


0.0 - 4.9 6 2.45

5.0 - 9.9 5 7.45

10.0 - 14.9 4

15.0 - 19.9 5

20.0 - 24.9 5



0.0 - 4.9 6 2.45

5.0 - 9.9 5 7.45

10.0 - 14.9 4 12.45

15.0 - 19.9 5 17.45

20.0 - 24.9 5 22.45


Class Boundaries

(Upper limit of one class + lower limit of next class)

divided by two

Finding Class Boundaries# of miles f class boundaries

0.0 - 4.9 6

5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4

15.0 - 19.9 5

20.0 - 24.9 5

Finding Class BoundariesFinding Class Boundaries

# of miles f class boundaries

0.0 - 4.9 6

5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4 9.95 - 14.95

15.0 - 19.9 5

20.0 - 24.9 5


0.0 - 4.9 6

5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4 9.95 - 14.95

15.0 - 19.9 5 14.95 - 19.95

20.0 - 24.9 5

Finding Class Boundaries


0.0 - 4.9 6 ??

5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4 9.95 - 14.95

15.0 - 19.9 5 14.95 - 19.95

20.0 - 24.9 5 19.95 - 24.95



0.0 - 4.9 6 ?? - 4.95 5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4 9.95 - 14.95

15.0 - 19.9 5 14.95 - 19.95

20.0 - 24.9 5 19.95 - 24.95



0.0 - 4.9 6 0.05 - 4.95 5.0 - 9.9 5 4.95 - 9.95

10.0 - 14.9 4 9.95 - 14.95

15.0 - 19.9 5 14.95 - 19.95

20.0 - 24.9 5 19.95 - 24.95


# of miles f

0.0 - 4.9 6

5.0 - 9.9 5

10.0 - 14.9 4

15.0 - 19.9 5

20.0 - 24.9 5

Constructing the Histogram

f

| | | | | |

6

5

4

3

2

1

0

-

-

-

-

-

-

--0.05 4.95 9.95 14.95 19.95 24.95 mi.

Relative Frequency

Relative frequency =

f = class frequency

n total of all frequencies

Relative Frequency

f = 6 = 0.24

n 25

f = 5 = 0.20

n 25

# of miles f relative frequency

0.0 - 4.9 6 0.24

5.0 - 9.9 5 0.20

10.0 - 14.9 4 0.16

15.0 - 19.9 5 0.20

20.0 - 24.9 5 0.20

Relative Frequency Histogram

| | | | | |

.24

.20

.16

.12

.08

.04

0

-

-

-

-

-

-

--0.05 4.95 9.95 14.95 19.95 24.95 mi.

Rel

ativ

e fr

eque

ncy

f/n

Common Shapes of HistogramsCommon Shapes of Histograms

Symmetrical

ff

When folded vertically, both sides are (more or less) the same.


Also Symmetrical

ff

Common Shapes of Histograms

Uniform

ff


Non-Symmetrical Histograms

These histograms are skewedskewed..


Skewed Histograms

Skewed left Skewed right


Bimodal

ff

The two largest rectangles are approximately equal in height and are separated by at least one class.

Frequency Polygon

A frequency polygon or line graph emphasizes the continuous rise or fall

of the frequencies.

Constructing the Frequency Polygon

• Dots are placed over the midpoints of each class.

• Dots are joined by line segments.

• Zero frequency classes are included at each end.

Weights(in pounds) f

2 - 4 6

5 - 7 5

8 - 10 4

11 - 13 5

Constructing the Frequency Polygon

f

| | | | | |

6

5

4

3

2

1

0

-

-

-

-

-

-

- 0 3 6 9 12 15

pounds

Cumulative Frequency

The sum of the frequencies for that class and all previous or later classes

Weights (in pounds) f

Greater than 1.5 20

Greater than 4.5 14

Greater than 7.5 9

Greater than 10.5 5

Greater than 13.5 0

Cumulative Frequency Table

Weights(in pounds) f

2 - 4 6

5 - 7 5

8 - 10 4

11 - 13 5

20

Ogive

Graph of a cumulative frequency table

Weights (in pounds) f

Greater than 1.5 20

Greater than 4.5 14

Greater than 7.5 9

Greater than 10.5 5

Greater than 13.5 0

Constructing the Ogive

Cu

mu

lati

ve f

req

uen

cy

| | | | | |

20

15

10

5

0

-

-

-

-

- 1.5 4.5 7.5 10.5 13.5 pounds

Exploratory Data Analysis

• A field of statistical study useful in detecting patterns and extreme data values

• Tools used include histograms and stem-and-leaf displays

Stem and Leaf Display

Raw Data:

35, 45, 42, 45, 41, 32, 25, 56, 67, 76, 65, 53, 53, 32, 34, 47, 43, 31

To make a Stem-and-Leaf Display

• Digits of each data value has two parts:

left = “stem,” right = “leaf”

• Align all stems in vertical column in increasing order with a vertical line to their right.

• Place all leaves with same stem on same row as the stem. Rearrange in increasing order

• Label

Stem and Leaf DisplayFirst data value = 35

Stem and Leaf DisplayFirst data value = 35

2

3

4

5

6

7

stems

5 leaf

Stem and Leaf DisplaySecond data value = 45

Stem and Leaf DisplaySecond data value = 45

2

3

4

5

6

7

5

5

Stem and Leaf DisplayThird data value = 42

Stem and Leaf DisplayThird data value = 42

2

3

4

5

6

7

5

5 2

Stem and Leaf DisplayNext data value = 45


2

3

4

5

6

7

5

5 2 5



2

3

4

5

6

7

5

5 2 5 1



2

3

4

5

6

7

5 2

5 2 5 1



2

3

4

5

6

7

5 2

5 2 5 1

5



2

3

4

5

6

7

5 2

5 2 5 1

5

6



2

3

4

5

6

7

5 2

5 2 5 1

5

6

7



2

3

4

5

6

7

5 2

5 2 5 1

5

6

7

6



2

3

4

5

6

7

5 2

5 2 5 1

5

6

7 5

6



2

3

4

5

6

7

5 2

5 2 5 1

5

6 3

7 5

6



2

3

4

5

6

7

5 2

5 2 5 1

5

6 3 3

7 5

6



2

3

4

5

6

7

5 2 2

5 2 5 1

5

6 3 3

7 5

6



2

3

4

5

6

7

5 2 2 4

5 2 5 1

5

6 3 3

7 5

6



2

3

4

5

6

7

5 2 2 4

5 2 5 1 7

5

6 3 3

7 5

6



2

3

4

5

6

7

5 2 2 4

5 2 5 1 7 3

5

6 3 3

7 5

6



2

3

4

5

6

7

5 2 2 4 1

5 2 5 1 7 3

5

6 3 3

7 5

6

Stem and Leaf DisplayStem and Leaf Display

2

3

4

5

6

7

5 2 2 4 1

5 2 5 1 7 3

5

6 3 3

7 5

6

Stem and Leaf Display with Leaves Rearranged

Stem and Leaf Display with Leaves Rearranged

2

3

4

5

6

7

1 2 2 4 5

1 2 3 5 5 7

5

3 3 6

5 7

6

Stem and Leaf Display with Labeling

Stem and Leaf Display with Labeling

2

3

4

5

6

7

1 2 2 4 5

1 2 3 5 5 7

5

3 3 6

5 7

6

Ages of Employees

2 5 represents 25 years

understandable statistics seventh edition by brase and brase prepared by: lynn smith gloucester...

Documents

lower class limit

class lower limit

class limits

class width1

upper class limit

convenient class width

round class width

twofinding class boundaries