Post on 19-Jan-2016
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
# of miles f 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
Finding Class Midpoints
# of miles f class midpoints
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
Finding Class Midpoints
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
# 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 14.95 - 19.95
20.0 - 24.9 5
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 9.95 - 14.95
15.0 - 19.9 5 14.95 - 19.95
20.0 - 24.9 5 19.95 - 24.95
Finding Class Boundaries
# of miles f class boundaries
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
Finding Class Boundaries
# of miles f class boundaries
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
Finding Class Boundaries
# 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.
Common Shapes of HistogramsCommon Shapes of Histograms
Also Symmetrical
ff
Common Shapes of Histograms
Uniform
ff
Common Shapes of HistogramsCommon Shapes of Histograms
Non-Symmetrical Histograms
These histograms are skewedskewed..
Common Shapes of HistogramsCommon Shapes of Histograms
Skewed Histograms
Skewed left Skewed right
Common Shapes of HistogramsCommon Shapes of Histograms
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
Stem and Leaf DisplayNext data value = 45
2
3
4
5
6
7
5
5 2 5
Stem and Leaf DisplayNext data value = 41
Stem and Leaf DisplayNext data value = 41
2
3
4
5
6
7
5
5 2 5 1
Stem and Leaf DisplayNext data value = 32
Stem and Leaf DisplayNext data value = 32
2
3
4
5
6
7
5 2
5 2 5 1
Stem and Leaf DisplayNext data value = 25
Stem and Leaf DisplayNext data value = 25
2
3
4
5
6
7
5 2
5 2 5 1
5
Stem and Leaf DisplayNext data value = 56
Stem and Leaf DisplayNext data value = 56
2
3
4
5
6
7
5 2
5 2 5 1
5
6
Stem and Leaf DisplayNext data value = 67
Stem and Leaf DisplayNext data value = 67
2
3
4
5
6
7
5 2
5 2 5 1
5
6
7
Stem and Leaf DisplayNext data value = 76
Stem and Leaf DisplayNext data value = 76
2
3
4
5
6
7
5 2
5 2 5 1
5
6
7
6
Stem and Leaf DisplayNext data value = 65
Stem and Leaf DisplayNext data value = 65
2
3
4
5
6
7
5 2
5 2 5 1
5
6
7 5
6
Stem and Leaf DisplayNext data value = 53
Stem and Leaf DisplayNext data value = 53
2
3
4
5
6
7
5 2
5 2 5 1
5
6 3
7 5
6
Stem and Leaf DisplayNext data value = 53
Stem and Leaf DisplayNext data value = 53
2
3
4
5
6
7
5 2
5 2 5 1
5
6 3 3
7 5
6
Stem and Leaf DisplayNext data value = 32
Stem and Leaf DisplayNext data value = 32
2
3
4
5
6
7
5 2 2
5 2 5 1
5
6 3 3
7 5
6
Stem and Leaf DisplayNext data value = 34
Stem and Leaf DisplayNext data value = 34
2
3
4
5
6
7
5 2 2 4
5 2 5 1
5
6 3 3
7 5
6
Stem and Leaf DisplayNext data value = 47
Stem and Leaf DisplayNext data value = 47
2
3
4
5
6
7
5 2 2 4
5 2 5 1 7
5
6 3 3
7 5
6
Stem and Leaf DisplayNext data value = 43
Stem and Leaf DisplayNext data value = 43
2
3
4
5
6
7
5 2 2 4
5 2 5 1 7 3
5
6 3 3
7 5
6
Stem and Leaf DisplayNext data value = 31
Stem and Leaf DisplayNext data value = 31
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