Transcript 12.1 – Visual Displays of Data

12.1 – Visual Displays of Data
In statistics:
A population includes all of the items of interest.
A sample includes some of the items in the population.
The study of statistics is usually divided into two main areas.
Descriptive statistics: has to do with collecting, organizing,
summarizing, and presenting data (information).
Inferential statistics: has to do with drawing inferences or
conclusions about populations based on information from
samples.
12.1 – Visual Displays of Data
Information that has been collected but not yet organized or
processed is called raw data.
Raw data are often quantitative (or numerical), but can also
be qualitative (or non-numerical).
Quantitative data: The number of siblings in ten different
families: 3, 1, 2, 1, 5, 4, 3, 3, 8, 2
Quantitative data can be sorted in mathematical order. The
number siblings can appear as 1, 1, 2, 2, 3, 3, 3, 4, 5, 8
Qualitative data: The makes of five different automobiles:
Toyota, Ford, Nissan, Chevrolet, Honda
12.1 – Visual Displays of Data
Frequency Distributions
Frequency Distribution is a method to organize data that
includes many repeated items.
It lists the distinct values (x) along with their frequencies (f ).
The relative frequency of each distinct item is the fraction, or
percentage, of the data set represented by each item.
12.1 – Visual Displays of Data
Example:
Ten students in a math class were polled as to the number of
siblings in their families {3, 2, 2, 1, 3, 4, 3, 3, 4, 2}. Construct
a frequency distribution and a relative frequency distribution
for the data.
Number x
1
2
3
4
Frequency f
Relative Frequency f /n
1
1/10 = 0.1
3
4
2
Sum = 10
3/10 = 0.3
4/10 = 0.4
2/10 = 0.2
Sum of f/n = 1
12.1 – Visual Displays of Data
Grouped Frequency Distributions
A Grouped Frequency Distribution is used when data sets
contain a large numbers of items.
The data are arranged into groups, or classes.
All data items are assigned to their appropriate classes, and
displayed in a table.
1.
2.
3.
4.
Guidelines for the Classes of a Grouped Frequency
Distribution
Make sure each data item will fit into one and only one,
class.
Make all the classes the same width.
Make sure that the classes do not overlap.
Use from 5 to 12 classes.
12.1 – Visual Displays of Data
Example:
Twenty students, selected randomly were asked to estimate
the number of hours that they had spent studying in the past
week (in and out of class). The responses are below.
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37
42
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Tabulate a grouped frequency distribution and a relative
frequency distribution.
The data contains values in the tens, twenties, thirties, forties
and fifties.
Five classes: 10-19, 20-29, 30-39, 40-49, 50-59
12.1 – Visual Displays of Data
15
42
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Hours
10 – 19
20 – 29
30 – 39
40 – 49
50 – 59
37
28
42
46
Frequency f
1
5
3
5
6
Sum = 20
20
29
27
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36
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29
40
Relative Frequency f /n
1/20 = 0.05 = 5%
5/20 = 0.25 = 25%
3/20 = 0.15 = 15%
5/20 = 0.25 = 25%
6/20 = 0.30 = 30%
Sum of f/n = 1 or 100%
12.1 – Visual Displays of Data
Histogram
The data from the frequency distribution or a grouped
frequency distribution can be graphically display using a
histogram.
A series of rectangles, whose lengths represent the
frequencies, are placed next to each other.
Frequenc
yf
1
1
2
3
3
4
= 0.1
3/10 = 0.3
4/10 = 0.4
4
2
2/10
Sum = 10
Relative
Frequency f /n
1/10
= 0.2
Sum of f/n = 1
5
Frequency
Number
x
4
3
2
1
0
1
2
3
Siblings
4
12.1 – Visual Displays of Data
Example: Histogram of a Grouped Frequency Distribution
10 – 19
20 – 29
30 – 39
Frequency f
1
5
3
7
6
Frequency
Hours
5
4
3
2
1
0
40 – 49
5
50 – 59
6
10-19
20-29
30-39
40-49 50-59
Hours
In the table, the numbers 10, 20, 30, 40, and 50 are called the lower class
limits.
The numbers 19, 29, 39, 49, and 59 are called the upper class limits.
The class width for the distribution is the difference of any two
successive lower (or upper) class limits. The class width is 10.
12.1 – Visual Displays of Data
Frequency Polygon
Data can also be displayed by a frequency polygon.
Plot a single point at the appropriate height for each frequency,
connect the points with a series of connected line segments
and complete the polygon with segments that trail down to the
axis.
Frequency
5
4
3
2
1
0
1
2
3
Siblings
4
12.1 – Visual Displays of Data
Line Graph
A line graph is used when it is important to show how data
changes with respect to another variable, such as time.
Example: Line Graph
The line graph below shows the stock price of company
PCWP over a 6-month span.
Price in dollars
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1
0
Jan
Feb
Mar
Apr
Month
May
June
12.1 – Visual Displays of Data
Line Graph
12.1 – Visual Displays of Data
Stem-and-Leaf Displays
The stem and leaf display is another method to present data.
It preserves the exact data values.
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1
5
2
0
7
8
3
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4
0 2 2 3 6
5
1
5
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9
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The digits to the left of the vertical line (blue region), are the “stems,”
The corresponding ones digits (green region) are the “leaves.”
12.1 – Visual Displays of Data
Bar Graphs
A frequency distribution of non-numerical data can be presented in the
form of a bar graph.
It is similar to a histogram except that the rectangles (bars) usually are
not touching each other and sometimes are arranged horizontally rather
than vertically.
Example: Bar Graph
A bar graph is given for the occurrence of vowels in this sentence.
Frequency
8
7
6
5
4
3
2
1
0
A
E
I
Vowel
O
U
12.1 – Visual Displays of Data
Circle Graphs or Pie Chart
A graphical alternative to the bar graph is the circle graph, or pie chart.
Each sector or wedge, shows the relative magnitude of the categories.
The entire circle measures 360°. The measure of each sector angle
should correspond to the percentage of the data being represented.
Example: Expenses
A general estimate of Amy’s monthly expenses are illustrated in the
circle graph below.
Clothing 10%
Other 35%
Food 30%
Rent25%