descriptive statistics 1 chapter 2. chapter outline 2 2.1 frequency distributions and their graphs...
TRANSCRIPT
Descriptive Statistics
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Chapter 2
Chapter Outline
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2.1 Frequency Distributions and Their Graphs
2.2 More Graphs and Displays
2.3 Measures of Central Tendency
2.4 Measures of Variation
2.5 Measures of Position
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Frequency Distributions and Their Graphs
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Section 2.1
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Section 2.1 Objectives
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Construct frequency distributionsConstruct frequency histograms, frequency
polygons, relative frequency histograms, and ogives
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Summarizing data/Frequency tables
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When data is collected from a survey or designed experiment, they must be organized into a manageable form. Data that is not organized is referred to as raw data.
Ways to Organize Data: Tables; Graphs; and Numerical Summaries.
A frequency distribution lists classes (or categories) of values, along with frequencies (or counts) of the number of values that fall into each class.
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Frequency tables: Definitions
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Lower Class Limits: are the smallest numbers that can actually belong to different classes.
Upper Class Limits: are the largest numbers that can actually belong to different classes.
Class Width: is the difference between two consecutive lower class limits or two consecutive upper class limits.
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Frequency Distribution
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Frequency Distribution
A table that shows classes or intervals of data with a count of the number of entries in each class.
The frequency, f, of a class is the number of data entries in the class.
Class Frequency, f
1 – 5 5
6 – 10 8
11 – 15 6
16 – 20 8
21 – 25 5
26 – 30 4
Lower classlimits
Upper classlimits
Class width 6 – 1 = 5
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Constructing a Frequency Distribution
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1. Decide on the number of classes. Usually between 5 and 20; otherwise, it may
be difficult to detect any patterns.
2. Find the class width. Determine the range of the data. Divide the range by the number of classes. Round up to the next convenient number.
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Constructing a Frequency Distribution
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3. Find the class limits. You can use the minimum data entry as the
lower limit of the first class. Find the remaining lower limits (add the class
width to the lower limit of the preceding class).
Find the upper limit of the first class. Remember that classes cannot overlap.
Find the remaining upper class limits.
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Constructing a Frequency Distribution
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4. Make a tally mark for each data entry in the row of the appropriate class.
5. Count the tally marks to find the total frequency f for each class.
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Guidelines for Frequency tables
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1. Be sure that the classes are mutually exclusive.
2. Include all classes, even if the frequency is zero.
3. Use the same width for all classes.
4. Select convenient numbers for class limits.
5. Use between 5 and 20 classes.
6. The sum of the class frequencies must equal the number of original data values.
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Example: Constructing a Frequency Distribution
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The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes.50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 8641 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 2018 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44
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Solution: Constructing a Frequency Distribution
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1. Number of classes = 7 (given)2. Find the class width
max min 86 711.29
#classes 7
Round up to 12
50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 86
41 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 20
18 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44
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Solution: Constructing a Frequency Distribution
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Lower limit
Upper limit
7Class width = 12
3. Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class.
7 + 12 = 19
Find the remaining lower limits.
19
31
43
5567
79
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Solution: Constructing a Frequency Distribution
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The upper limit of the first class is 18 (one less than the lower limit of the second class). Add the class width of 12 to get the upper limit of the next class.
18 + 12 = 30Find the remaining upper limits.
Lower limit
Upper limit
7
19
31
43
55
67
79
Class width = 1230
42
54
66
78
90
18
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Solution: Constructing a Frequency Distribution
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4. Make a tally mark for each data entry in the row of the appropriate class.
5. Count the tally marks to find the total frequency f for each class.Class Tally Frequenc
y, f
7 – 18 IIII I 6
19 – 30 IIII IIII 10
31 – 42 IIII IIII III 13
43 – 54 IIII III 8
55 – 66 IIII 5
67 – 78 IIII I 6
79 – 90 II 2
Σf = 5016
Determining the Midpoint
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Midpoint of a class
(Lower class limit) (Upper class limit)
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Class Midpoint Frequency, f
7 – 18 6
19 – 30 10
31 – 42 13
7 1812.5
2
19 3024.5
2
31 4236.5
2
Class width = 12
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Determining the Relative Frequency
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Relative Frequency of a class Portion or percentage of the data that falls in
a particular class.
n
f
sizeSample
frequencyclassfrequencyrelative
Class Frequency, f
Relative Frequency
7 – 18 6
19 – 30 10
31 – 42 13
60.12
50
100.20
50
130.26
50
•
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Determining the Cumulative Frequency
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Cumulative frequency of a classThe sum of the frequency for that class and all
previous classes.
Class Frequency, f Cumulative frequency
7 – 18 6
19 – 30 10
31 – 42 13
+
+
6
1629
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Expanded Frequency Distribution
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Class Frequency, f
MidpointRelative
frequencyCumulative frequency
7 – 18 6 12.5 0.12 6
19 – 30 10 24.5 0.20 16
31 – 42 13 36.5 0.26 29
43 – 54 8 48.5 0.16 37
55 – 66 5 60.5 0.10 42
67 – 78 6 72.5 0.12 48
79 – 90 2 84.5 0.04 50Σf = 50 1
n
f
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Graphs of Frequency Distributions
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Frequency HistogramA bar graph that represents the frequency
distribution.The horizontal scale is quantitative and
measures the data values.The vertical scale measures the
frequencies of the classes.Consecutive bars must touch.
data values
freq
uen
cy
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Class Boundaries
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Class boundariesThe numbers that separate classes without
forming gaps between them.Class
ClassBoundarie
s
Frequency, f
7 – 18 6
19 – 30
10
31 – 42
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• The distance from the upper limit of the first class to the lower limit of the second class is 19 – 18 = 1.
• Half this distance is 0.5. • First class lower boundary = 7 – 0.5 = 6.5
• First class upper boundary = 18 + 0.5 = 18.5
6.5 – 18.5
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Class Boundaries
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ClassClass
boundariesFrequency
, f
7 – 18 6.5 – 18.5 6
19 – 30 18.5 – 30.5 10
31 – 42 30.5 – 42.5 13
43 – 54 42.5 – 54.5 8
55 – 66 54.5 – 66.5 5
67 – 78 66.5 – 78.5 6
79 – 90 78.5 – 90.5 2
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Example: Frequency Histogram
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Construct a frequency histogram for the Internet usage frequency distribution.
ClassClass
boundaries
MidpointFrequen
cy, f
7 – 18 6.5 – 18.5
12.5 6
19 – 30 18.5 – 30.5
24.5 10
31 – 42 30.5 – 42.5
36.5 13
43 – 54 42.5 – 54.5
48.5 8
55 – 66 54.5 – 66.5
60.5 5
67 – 78 66.5 – 78.5
72.5 6
79 – 90 78.5 – 90.5
84.5 2
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Solution: Frequency Histogram (using Midpoints)
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Solution: Frequency Histogram (using class boundaries)
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6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5
You can see that more than half of the subscribers spent between 19 and 54 minutes on the Internet during their most recent session.
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Graphs of Frequency Distributions
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Frequency PolygonA line graph that emphasizes the continuous change in frequencies.The line segments are extended to the right and left so that the
graph begins and ends on the x-axis.
data values
freq
uen
cy
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Example: Frequency Polygon
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Construct a frequency polygon for the Internet usage frequency distribution.
Class Midpoint Frequency, f
7 – 18 12.5 6
19 – 30 24.5 10
31 – 42 36.5 13
43 – 54 48.5 8
55 – 66 60.5 5
67 – 78 72.5 6
79 – 90 84.5 2
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Solution: Frequency Polygon
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02468101214
0.5 12.5 24.5 36.5 48.5 60.5 72.5 84.5 96.5
Freq
uenc
y
Time online (in minutes)
Internet Usage
You can see that the frequency of subscribers increases up to 36.5 minutes and then decreases.
The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.
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Graphs of Frequency Distributions
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Relative Frequency HistogramHas the same shape and the same horizontal
scale as the corresponding frequency histogram.
The vertical scale measures the relative frequencies, not frequencies.
data values
rela
tive
freq
uen
cy
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Example: Relative Frequency Histogram
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Construct a relative frequency histogram for the Internet usage frequency distribution.
ClassClass
boundariesFrequency,
fRelative
frequency
7 – 18 6.5 – 18.5 6 0.12
19 – 30 18.5 – 30.5 10 0.20
31 – 42 30.5 – 42.5 13 0.26
43 – 54 42.5 – 54.5 8 0.16
55 – 66 54.5 – 66.5 5 0.10
67 – 78 66.5 – 78.5 6 0.12
79 – 90 78.5 – 90.5 2 0.04
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Solution: Relative Frequency Histogram
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6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5
From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online.32
Graphs of Frequency Distributions
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Cumulative Frequency Graph or OgiveA line graph that displays the cumulative frequency of
each class at its upper class boundary.The upper boundaries are marked on the horizontal
axis.The cumulative frequencies are marked on the vertical
axis.Ogives are useful for determining the number of values
less than some particular class boundary.
data values
cum
ula
tiv
e
freq
uen
cy
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Constructing an Ogive
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1. Construct a frequency distribution that includes cumulative frequencies as one of the columns.
2. Specify the horizontal and vertical scales. The horizontal scale consists of the upper
class boundaries. The vertical scale measures cumulative
frequencies.
3. Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.
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Constructing an Ogive
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4. Connect the points in order from left to right.5. The graph should start at the lower boundary
of the first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size).
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Example: Ogive
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Construct an ogive for the Internet usage frequency distribution.
ClassClass
boundariesFrequency,
fCumulative frequency
7 – 18 6.5 – 18.5 6 6
19 – 30 18.5 – 30.5 10 16
31 – 42 30.5 – 42.5 13 29
43 – 54 42.5 – 54.5 8 37
55 – 66 54.5 – 66.5 5 42
67 – 78 66.5 – 78.5 6 48
79 – 90 78.5 – 90.5 2 50
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Solution: Ogive
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0
10
20
30
40
50
60C
um
ula
tive
freq
uen
cy
Time online (in minutes)
Internet Usage
6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5
From the ogive, you can see that about 40 subscribers spent 60 minutes or less online during their last session. The greatest increase in usage occurs between 30.5 minutes and 42.5 minutes.
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Practice Questions
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Q. (2.1)The average quantitative GRE scores for the top 30 graduate schools of engineering are listed below. Construct a frequency distribution with six classes.767 770 761 760 771 768 776 771 756 770
763 760 747 766 754 771 771 778 766 762
780 750 746 764 769 759 757 753 758 746
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Practice Questions
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Q. (2.2)The ages of the signers of the Declaration of Independence
are shown below (age is approximate since only the birth
year appeared in the source and one has been omitted since
his birth year is unknown). Construct a frequency
distribution for the data using seven classes.41 54 47 40 39 35 50 37 49 42 70 32
44 52 39 50 40 30 34 69 39 45 33 42
44 63 60 27 42 34 50 42 52 38 36 45
35 43 48 46 31 27 55 63 46 33 60 62
35 46 45 34 53 50 50
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Practice Questions
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Q. (2.3)For 108 randomly selected college applicants, the
following frequency distribution for entrance exam scores
was obtained. Construct a histogram, frequency polygon,
and ogive for the data. Class limits
126 – 134 9
28117 – 125
43108 – 116
2299 – 107
690 – 98
Frequency
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Practice Questions
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Q. (2.4)The number of calories per serving for selected ready –
to - eat cereals is listed here. Construct a frequency
distribution using seven classes. Draw a histogram,
frequency polygon, and ogive for the data, using relative
frequencies. Describe the shape of the histogram.
130 190 140 80 100 120 220 220 110 100
210 130 100 90 210 120 200 120 180 120
190 210 120 200 130 180 260 270 100 160
190 240 80 120 90 190 200 210 190 180
115 210 110 225 190 130
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Section 2.1 Summary
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Constructed frequency distributionsConstructed frequency histograms,
frequency polygons, relative frequency histograms and ogives
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