Section 2.1 Frequency Distributions and Their Graphs Larson/Farber 4th ed.
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Transcript of Section 2.1 Frequency Distributions and Their Graphs Larson/Farber 4th ed.
Section 2.1
Frequency Distributions
and Their Graphs
Larson/Farber 4th ed.
Section 2.1 Objectives
• Construct frequency distributions• Construct frequency histograms, frequency polygons,
relative frequency histograms, and ogives
Larson/Farber 4th ed.
Frequency Distribution
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.
Larson/Farber 4th ed.
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
Constructing a Frequency Distribution
Larson/Farber 4th ed.
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.
Constructing a Frequency Distribution
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.
Larson/Farber 4th ed.
Constructing a Frequency Distribution
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.
Larson/Farber 4th ed.
Example: Constructing a Frequency Distribution
The following sample data set lists the amount spent on books for a semester.
Construct a frequency distribution that has seven classes.
Larson/Farber 4th ed.
91 472 279 249 530 376 188 341 266
199 142 273 189 130 489 266 248 101 375
486 190 398 188 269 43 30 127 354 84
Solution: Constructing a Frequency Distribution
1. Number of classes = 7 (given)
2. Find the class width
Round up to 72
91 472 279 249 530 376 188 341 266
199 142 273 189 130 489 266 248 101 375
486 190 398 188 269 43 30 127 354 84
Solution: Constructing a Frequency Distribution
Lower limit
Upper limit
30
102
174
246
318
390
462
Class width = 72
3. Use 30 (minimum value) as first lower limit. Add the class width of 72 to get the lower limit of the next class.
30 + 72 = 102
Find the remaining lower limits.
Solution: Constructing a Frequency Distribution
The upper limit of the first class is 101 (one less than the lower limit of the second class).
Add the class width of 72 to get the upper limit of the next class.
101 + 72 = 173
Find the remaining upper limits.
Larson/Farber 4th ed.
Lower limit
Upper limit
30 101
102 173
174 245
246 317
318 389
390 461
462 533
Class width = 72
Solution: Constructing a Frequency Distribution
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.
Larson/Farber 4th ed.
Class Tally Frequency, f
30 - 101 lllll 5
102 - 173 lll 3
174 - 245 lllll 5
246 - 317 lllll ll 7
318 - 389 llll 4
390 - 461 l 1
462 - 533 llll 4Σf = 29
Determining the Midpoint
Midpoint of a class
Class width = 72
Class Midpoint Frequency, f
30 - 101 65.5 5
102 - 173 137.5 3
174 - 245 209.5 5
246 - 317 281.5 7
318 - 389 353.5 4
390 - 461 425.5 1
462 - 533 497.5 4
Determining the Relative Frequency
Relative Frequency of a class • Portion or percentage of the data that falls in a
particular class.
Larson/Farber 4th ed.
Determining the Relative Frequency continued
Class Midpoint Frequency, f Relative Frequency
30 - 101 65.5 5 0.172
102 - 173 137.5 3 0.103
174 - 245 209.5 5 0.172
246 - 317 281.5 7 0.241
318 - 389 353.5 4 0.138
390 - 461 425.5 1 0.034
462 - 533 497.5 4 0.138
Σf = 29 1.000
Sum of Rel. Freq.
Determining the Cumulative FrequencyCumulative frequency of a class• The sum of the frequency for that class and all
previous classes.Class Midpoint Frequency, f Cumulative
Frequency
30 - 101 65.5 5 5
102 - 173 137.5 3 8
174 - 245 209.5 5 13
246 - 317 281.5 7 20
318 - 389 353.5 4 24
390 - 461 425.5 1 25
462 - 533 497.5 4 29
Σf = 29
Sum of class and previous class.
Expanded Frequency Distribution
Class Midpoint Frequency, f Relative Frequency
Cumulative Frequency
30 - 101 65.5 5 0.172 5
102 - 173 137.5 3 0.103 8
174 - 245 209.5 5 0.172 13
246 - 317 281.5 7 0.241 20
318 - 389 353.5 4 0.138 24
390 - 461 425.5 1 0.034 25
462 - 533 497.5 4 0.138 29
Σ = 29 1
Graphs of Frequency Distributions
Frequency Histogram• A 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.
Larson/Farber 4th ed.
data valuesfr
eque
ncy
Class Boundaries
Class boundaries• The numbers that separate classes without forming
gaps between them.
Larson/Farber 4th ed.
ClassClass
Boundaries
30 - 101
102 - 173
174 - 245
• The distance from the upper limit of the first class to the lower limit of the second class is 102 – 101 = 1.
• Half this distance is 0.5.
• First class lower boundary = 30 – 0.5 = 29.5• First class upper boundary = 101 + 0.5 = 101.5
29.5 – 101.5
Class Boundaries
Larson/Farber 4th ed.
ClassClass boundaries Frequency, f
30 - 101 29.5 - 101.5 5
102 - 173 101.5 - 173.5 3
174 - 245 173.5 - 245.5 5
246 - 317 245.5 - 317.5 7
318 - 389 317.5 - 389.5 4
390 - 461 389.5 - 461.5 1
462 - 533 461.5 - 533.5 4
Example: Frequency Histogram
Construct a frequency histogram for the book costs frequency distribution.
Larson/Farber 4th ed. 21
ClassClass
boundaries MidpointFrequency,
f
30 - 101 29.5 - 101.5 65.5 5
102 - 173 101.5 - 173.5 137.5 3
174 - 245 173.5 - 245.5 209.5 5
246 - 317 245.5 - 317.5 281.5 7
318 - 389 317.5 - 389.5 353.5 4
390 - 461 389.5 - 461.5 425.5 1
462 - 533 461.5 - 533.5 497.5 4
Solution: Frequency Histogram (using Midpoints)
Graphs of Frequency Distributions
Frequency Polygon• A line graph that emphasizes the continuous change
in frequencies.
Larson/Farber 4th ed.
data values
freq
uenc
y
Example: Frequency Polygon
Construct a frequency polygon for the Books costs frequency distribution.
Larson/Farber 4th ed.
Class MidpointFrequency, f
30 - 101 65.5 5
102 - 173 137.5 3
174 - 245 209.5 5
246 - 317 281.5 7
318 - 389 353.5 4
390 - 461 425.5 1
462 - 533 497.5 4
Solution: Frequency Polygon
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.
This is bull! In most cases it is better to just show the data and not have false markers!
Graphs of Frequency Distributions
Relative Frequency Histogram• Has the same shape and the same horizontal scale as
the corresponding frequency histogram.• The vertical scale measures the relative frequencies,
not frequencies.
Larson/Farber 4th ed.
data valuesre
lativ
e fr
eque
ncy
Graphs of Frequency Distributions
Cumulative Frequency Graph or Ogive• A 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.
Larson/Farber 4th ed.
data valuescu
mul
ativ
e fr
eque
ncy
Constructing an Ogive
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 or upper limit. The vertical scale measures cumulative
frequencies.
3. Plot points that represent the upper class boundaries and their corresponding cumulative frequencies.
Larson/Farber 4th ed.
Constructing an Ogive
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).
Larson/Farber 4th ed.
Example: Ogive
Construct an ogive for the book cost frequency distribution.
Class Midpoint Frequency, f Cumulative Frequency
30 - 101 65.5 5 5
102 - 173 137.5 3 8
174 - 245 209.5 5 13
246 - 317 281.5 7 20
318 - 389 353.5 4 24
390 - 461 425.5 1 25
462 - 533 497.5 4 29
Solution: Ogive
Larson/Farber 4th ed.
From the ogive, you can see that about 25 students spent $461 or less. The greatest increase in in cost occurs between $245 and $389.
Section 2.1 Summary
• Constructed frequency distributions• Constructed frequency histograms, frequency
polygons, relative frequency histograms and ogives
Larson/Farber 4th ed.