Section 2.1

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


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.


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


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.


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.



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.


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.


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


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.


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.


Lower limit

Upper limit

30 101

102 173

174 245

246 317

318 389

390 461

462 533

Class width = 72


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 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.


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.


data valuesfr

eque

ncy

Class Boundaries

Class boundaries• The numbers that separate classes without forming

gaps between them.


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


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)


Frequency Polygon• A line graph that emphasizes the continuous change

in frequencies.


data values

freq

uenc

y

Example: Frequency Polygon

Construct a frequency polygon for the Books costs frequency distribution.


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!


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.


data valuesre

lativ

e fr

eque

ncy


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.


data valuescu

mul

ativ

e fr

eque

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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.


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).


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


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


Documents

Section 2.1