Sect. 2-1 Frequency Distributions and Their graphsObjective SWBAT construct a frequency distribution including limits ,boundaries , midpoints, relative frequencies, and cumulative frequenciesAlso how to construct frequency histograms, frequency polygons, , relative frequency histograms, and ogives.

Frequency Distribution

Definition A frequency distribution is a table that shows classes or intervals of data entries 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.

Example of a frequency Distribution

Class Frequency 1- 5 6-10 11-15 16-20 21- 25 26-30

5 6 8 8 5 4

GuidelinesConstructing a Frequency Distribution from a Data Set1. Decide on the number of classes to include in the frequency

Distribution. The number of classes should be between five and twenty; otherwise it may be difficult to detect any patterns.

2. Find the class width as follows . Determine the range of the data . Divide the range by the number of classes, and 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. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper limit of the first class. Remember that classes can not 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 chart from a Data Set• 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 chart that has 7 classes.• 50 40 41 17 11 7 22 44 28 21 19 23

37 51 54 42 88• 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

Lower Limit

Upper Limit

7 19 31 43 55 67 79

18 30 42 54 66 78 90

Solution• The number4 of classes (7) is stated in the problem.

The minimum data entry is 7 and the maximum is 88 so the range is 81.

11.57 or 12

88 7

7

81

7

min

#

Max entry

of classes

range

No. of classes

Round up to 12

Class Tally Frequency

7-1819-3031-4243-5455-6667-7879-90

6 10 13 8 5 6 2

∑ f = 50

Try it Yourself

Construct a frequency distribution using the ages of the residents of Akhiok given in the opening on page 30. Use six classes.a. State the number of classesb. Find the minimum and maximum values and the class widthc. Find the class limits.d. Tally the data entriese. Write the fr4equency for each class.

DefinitionThe midpoint of a class is the sum of the lower and upper limit of the class divided by 2 . The midpoint is sometimes called the class mark. Midoint = The relative frequency of a class is the portion or percent of the data that falls in that class. To find the relative frequency of a class , divide the frequency f by the sample size n. Relative frequency =

The cumulative frequency is the sum of the frequency for that class and all previous classes. The cumulative frequency for the last class is equal to the sample size n.

Lower class limit + Upper class limit

2

Class frequency

Sample size

ExampleMidpoints, Relative and Cumulative frequenciesUsing the frequency distribution constructed in example 1, find the midpoint , relative frequency, and cumulative frequency for each class. Identify any patterns.SOLUTION The midpoint , relative, and cumulative frequency for the first three classes are calculated as follows. Relative CumulativeClass f Midpoint frequency frequency7 – 18 6 7-18 ∕ 2 = 12.5 6 / 50 = 0.12 6

19-30 10 19+ 30 / 2 = 24.5 10/50 = 0.2 6+10 = 16

31 – 42 13 31+ 42 / 2 = 36.5 13/50 = 0.26 16+13 = 29

Frequency Distribution for Internet usage (in minutes)

Class Frequency f

Midpoint Relative Frequency

CumulativeFrequency

7- 18 19-30 31-42 43- 54 55- 66 67-78 79- 90

6 10 13 8 5 6 2

12.5 24.5 36.5 48.5 60.5 72.5 84.5

0.12 0.2 0.26 0.16 0.1 0.12 0.04

6 16 29 37 42 48 50

∑ f = 50 ∑ f = 1 n

There are several patterns in the data set. For instance, the most common time span that users spent online was 31 to 42 minutes.

Try it Yourself 2

Using the frequency distribution constructed in try it Yourself1, find the midpoint, relative frequency, and cumulative frequency for each class. Identify any patterns.

a. Use the formulas to find each midpoint, relative frequency, and cumulative frequency..

b. Organize your results in a frequency distribution.

c. Identify patterns that emerge from the data.

Graphs of frequency Distributions

Definition A frequency histogram is a bar graph that represents the frequency distribution of a data set. A histogram has the following properties. 1. The horizontal scale is quantitative and measures the data values.2. The vertical scale measures the frequencies of the classes.3. Consecutive bars must touch.

Example : Constructing a frequency Histogram

Time0

2

4

6

8

10

12

14

12.524.536.548.560.572.584.5

Example 3 Constructing a frequency HistogramClass Class

BoundariesFrequency f

7- 1819 – 3031 - 4243 – 5455 - 6667 – 7879 - 90

6.5 – 18.5 18.5 - 30.5 30.5 - 42.5 42.5 - 54.5 54.5 - 66.5 66.5 - 78.5 78.5 - 90.5

6 10 13 8 5 6 2

Draw a histogram for the frequency Distribution in Example 2 describe any patterns.

First find the class boundaries. 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. So the lower and upper boundaries of the first class are as follows. First Class boundary = 7- 0.5 = 6.5First class upper boundary = 18 + 0.5 = 18.5

The boundaries of the remaining classes are shown in the table at the left.

Try it yourself

Use the frequency distribution form try it yourself 1 to construct a frequency histogram that represents the ages of the residents of Akhiok. Describe any patterns.

a. Find the class boundariesb. Choose the appropriate horizontal and vertical axes.c. Use the frequency distribution to find the height of each bar.d. Describe any patterns for the data.

Example 4 Constructing a frequency PolygonTo construct a frequency polygon use the same horizontal and

vertical scales that were used in the histogram labeled with class midpoints in example 3. Then plot the points that represent the midpoint and frequency of each class and connect the points in order from left to right. Because the graph should begin and end on the horizontal axis, 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. You can see that the frequency of subscribers increase up to 36.5 minutes and then decreases.

Time on (in minutes)0

2

4

6

8

10

12

14

0.512.524.536.548.560.572.584.590.5

Try it yourselfConstruct a frequency polygon that represents the ages of the

residents of Akhiok. Describe any patternsa. Choose the appropriate horizontal and vertical scalesb. Plot points that represent the midpoint and frequency for each

class.c. Connect the points and extend the sides as necessary.d. Describe any patterns fore the data.

Example 5 Constructing a relative frequency HistogramDraw a relative frequency histogram for the frequency

distribution in example 2.Solution : The relative frequency histogram is shown. Notice

that the shape of the histogram is the same as the frequency histogram constructed in example 3.

Time Online (in Minutes)0

0.05

0.1

0.15

0.2

0.25

0.3

6.518.530.5t42.554.566.578.590.5

Try it Yourself

Construct a relative frequency Histogram that represents the ages of the residents of Akhiok.

a. Use the same horizontal scale as used in the frequency histogram.

b. Revise the vertical scale to reflect relative frequencies.c. Use the relative frequencies to find the height of each bar.

DefinitionA cumulative frequency graph or ogive is 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 and the cumulative frequencies are marked on the vertical axis.

GUIDELINES Constructing an OGIVE (cumulative frequency Graph)1. Construct a frequency Distribution that includes

cumulative frequencies as one of its columns.2. Specify the horizontal and vertical scales. The horizontal

consists of upper class boundaries and the vertical scale measures cumulative frequencies.

3. Plot points that represents the upper class boundaries and their corresponding fre3quencies.

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 6; Constructing an Ogive

Category 10

10

20

30

40

50

60

6.518.530.542.554.566.578.590.5

Constructing an OgiveDraw an Ogive for the frequency distribution in example 2.

Estimate how many subscribers spent less than 60 minutes during their last session. Also use the graph to estimate when the greatest increase in usage occurs.

Upper Class Boundaries

f

Cumulative Frequencies

18.5 30.5 42.5 54.5 66.5 78.5 90.5

61013 8 5 6 2

6 16 29 37 42 48 50

Using the frequency distribution you can construct the ogive shown . The upper class boundaries, frequencies, and the cumulative frequencies are listed in the table. Notice that the graph starts at 6.5 where the cumulative frequency is zero and the grasph ends at 90.5 where the cumulative frequency is 50.From the ogive you can see that subscribers spent less than 60 minutes online during their last session.

Try it yourselfConstruct an ogive that represents the ages of the residents of

Akhiok. Estimate the number of residents who are less than 45 years old.

a. Specify the horizontal and vertical scales.b. Plot the points given by the upper class boundaries and the

cumulative frequencies.c. Construct the graph.d. Estimate the number of residents who are less than 45 years old.

Example 7: Using technology to construct Histograms • Use a calculator to construct a Histogram for the frequency

distribution in Example 2.Solution : Excel and TI – 83 have features for graphing

histograms..

Homework 1-8, 9-31odd, ex.cr. 33 Pgs.41-45

Frequency Distributions and Their Graphs

Section 2.1

Frequency Distributions

102 124 108 86 103 82 71 104 112 118 87 95103 116 85 122 87 100105 97 107 67 78 125109 99 105 99 101 92

Make a frequency distribution table with five classes.

Minutes Spent on the Phone

Key values: Minimum value =Maximum value =

67

125

4. Mark a tally | in appropriate class for each data value.

Steps to Construct aFrequency Distribution

1. Choose the number of classes

2. Calculate the Class Width

3. Determine Class Limits

Should be between 5 and 15. (For this problem use 5)

Find the range = maximum value – minimum. Then dividethis by the number of classes. Finally, round up to aconvenient number. (125 - 67) / 5 = 11.6 Round up to 12.

The lower class limit is the lowest data value that belongs ina class and the upper class limit is the highest. Use theminimum value as the lower class limit in the first class. (67)

After all data values are tallied, count the tallies in eachclass for the class frequencies.

78

90

102

114

126

35

8

9

5

67

79

91

103

115

Do all lower class limits first.

Class Limits Tally

Construct a Frequency Distribution

Minimum = 67, Maximum = 125Number of classes = 5Class width = 12

126.5114.5102.590.578.566.5

9

8

7

6

5

4

3

2

1

0

5

98

5

3

Boundaries

66.5 - 78.5

78.5 - 90.5

90.5 - 102.5

102.5 -114.5

114.5 -126.5

Frequency Histogram

Time on Phone

minutes

Class

67 - 78

79 - 90

91 - 102

103 -114

115 -126

35

8

9

5

Frequency Polygon

9

8

7

6

5

4

3

2

1

0

5

98

5

3

Time on Phone

minutes

Class

67 - 78

79 - 90

91 - 102

103 -114

115 -126

35

8

9

5

72.5 84.5 96.5 108.5 120.5

Mark the midpoint at the top of each bar. Connect consecutive midpoints. Extend the frequency polygon to the axis.

67 - 78 79 - 90 91 - 102103 - 114115 - 126

35895

Midpoint: (lower limit + upper limit) / 2

Relative frequency: class frequency/total frequency

Cumulative frequency: number of values in that class or in lower

Midpoint Relative Frequency Class

72.5 84.5 96.5108.5120.5

0.10 0.17 0.27 0.30 0.17

3 8 16 25 30

Other Information

CumulativeFrequency

(67 + 78)/2 3/30

Relative Frequency Histogram

Time on Phone

minutes

Relative frequency on vertical scale

Rela

tive

freq

uenc

y

Ogive

03

8

16

25

30

An ogive reports the number of values in the data set thatare less than or equal to the given value, x.

66.5 78.5 90.5 102.5 114.5 126.50

10

20

30

Cum

ulati

ve F

requ

ency

minutes

Minutes on Phone