Sect 2 1 Frequency Distributions and Their graphs

Sect. 2 -1 Frequency Distributions and Their graphs Objective SWBAT construct a frequency distribution including limits , boundaries , midpoints, relative frequencies, and cumulative frequencies Also 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 1 - 5 6 -10 11 -15 16 -20 21 - 25 26 -30 Frequency 5 6 8 8 5 4

Guidelines Constructing a Frequency Distribution from a Data Set 1. 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 Upper Limit 7 19 31 43 55 67 79 18 30 42 54 66 78 90

Solution • The number 4 of classes (7) is stated in the problem. The minimum data entry is 7 and the maximum is 88 so the range is 81. Class 7 -18 19 -30 31 -42 range 43 -54 55 -66 No. of classes 67 -78 79 -90 Round up to 12 Tally Frequency 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 classes b. Find the minimum and maximum values and the class width c. Find the class limits. d. Tally the data entries e. Write the fr 4 equency for each class.

Definition The 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. Lower class limit + Upper class limit Midoint = 2 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. Class frequency Relative frequency = Sample size 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.

Example Midpoints, Relative and Cumulative frequencies Using 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 Cumulative Class f Midpoint frequency 7 – 18 6 7 -18 ∕ 2 = 12. 5 6 / 50 = 0. 12 6 19 -30 10 31 – 42 13 19+ 30 / 2 = 24. 5 31+ 42 / 2 = 36. 5 10/50 = 0. 2 13/50 = 0. 26 6+10 = 16 16+13 = 29

Frequency Distribution for Internet usage (in minutes) Class 7 - 18 19 -30 31 -42 43 - 54 55 - 66 67 -78 79 - 90 Frequency f 6 10 13 8 5 6 2 ∑ f = 50 Midpoint Relative Frequency 12. 5 24. 5 36. 5 48. 5 60. 5 72. 5 84. 5 0. 12 0. 26 0. 12 0. 04 Cumulative Frequency 6 16 29 37 42 48 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 Yourself 1, 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 14 12 12. 5 10 24. 5 8 36. 5 6 48. 5 60. 5 4 72. 5 2 0 84. 5 Time

Example 3 Constructing a frequency Histogram Class 7 - 18 19 – 30 31 - 42 43 – 54 55 - 66 67 – 78 79 - 90 Class Boundaries Frequency f 6. 5 – 18. 5 - 30. 5 - 42. 5 - 54. 5 - 66. 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. The boundaries of the remaining classes are shown in the table at the left. 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. 5 First class upper boundary = 18 + 0. 5 = 18. 5

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 boundaries b. 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 Polygon To 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.

14 12 0. 5 10 12. 5 24. 5 8 36. 5 48. 5 6 60. 5 4 72. 5 84. 5 2 0 90. 5 Time on (in minutes)

Try it yourself Construct a frequency polygon that represents the ages of the residents of Akhiok. Describe any patterns a. Choose the appropriate horizontal and vertical scales b. 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 Histogram Draw 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.

0. 3 0. 25 6. 5 18. 5 0. 2 30. 5 t 42. 5 0. 15 54. 5 66. 5 0. 1 78. 5 0. 05 0 90. 5 Time Online (in Minutes)

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.

Definition A 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 fre 3 quencies. 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 60 50 6. 5 18. 5 40 30. 5 30 42. 5 54. 5 20 66. 5 78. 5 10 0 90. 5 Category 1

Constructing an Ogive Draw 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 f Boundaries 18. 5 30. 5 42. 5 54. 5 66. 5 78. 5 90. 5 6 10 13 8 5 6 2 Cumulative Frequencies 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 yourself Construct 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 -31 odd, ex. cr. 33 Pgs. 41 -45

Section 2. 1 Frequency Distributions and Their Graphs

Frequency Distributions Minutes Spent on the Phone 102 124 71 104 103 116 105 97 109 99 108 112 85 107 105 86 118 122 67 99 103 82 87 95 87 100 78 125 101 92 Make a frequency distribution table with five classes. Key values: Minimum value = Maximum value = 67 125

Steps to Construct a Frequency Distribution 1. Choose the number of classes Should be between 5 and 15. (For this problem use 5) 2. Calculate the Class Width Find the range = maximum value – minimum. Then divide this by the number of classes. Finally, round up to a convenient number. (125 - 67) / 5 = 11. 6 Round up to 12. 3. Determine Class Limits The lower class limit is the lowest data value that belongs in a class and the upper class limit is the highest. Use the minimum value as the lower class limit in the first class. (67) 4. Mark a tally | in appropriate class for each data value. After all data values are tallied, count the tallies in each class for the class frequencies.

Construct a Frequency Distribution Minimum = 67, Maximum = 125 Number of classes = 5 Class width = 12 Class Limits Tally 67 78 79 90 91 102 103 114 115 126 Do all lower class limits first. 3 5 8 9 5

Frequency Histogram Boundaries Class 66. 5 - 78. 5 67 - 78 3 78. 5 - 90. 5 79 - 90 5 90. 5 - 102. 5 91 - 102 8 102. 5 -114. 5 103 -114 9 114. 5 -126. 5 115 -126 5 Time on Phone 9 9 8 8 7 6 5 5 5 4 3 3 2 1 0 66. 5 78. 5 90. 5 102. 5 minutes 114. 5 126. 5

Frequency Polygon Class 67 - 78 79 - 90 91 - 102 3 5 8 Time on Phone 9 9 8 8 7 6 103 -114 9 5 115 -126 5 3 4 5 5 3 2 1 0 72. 5 84. 5 96. 5 108. 5 minutes 120. 5 Mark the midpoint at the top of each bar. Connect consecutive midpoints. Extend the frequency polygon to the axis.

Other Information Midpoint: (lower limit + upper limit) / 2 Relative frequency: class frequency/total frequency Cumulative frequency: number of values in that class or in lower Midpoint Class Relative Frequency (67 + 78)/2 3/30 Cumulative Frequency 67 - 78 3 72. 5 0. 10 3 79 - 90 5 84. 5 0. 17 8 91 - 102 8 96. 5 0. 27 16 103 - 114 9 108. 5 0. 30 25 115 - 126 5 120. 5 0. 17 30

Relative Frequency Histogram Relative frequency Time on Phone Relative frequency on vertical scale minutes

Ogive An ogive reports the number of values in the data set that are less than or equal to the given value, x. Cumulative Frequency Minutes on Phone 30 30 25 20 16 10 8 3 0 0 66. 5 78. 5 90. 5 102. 5 minutes 114. 5 126. 5