Section 6.2: Frequency Distributions

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Learning Objectives

• Contrast descriptive and inferential statistics
• Construct an ungrouped frequency distribution for a data set
• Construct a grouped frequency distribution for a data set

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Statistics can be divided into two main branches, descriptive statistics and inferential statistics, each serving different purposes in data analysis.

 

Definition: Descriptive Statistics

Descriptive statistics involves methods for organizing, summarizing, and presenting data in a clear and understandable way. It describes what the data shows without drawing conclusions beyond the data itself.

 

Common Tools & Techniques for Descriptive Statistics

• Measures of center: Mean, median, mode
• Measures of spread: Range, standard deviation, variance, interquartile range
• Graphical displays: Histograms, stem-and-leaf plots, pie charts, bar graphs, box plots
• Tables: Frequency distributions, contingency tables
• Percentages and proportions: Summary statistics

Note: Most of the tools and techniques mentioned above will be discussed further throughout the chapter.

Examples of Descriptive Statistics

• The average (mean) test score in a class is 82
• 65% of survey respondents prefer option A
• The temperature ranged from 68°F to 83°F over 20 days
• Creating a stem-and-leaf plot of customer ages
• Calculating that the median household income in a city is $55,000

Descriptive statistics only describes the data you have collected. It makes no predictions or generalizations beyond that specific data set.

 

Definition: Inferential Statistics

Inferential statistics involves methods for making predictions, decisions, or generalizations about a population based on sample data. It goes beyond the data to draw conclusions.

 

Common Tools & Techniques for Inferential Statistics

• Hypothesis testing: t-tests, chi-square tests, ANOVA
• Confidence intervals: Estimating population parameters with a margin of error
• Regression analysis: Predicting relationships between variables
• Probability theory: Foundation for making inferences
• Sampling distributions: Understanding variability in estimates
• P-values and significance levels: Determining statistical significance

Note: All of the tools and techniques mentioned above will NOT be discussed in this text. These topics are discussed in an Elementary Statistics class such as a STAT C1000 Introduction To Statistics course that can be taken at any California Community College.

Examples of Inferential Statistics

• Based on a sample of 500 voters, we estimate that 52% ± 4% of all voters support the candidate
• A clinical trial concludes that a new drug is significantly more effective than a placebo (P < 0.05)
• Predicting that increasing study time by 1 hour will raise test scores by 5 points on average
• Testing whether there's a significant difference in customer satisfaction between two store locations
• Using sample data to determine if a new teaching method improves student performance

Inferential statistics uses sample data to make generalizations about populations. It involves uncertainty and requires probability theory to assess reliability.

 

Now, that we are familiar with descriptive statistics as the broad field of organizing and summarizing data, frequency distributions provide the first concrete tool we need to learn. Frequency distributions are often the starting point for all data analysis. Before you can calculate a mean, create a histogram, or understand a distribution's shape, you need to know how often each value (or range of values) occurs. This makes frequency distributions a logical "first step" in the descriptive statistics toolkit.

 

Definition: Frequency Distributions

A frequency distribution is a table that organizes data by showing how often each value (or group of values) occurs.

 

Two Types of Frequency Distributions

• Ungrouped frequency distributions - Used for either qualitative data (categorical data) or single value data
• Grouped frequency distributions - Used for quantitative data (numerical data) where the data needs to be grouped into class intervals

Recall from the previous section that qualitative or categorical variables yields descriptive information (e.g., colors, names, categories), whereas quantitative variables yields numerical measurements (e.g., heights, weights, temperatures). Generally speaking if you have a small data set with relatively few distinct values (typically fewer than 15-20 unique values), then its best to use ungrouped frequency distributions. On the other hand, if you have a large data set or many unique values, it is best to use grouped frequency distributions where the data needs to be grouped into class intervals.

 

Constructing Ungrouped Frequency Distributions

• Step 1 — Organize your data (optional, if not using tallies)
• Step 2 — Identify all unique values in your data sets
• Step 3 — Create a two-column table (without tallies) or a three-column table (with tallies):
  • Two-column table:
    • Column 1: "Value" or "Data Value" (list each unique value)
    • Column 2: "Frequency" (count how many times each appears)
  • Three-column table:
    • Column 1: "Value" or "Data Value" (list each unique value)
    • Column 2: "Tally" (tally each observation; this is best when you did not arrange the data in order in Step 1)
    • Column 3: "Frequency" (count how many times each appears)
• Step 4 — Count or use tally marks
• Step 5 — Record the count
• Step 6 — Check your work

 

Let's see several examples for the simpler of the two, ungrouped frequency distributions.

 

Example #6.2.1 🤔

Consider the data set for twenty overall ratings from 1 through 10 of an exclusive Disney+ show:

4, 7, 5, 9, 10, 2, 8, 1, 7, 9, 7, 6, 1, 9, 7, 6, 4, 1, 8, 7

Construct an ungrouped frequency distribution for the data set.

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✅ Solution:

• Step 1 — Organize your data (optional, if not using tallies): 1, 1, 1, 2, 4, 4, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 10
• Step 2 — Identify all unique values in your data sets: 1, 2, 4, 5, 6, 7, 8, 9, 10
• Step 3 — Create a two-column table (without tallies) or a three-column table (with tallies):
  • Two-column table:
    • Column 1: "Value" or "Data Value" (list each unique value)
    • Column 2: "Frequency" (count how many times each appears)
  • Three-column table:
    • Column 1: "Value" or "Data Value" (list each unique value)
    • Column 2: "Tally" (tally each observation; this is best when you did not arrange the data in order in Step 1)
    • Column 3: "Frequency" (count how many times each appears)
• Step 4 — Count or use tally marks: Count how many times each value appears in your data.
• Step 5 — Record the count: Place each count in the frequency column.
• Step 6 — Check your work: The sum of all frequencies should equal total number of data values.

Ratings	Tally	Frequency
1	\(|~|~|\)	3
2	\(|\)	1
3		0
4	\(|~|\)	2
5	\(|\)	1
6	\(|~|\)	2
7	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
8	\(|~|\)	2
9	\(|~|~|\)	3
10	\(|\)	1
Total		20 ✓

Example #6.2.2 🤔

A parking lot attendant recorded the number of cars parked each hour for 25 hours:

15, 18, 15, 20, 18, 15, 20, 18, 15, 20, 18, 20, 22, 17, 19, 18, 20, 18, 18, 17, 18, 23, 18, 20, 18

Construct an ungrouped frequency distribution for the data set.

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✅ Solution:

• Step 1 — Organize your data (optional, if not using tallies): Since there are many observations, it is best to skip this step and use tallies going forward.
• Step 2 — Identify all unique values in your data sets: 15, 17, 18, 19, 20, 22, 23
• Step 3 — Create a three-column table (with tallies):
  • Three-column table:
    • Column 1: "Value" or "Data Value" (list each unique value)
    • Column 2: "Tally" (tally each observation; this is best when you did not arrange the data in order in Step 1)
    • Column 3: "Frequency" (count how many times each appears)
• Step 4 — Count or use tally marks: Using the unordered data set, tally each observation starting with first observation, then the second observation, and so on, until all observations are accounted for.
• Step 5 — Record the count: Count the tallies and place each count in the frequency column.
• Step 6 — Check your work: The sum of all frequencies should equal total number of data values.

No. of Cars	Tally	Frequency
15	\(|~|~|~|\)	4
16		0
17	\(|~|\)	2
18	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~\color{black}\bcancel{\color{black}\text{| | | |}}\)	10
19	\(|\)	1
20	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
21		0
22	\(|\)	1
23	\(|\)	1
Total		25 ✓

Example #6.2.3 🤔

A single dice was rolled 30 times with the following results:

3, 5, 2, 6, 4, 1, 3, 5, 6, 2, 4, 3, 6, 1, 5, 3, 4, 2, 6, 5, 3, 5, 1, 6, 2, 5, 3, 4, 6, 5

Construct an ungrouped frequency distribution for the data set.

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✅ Solution:

• Step 1 — Organize your data (optional, if not using tallies): Since there are many observations, it is best to skip this step and use tallies going forward.
• Step 2 — Identify all unique values in your data sets: 1, 2, 3, 4, 5, 6
• Step 3 — Create a three-column table (with tallies):
  • Three-column table:
    • Column 1: "Value" or "Data Value" (list each unique value)
    • Column 2: "Tally" (tally each observation; this is best when you did not arrange the data in order in Step 1)
    • Column 3: "Frequency" (count how many times each appears)
• Step 4 — Count or use tally marks: Using the unordered data set, tally each observation starting with first observation, then the second observation, and so on, until all observations are accounted for.
• Step 5 — Record the count: Count the tallies and place each count in the frequency column.
• Step 6 — Check your work: The sum of all frequencies should equal total number of data values.

Outcome	Tally	Frequency
1	\(|~|~|\)	3
2	\(|~|~|~|\)	4
3	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
4	\(|~|~|~|\)	4
5	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|~|\)	7
6	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
Total		30 ✓

 

Now, let's consider a data set consisting of temperatures (in degrees Fahrenheit) recorded in 20 cities on a particular day, rounded to the nearest degree: 58, 68, 65, 79, 51, 94, 64, 80, 69, 113, 63, 70, 102, 70, 92, 87, 77, 64, 64, 72. If we were to arrange the data in ascending order, we have 51, 58, 63, 64, 64, 64, 65, 68, 69, 70,
70, 72, 77, 79, 80, 87, 92, 94, 102, 113. Notice that almost all of the values are unique, which means most values have a frequency of 1. The only observation that does not have a frequency of 1 is '64', which has a frequency of 3.

An ungrouped frequency distribution would be ineffective for this data set. Since most values are unique, the resulting distribution would consist primarily of frequencies equal to one, appearing as a little to nothing more than a vertical listing of the original data set that was initially written horizontally. Such a representation provides minimal insight into the data's structure or patterns.

When data contain many distinct values or span a wide range, it becomes necessary to organize the values into class intervals. This grouping process creates a grouped frequency distribution, which reveals patterns and trends that would otherwise remain obscured in the raw data.

To construct a grouped frequency distribution, we must first establish appropriate class intervals, unless a class width and starting point are explicitly provided. This process involves several key decisions and calculations. For all intent purposes in this text, a class width be given as well as a starting value to use for the lower limit of the first class. If you would prefer to learn the process when a class width and starting point is not provided, use the interactive pull-down menu below. Otherwise, skip to the next one.

 

Constructing Grouped Frequency Distributions When a Class Width and Starting Point Are Not Given

• Step 1 — Find the Range
  • Range = Maximum value − Minimum value
• Step 2 — Selecting the Number of Classes
  • The number of classes should be chosen to provide a clear picture of the distribution without excessive detail or oversimplification. Generally, 5 to 20 classes are appropriate, though 5 to 10 classes work well for most introductory applications. Several guidelines exist for determining the number of classes:
    • For small data sets (\(n\) < 50), use 5-7 classes
    • For moderate data sets (50 ≤ \(n\) ≤ 100), use 7-10 classes
    • A common rule of thumb: \(k\approx\sqrt{n}\), where \(k\) is the number of classes and \(n\) is the sample size
• Step 3 — Calculating Class Width
  • Once the number of classes is determined, calculate the class width using the formula: Class Width = Range ÷ Number of Classes
  • It is standard practice to round up to a convenient whole number or decimal value, even if the result is not exact.
• Step 4 — Determine the Lower Limits of Each Class Interval
  • With the class width determined, we construct the class intervals beginning with the minimum value (or a convenient value slightly below it).
  • Lower Class Limit: The smallest value that can belong to a class
  • Beginning with the starting point, create the other lower limits of each class by adding the class width repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
• Step 5 — Determine the Upper Limits of Each Class Interval
  • Upper Class Limit: The largest value that can belong to a class
  • Beginning with the lower limit, add one less than the class width to obtain the upper limit. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
• Step 6 — Create the Frequency Distribution
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

 

Example #6.2.4 🤔

Let's use the data set from above of the temperatures (in degrees Fahrenheit) recorded in 20 cities on a particular day, rounded to the nearest degree:

58, 68, 65, 79, 51, 94, 64, 80, 69, 113, 63, 70, 102, 70, 92, 87, 77, 64, 64, 72.

If we were to arrange the data in ascending order, we have

Construct a grouped frequency distribution for the data set.

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✅ Solution:

• Step 1 — Find the Range
  • Range = Maximum value − Minimum value
  • Maximum temperature = 113°F
  • Minimum temperature = 51°F
  • Range = 113 − 51 = 62°F
• Step 2 — Selecting the Number of Classes
  • The number of classes should be chosen to provide a clear picture of the distribution without excessive detail or oversimplification. Generally, 5 to 20 classes are appropriate, though 5 to 10 classes work well for most introductory applications. Several guidelines exist for determining the number of classes:
    • For small data sets (\(n\) < 50), use 5-7 classes. For our temperature data set with n = 20, we might choose 6 or 7 classes. Let's choose 6. 
• Step 3 — Calculating Class Width
  • Class Width = Range ÷ Number of Classes 
  • Class Width = 62 ÷ 6 = 10.33...
  • Rounding up to a convenient value: Class Width = 11°F.
• Step 4 — Determine the Lower Limits of Each Class Interval
  • With the class width determined, we construct the class intervals beginning with the minimum value (or a convenient value slightly below it).
    • Lower Class Limit: The smallest value that can belong to a class. In this data set, either 50 or 51 (the actual lowest value in the data set) is a good choice. Here we will choose out starting point at 51.
    • Beginning with the observation of 51, we create the other lower limits of each class by adding the class width of 11 repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
      • First class: Lower limit = 51
      • Second class: Lower limit = 62
      • Third class: Lower limit = 73
      • Fourth class: Lower limit = 84
      • Fifth class: Lower limit = 95
      • Sixth class: Lower limit = 106
    • We continue creating classes until all data values (up to 113°F) are covered. Note that if there was a seventh class, the lower limit would have been 117. Since the maximum value of 113 is less than 117, there is no need for the seventh class.
• Step 5 — Determine the Upper Limits of Each Class Interval
  • Upper Class Limit: The largest value that can belong to a class.
  • Beginning with the lower limit of 51, add one less than the class width to obtain the upper limit. So, 51 + 11 − 1 = 61. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width of 11 repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
    • First class: Upper limit = 61
    • Second class: Upper limit = 72
    • Third class: Upper limit = 83
    • Fourth class: Upper limit = 94
    • Fifth class: Upper limit = 105
    • Sixth class: Upper limit = 116
• Step 6 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

Temperature	Tally	Frequency
51—61	\(|~|\)	2
62—72	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	10
73—83	\(|~|~|\)	3
84—94	\(|~|~|\)	3
95—105	\(|\)	1
106—116	\(|\)	1
Total		20 ✓

 

By applying the guidelines above, one can see that the frequency distributions will vary depending on the desired class width. A much simpler approach would to have the lower class limit given along with the desired class width. If that was to be the case, then there would be only one correct frequency distribution made under those conditions.

 

Constructing Grouped Frequency Distributions When a Class Width and Starting Point Are Given

• Step 1 — Identify the Given Information
  • Class width
  • Starting point
  • Data range
• Step 2 — Determine the Lower Limits of Each Class Interval
  • Lower Class Limit: The smallest value that can belong to a class
  • Beginning with the starting point, create the other lower limits of each class by adding the class width repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
• Step 3 — Determine the Upper Limits of Each Class Interval
  • Upper Class Limit: The largest value that can belong to a class
  • Beginning with the lower limit, add one less than the class width to obtain the upper limit. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
• Step 4 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column
  • Verify the total. The sum of all frequencies should equal total number of data values.

 

Example #6.2.5 🤔

Let's use the data set from above of the temperatures (in degrees Fahrenheit) recorded in 20 cities on a particular day, rounded to the nearest degree:

58, 68, 65, 79, 51, 94, 64, 80, 69, 113, 63, 70, 102, 70, 92, 87, 77, 64, 64, 72.

Construct a grouped frequency distribution with a class width of 9 and a starting point of the lowest observation in the data set.

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✅ Solution:

• Step 1 — Identify the Given Information
  • Class width: 8
  • Starting point: 51 (the minimum value in our data set)
  • Data range: From 51 to 113
• Step 2 — Determine the Lower Limits of Each Class Interval
  • Beginning with the lower observation of 51, we create the other lower limits of each class by adding the class width of 8 repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
    • First class: Lower limit = 51
    • Second class: Lower limit = 59
    • Third class: Lower limit = 67
    • Fourth class: Lower limit = 75
    • Fifth class: Lower limit = 83
    • Sixth class: Lower limit = 91
    • Seventh class: Lower limit = 99
    • Eighth class: Lower limit = 107
  • We continue creating classes until all data values (up to 113°F) are covered. Note that if there was a ninth class, the lower limit would have been 115. Since the maximum value of 113 is less than 115, there is no need for the ninth class.
• Step 3 — Determine the Upper Limits of Each Class Interval
  • Beginning with the lower limit of 51, add one less than the class width to obtain the upper limit. So, 51 + 8 − 1 = 58. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width of 8 repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
    • First class: Upper limit = 58
    • Second class: Upper limit = 66
    • Third class: Upper limit = 74
    • Fourth class: Upper limit = 82
    • Fifth class: Upper limit = 90
    • Sixth class: Upper limit = 98
    • Seventh class: Upper limit = 106
    • Eighth class: Upper limit = 114
• Step 4 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

Temperature	Tally	Frequency
51—58	\(|~|\)	2
59—66	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
67—74	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
75—82	\(|~|~|\)	3
83—90	\(|\)	1
91—98	\(|~|\)	2
99—106	\(|\)	1
107—114	\(|\)	1
Total		20 ✓

 

The grouped frequency distribution from above reveals several patterns:

• The highest concentration of temperatures occurs in the 59 — 66°F and 67 — 74°F ranges (5 cities each)
• Temperatures are relatively spread across the range
• There are some unusually high temperatures (102°F and 113°F) in the upper classes
• The distribution shows some clustering in the middle ranges with a few unusual outliers* at both ends

* Note: An outlier is a value that is much larger or much smaller than most other observations in the data set.

 

Example #6.2.6 🤔

A survey recorded the number of hours 30 students spent on homework per week:

12, 4, 18, 7, 15, 22, 9, 14, 20, 11, 16, 8, 19, 13, 10, 17, 21, 6, 23, 14, 12, 15, 18, 11, 16, 9, 20, 13, 17, 10

Construct a grouped frequency distribution with a class width of 5 and a starting point of the lowest observation in the data set.

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✅ Solution:

• Step 1 — Identify the Given Information
  • Class width: 5
  • Starting point: 4 (the minimum value in our data set)
  • Data range: From 4 to 23
• Step 2 — Determine the Lower Limits of Each Class Interval
  • Beginning with the lower observation of 4, we create the other lower limits of each class by adding the class width of 5 repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
    • First class: Lower limit = 4
    • Second class: Lower limit = 9
    • Third class: Lower limit = 14
    • Fourth class: Lower limit = 19
  • We continue creating classes until all data values (up to 23) are covered. Note that if there was a fifth class, the lower limit would have been 24. Since the maximum value of 25 is less than 24, there is no need for the fifth class.
• Step 3 — Determine the Upper Limits of Each Class Interval
  • Beginning with the lower limit of 4, add one less than the class width to obtain the upper limit. So, 4 + 5 − 1 = 8. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width of 5 repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
    • First class: Upper limit = 8
    • Second class: Upper limit = 13
    • Third class: Upper limit = 18
    • Fourth class: Upper limit = 23
• Step 4 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

No. of Hours	Tally	Frequency
4—8	\(|~|~|~|\)	4
9—13	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	10
14—18	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	10
19—23	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
Total		30 ✓

Example #6.2.7 🤔

The weights (in pounds) of 22 packages shipped by a company:

8, 15, 23, 12, 19, 27, 10, 18, 24, 14, 21, 9, 16, 25, 11, 20, 28, 13, 22, 17, 26, 15

Construct a grouped frequency distribution with a class width of 4 and a starting point of the lowest observation in the data set.

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✅ Solution:

• Step 1 — Identify the Given Information
  • Class width: 4
  • Starting point: 8 (the minimum value in our data set)
  • Data range: From 8 to 28
• Step 2 — Determine the Lower Limits of Each Class Interval
  • Beginning with the lower observation of 8, we create the other lower limits of each class by adding the class width of 4 repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
    • First class: Lower limit = 8
    • Second class: Lower limit =12
    • Third class: Lower limit = 16
    • Fourth class: Lower limit = 20
    • Fifth class: Lower limit = 24
    • Sixth class: Lower limit = 28
  • We continue creating classes until all data values (up to 28) are covered. Note that if there was a seventh class, the lower limit would have been 32. Since the maximum value of 28 is less than 32, there is no need for the seventh class.
• Step 3 — Determine the Upper Limits of Each Class Interval
  • Beginning with the lower limit of 51, add one less than the class width to obtain the upper limit. So, 8 + 4 − 1 = 11. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width of 4 repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
    • First class: Upper limit = 11
    • Second class: Upper limit = 15
    • Third class: Upper limit = 19
    • Fourth class: Upper limit = 23
    • Fifth class: Upper limit = 27
    • Sixth class: Upper limit = 31
• Step 4 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

No. of Pounds	Tally	Frequency
8—11	\(|~|~|~|\)	4
12—15	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
16—19	\(|~|~|~|\)	4
20—23	\(|~|~|~|\)	4
24—27	\(|~|~|~|\)	4
28—31	\(|\)	1
Total		22 ✓

Example #6.2.8 🤔

The following data shows the prices (in dollars) of 20 textbooks:

78, 92, 65, 88, 110, 73, 85, 82, 67, 105, 89, 76, 98, 71, 85, 45, 102, 53, 91, 80

Construct a grouped frequency distribution with a class width of 9 and a starting point of the lowest observation in the data set.

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✅ Solution:

• Step 1 — Identify the Given Information
  • Class width: 9
  • Starting point: 45 (the minimum value in our data set)
  • Data range: From 45 to 110
• Step 2 — Determine the Lower Limits of Each Class Interval
  • Beginning with the lower observation of 45, we create the other lower limits of each class by adding the class width of 9 repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
    • First class: Lower limit = 45
    • Second class: Lower limit = 54
    • Third class: Lower limit = 63
    • Fourth class: Lower limit = 72
    • Fifth class: Lower limit = 81
    • Sixth class: Lower limit = 90
    • Seventh class: Lower limit = 99
    • Eighth class: Lower limit = 108
  • We continue creating classes until all data values (up to 110) are covered. Note that if there was a ninth class, the lower limit would have been 117. Since the maximum value of 110 is less than 117, there is no need for the ninth class.
• Step 3 — Determine the Upper Limits of Each Class Interval
  • Beginning with the lower limit of 45, add one less than the class width to obtain the upper limit. So, 45 + 9 − 1 = 53. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width of 9 repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
    • First class: Upper limit = 53
    • Second class: Upper limit = 62
    • Third class: Upper limit = 71
    • Fourth class: Upper limit = 80
    • Fifth class: Upper limit = 89
    • Sixth class: Upper limit = 98
    • Seventh class: Upper limit = 107
    • Eighth class: Upper limit = 116
• Step 4 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

Prices	Tally	Frequency
45—53	\(|~|\)	2
54—62		0
63—71	\(|~|~|\)	3
72—80	\(|~|~|~|\)	4
81—89	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
90—98	\(|~|~|\)	3
99—107	\(|~|\)	2
108—116	\(|\)	1
Total		20 ✓

Example #6.2.9 🤔

The number of customers served per day at a restaurant over 31 days:

78, 105, 132, 89, 115, 145, 95, 122, 152, 85, 110, 138, 92, 118, 148, 111
82, 108, 135, 98, 125, 155, 88, 112, 142, 102, 128, 158, 75, 120, 140

Construct a grouped frequency distribution with a class width of 15 and a starting point of the lowest observation in the data set.

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✅ Solution:

• Step 1 — Identify the Given Information
  • Class width: 15
  • Starting point: 75 (the minimum value in our data set)
  • Data range: From 75 to 158
• Step 2 — Determine the Lower Limits of Each Class Interval
  • Beginning with the lower observation of 75, we create the other lower limits of each class by adding the class width of 15 repeatedly. The class width is defined as the difference of any two consecutive lower limits. So, simply add the class width to any lower limit of a class to get the lower limit for the very next class.
    • First class: Lower limit = 75
    • Second class: Lower limit = 90
    • Third class: Lower limit = 105
    • Fourth class: Lower limit = 120
    • Fifth class: Lower limit = 135
    • Sixth class: Lower limit = 150
  • We continue creating classes until all data values (up to 158) are covered. Note that if there was a seventh class, the lower limit would have been 165. Since the maximum value of 158 is less than 165, there is no need for the seventh class.
• Step 3 — Determine the Upper Limits of Each Class Interval
  • Beginning with the lower limit of 75, add one less than the class width to obtain the upper limit. So, 75 + 15 − 1 = 89. Continue to do this for the rest of the upper limits. You could also create the other upper limits of each class by adding the class width of 15 repeatedly after you get the upper class limit of the first class. The class width can also be defined as the difference of any two consecutive upper limits.
    • First class: Upper limit = 89
    • Second class: Upper limit = 104
    • Third class: Upper limit = 119
    • Fourth class: Upper limit = 134
    • Fifth class: Upper limit = 149
    • Sixth class: Upper limit = 164
• Step 4 — Create a three-column table (with tallies) (Exactly the same as ungrouped frequency distributions earlier)
  • Create the classes by writing the lower limit then a dash followed by the upper limit, for each respective class.
  • Place the classes in the first column.
  • Use tallies and record the frequency/counts in the last column.
  • Verify the total. The sum of all frequencies should equal total number of data values.

No. of Customers	Tally	Frequency
75—89	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
90—104	\(|~|~|~|\)	4
105—119	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|~|\)	7
120—134	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
135—149	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
150—164	\(|~|~|\)	3
Total		31 ✓

Section 6.2: Frequency Distributios [In-Class Exercises]

1. Twenty overall ratings from 1 through 10 of an exclusive Netflix show is given below:

   4, 2, 5, 9, 10, 2, 8, 1, 7, 9, 7, 4, 1, 9, 7, 6, 4, 1, 8, 7

   Construct an ungrouped frequency distribution for the data set.

2. The heights (in inches) of 28 adult males is recorded.

   64, 65, 66, 66, 67, 67, 68, 68, 68, 69, 69, 69, 70, 70,
   70, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 76

   Construct a grouped frequency distribution for the data set.

3. A random sample of annual salaries (in thousands of dollars) of 18 parents who just had their first child are given below:

   48, 50, 53, 55, 56, 60, 60, 60, 61,
   63, 68, 70, 72, 75, 78, 80, 85, 90

   Construct a grouped frequency distribution using a range of 7 and start the first class with the lowest value.

4. Twenty AA batteries were tested to determine how long they would last. The results, to the nearest minute, were recorded as follows:

   423, 369, 387, 411, 393, 399, 371, 377, 409, 392, 417, 431, 401, 363, 391, 405, 425, 400, 381, 399

   Construct a grouped frequency distribution using a range of 9 and start the first class with the lowest value.

Answers

1.  

Ratings	Tally	Frequency
1	\(|~|~|\)	3
2	\(|~|\)	2
3		0
4	\(|~|~|\)	3
5	\(|\)	1
6	\(|\)	1
7	\(|~|~|~|\)	4
8	\(|~|\)	2
9	\(|~|~|\)	3
10	\(|\)	1
Total		20 ✓

 

2. Answers may vary. Answer below is using 5 classes with a class width of 3.

Heights	Tally	Frequency
64—66	\(|~|~|~|\)	4
67—69	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|~|~|\)	8
70—72	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|~|~|~|\)	9
73—75	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
76—78	\(|\)	1
Total		28 ✓

 

3.  

Salaries	Tally	Frequency
48—54	\(|~|~|\)	3
55—61	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}~|\)	6
62—68	\(|~|\)	2
69—75	\(|~|~|\)	3
76—82	\(|~|\)	2
83—89	\(|\)	1
90—96	\(|\)	1
Total		18 ✓

 

4.  

Minutes	Tally	Frequency
363—371	\(|~|~|\)	3
372—380	\(|\)	1
381—389	\(|~|\)	2
390—398	\(|~|~|\)	3
399—407	\(\color{black}\bcancel{\color{black}\text{|~|~|~|}}\)	5
408—416	\(|~|\)	2
417—425	\(|~|~|\)	3
426—434	\(|\)	1
Total		20 ✓

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