The Central Idea
A frequency tells us how many observations fall into a particular category, value, or interval.
A frequency table organizes these counts so that a dataset can be read as a distribution rather than as an unstructured list.
The essential operation is simple:
\[ \boxed{\text{frequency}=\text{number of observations in a specified group}}. \]From Raw Data to Organized Data
Suppose the numbers of support requests received during twelve consecutive hours are
\[ 2,\ 4,\ 3,\ 2,\ 5,\ 4,\ 2,\ 3,\ 4,\ 4,\ 1,\ 2. \]The raw list contains all the information, but repeated values are not immediately summarized.
A frequency table makes the repetition visible.
Basic Frequency Table
| Requests | Frequency |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 2 |
| 4 | 4 |
| 5 | 1 |
The frequencies sum to the total number of observations:
\[ 1+4+2+4+1=12. \]This total provides an immediate check on the table.
Frequency Notation
Frequency is often represented by
\[ f. \]If the possible values are \(x_1,x_2,\ldots,x_k\), their frequencies may be written
\[ f_1,f_2,\ldots,f_k. \]For a dataset of size \(n\),
\[ \boxed{\sum_{i=1}^{k}f_i=n}. \]Frequency Tables for Categorical Data
Frequency tables are especially natural for categorical variables.
Suppose \(20\) customers select a preferred delivery option. The responses are summarized as follows:
| Delivery Option | Frequency |
|---|---|
| Standard | 9 |
| Expedited | 7 |
| Pickup | 4 |
The categories do not need to be numerical. The table simply counts how many cases belong to each category.
Frequency Tables for Discrete Quantitative Data
Discrete quantitative variables can often be summarized by listing each observed numerical value separately.
Suppose the number of defects found in \(15\) inspected products is
\[ 0,\ 1,\ 0,\ 2,\ 1,\ 0,\ 0,\ 3,\ 1,\ 2,\ 0,\ 1,\ 2,\ 0,\ 1. \]| Defects | Frequency |
|---|---|
| 0 | 6 |
| 1 | 5 |
| 2 | 3 |
| 3 | 1 |
Relative Frequency
A relative frequency expresses a category's frequency as a fraction of the entire dataset.
If a category has frequency \(f\) in a sample of size \(n\), then
\[ \boxed{\text{relative frequency}=\frac{f}{n}}. \]Relative frequency answers the question: What share of the observations belong to this group?
Example: Relative Frequencies
Return to the delivery-option data:
| Delivery Option | Frequency | Relative Frequency |
|---|---|---|
| Standard | 9 | \(\frac{9}{20}=0.45\) |
| Expedited | 7 | \(\frac{7}{20}=0.35\) |
| Pickup | 4 | \(\frac{4}{20}=0.20\) |
The relative frequencies sum to
\[ 0.45+0.35+0.20=1. \]Percentage Frequency
Relative frequency can be expressed as a percentage:
\[ \boxed{\text{percentage}=100\left(\frac{f}{n}\right)\%}. \]For the standard-delivery category,
\[ 100\left(\frac{9}{20}\right)\%=45\%. \]Relative frequencies sum to \(1\), while percentage frequencies sum to \(100\%\), apart from small rounding differences.
Why Relative Frequency Is Useful
Absolute counts can be difficult to compare when groups have different total sizes.
Suppose Store A records \(40\) returns among \(500\) sales, while Store B records \(30\) returns among \(250\) sales.
Store A has more returns in absolute terms:
\[ 40>30. \]But the relative return rates are
\[ \frac{40}{500}=0.08 \]and
\[ \frac{30}{250}=0.12. \]Store B has the larger return proportion.
Cumulative Frequency
For ordered data, a cumulative frequency gives the number of observations at or below a specified value or category.
Using the support-request table:
| Requests | Frequency | Cumulative Frequency |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 5 |
| 3 | 2 | 7 |
| 4 | 4 | 11 |
| 5 | 1 | 12 |
Building Cumulative Frequencies
Cumulative frequencies are obtained by successive addition.
\[ 1 \] \[ 1+4=5 \] \[ 1+4+2=7 \] \[ 1+4+2+4=11 \] \[ 1+4+2+4+1=12. \]The final cumulative frequency must equal the total sample size.
Cumulative Relative Frequency
We can also accumulate relative frequencies.
| Requests | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|
| 1 | 0.083 | 0.083 |
| 2 | 0.333 | 0.416 |
| 3 | 0.167 | 0.583 |
| 4 | 0.333 | 0.916 |
| 5 | 0.083 | 0.999 |
The final value is approximately \(1\) rather than exactly \(1\) because the displayed relative frequencies were rounded.
When Cumulative Frequency Makes Sense
Cumulative frequency requires an ordering.
It is meaningful for:
- numerical values;
- ordered categories;
- class intervals.
It is generally not meaningful for nominal categories such as eye color or country because those categories have no inherent order.
Ungrouped Frequency Tables
An ungrouped frequency table lists individual values separately.
This works well when:
- the number of distinct values is small;
- the values are discrete;
- preserving the exact observed values is useful.
If there are many distinct values, an ungrouped table may become too long to reveal the overall pattern.
Grouped Frequency Tables
A grouped frequency table combines numerical observations into intervals called classes.
Suppose \(30\) delivery times range from about \(10\) to \(59\) minutes. Instead of listing every distinct time, we might use:
| Delivery Time (minutes) | Frequency |
|---|---|
| 10–19 | 4 |
| 20–29 | 8 |
| 30–39 | 10 |
| 40–49 | 6 |
| 50–59 | 2 |
Class Intervals
A class interval is the range of values assigned to one group in a grouped frequency table.
For example,
\[ 20\text{--}29 \]is one class interval.
Every observation should belong to exactly one class.
Class Limits
In a class written
\[ 20\text{--}29, \]\(20\) is the lower class limit and \(29\) is the upper class limit.
These are the smallest and largest displayed values assigned to the class under that convention.
Class Width
The class width is commonly measured by the difference between consecutive lower class limits.
For the classes
\[ 10\text{--}19,\quad 20\text{--}29,\quad 30\text{--}39, \]the class width is
\[ \boxed{20-10=10}. \]Class Boundaries for Continuous Data
Continuous measurements require special care because adjacent intervals must cover all possible values without gaps or overlap.
A useful mathematical convention is to use half-open intervals such as
\[ 20\le x<30, \] \[ 30\le x<40. \]Then a value of exactly \(30\) belongs to the second class and no value is counted twice.
Class Midpoints
The class midpoint is halfway between the class endpoints.
For a class from \(20\) to \(30\),
\[ \text{midpoint} = \frac{20+30}{2} = 25. \]Midpoints are sometimes used as representative values for grouped data.
Because the original observations are no longer individually visible, calculations based on class midpoints are generally approximations.
Choosing the Number of Classes
There is no single class count that is best for every dataset.
Too few classes can hide important structure.
Too many classes can make the table nearly as complicated as the raw data.
A practical grouped table usually uses enough classes to reveal shape while keeping the summary readable.
A Simple Class-Width Estimate
If the approximate range is known and \(k\) classes are desired, a starting estimate for class width is
\[ \boxed{ \text{class width} \approx \frac{\text{maximum}-\text{minimum}}{k} }. \]The result is often rounded upward to a convenient value so that the classes cover the entire dataset.
Example: Choosing a Class Width
Suppose values range from \(12\) to \(68\), and we want approximately \(6\) classes.
\[ \frac{68-12}{6} = \frac{56}{6} \approx 9.33. \]A convenient class width is
\[ 10. \]Possible classes could begin
\[ 10\text{--}19,\quad 20\text{--}29,\quad \ldots,\quad 60\text{--}69. \]Grouped Tables Trade Detail for Structure
Once data are grouped, we know how many observations fall in each interval but may no longer know their exact original values.
For example, a frequency of \(8\) in the class
\[ 20\le x<30 \]tells us that eight observations lie in that interval.
It does not tell us whether those values were concentrated near \(20\), near \(30\), or spread throughout the class.
Complete Grouped Frequency Table
A grouped table can include several columns at once.
| Time (min) | Frequency | Relative Frequency | Cumulative Frequency |
|---|---|---|---|
| 10–19 | 4 | 0.133 | 4 |
| 20–29 | 8 | 0.267 | 12 |
| 30–39 | 10 | 0.333 | 22 |
| 40–49 | 6 | 0.200 | 28 |
| 50–59 | 2 | 0.067 | 30 |
Reading a Frequency Table
A frequency table supports direct questions such as:
- Which value or class occurs most often?
- How many observations fall in a specified group?
- What proportion of observations belong to that group?
- How many observations lie at or below a given threshold?
- Where is the distribution concentrated?
These questions turn raw data into interpretable structure.
The Modal Category or Value
The value or category with the greatest frequency is called the mode.
In the delivery-option example,
| Option | Frequency |
|---|---|
| Standard | 9 |
| Expedited | 7 |
| Pickup | 4 |
Standard delivery is the modal category because \(9\) is the largest frequency.
Frequencies and Probabilities
Relative frequencies often provide empirical estimates of probabilities.
If an event occurs \(f\) times in \(n\) observations, its empirical frequency is
\[ \frac{f}{n}. \]In repeated sampling or repeated trials, these observed proportions can be compared with theoretical probability models.
Frequency Tables and Missing Data
Missing values should be handled explicitly rather than silently ignored.
A table may include a category such as:
- missing;
- not reported;
- unknown;
- not applicable.
Whether missing observations belong in the denominator of a relative frequency depends on the question being answered.
Example: Relative Frequency with Missing Responses
Suppose \(100\) people are surveyed, but \(8\) do not answer a particular question.
If \(46\) of the \(92\) valid responses are "yes," then the percentage among respondents is
\[ \frac{46}{92}=0.50=50\%. \]But the percentage among all sampled individuals is
\[ \frac{46}{100}=0.46=46\%. \]The denominator must match the interpretation.
Common Error: Frequencies Do Not Sum to the Sample Size
If a table describes \(n=50\) observations, its ordinary category frequencies should sum to \(50\) unless missing or excluded observations are explicitly handled separately.
Always check
\[ \sum f_i=n. \]Common Error: Overlapping Classes
Consider the classes
\[ 10\text{--}20 \]and
\[ 20\text{--}30. \]If both endpoints are included, the value \(20\) belongs to both classes.
Classes must be defined so that every observation belongs to exactly one class.
Common Error: Leaving Gaps Between Classes
Suppose one class ends below the next class:
\[ 10\le x<20 \]followed by
\[ 21\le x<30. \]Values from \(20\) up to but not including \(21\) have nowhere to go.
Class intervals should cover the intended data range continuously.
Common Error: Using Cumulative Frequency for Nominal Data
If categories are
\[ \text{red},\quad \text{blue},\quad \text{green}, \]there is no natural meaning to "at or below blue."
Cumulative frequency is appropriate only when the categories or values have a meaningful order.
Common Error: Comparing Raw Frequencies Across Unequal Groups
A larger group will often produce a larger count simply because it contains more cases.
When comparing groups of different sizes, relative frequencies or percentages are usually more informative than raw frequencies alone.
Common Error: Too Many or Too Few Classes
A grouped table with only two very wide classes may conceal the distribution's structure.
A table with dozens of narrow classes may become unnecessarily fragmented.
Class construction should balance detail with readability.
A Reliable Procedure for an Ungrouped Frequency Table
- Identify the variable being summarized.
- List each distinct value or category once.
- Count how many observations belong to each value or category.
- Record those counts as frequencies.
- Verify that the frequencies sum to the number of observations.
- Compute relative frequencies or percentages when useful.
- For ordered data, compute cumulative frequencies when useful.
A Reliable Procedure for a Grouped Frequency Table
- Identify the minimum and maximum observed values.
- Choose a reasonable number of classes.
- Select a convenient class width that covers the full data range.
- Define nonoverlapping class intervals.
- Count observations in each class.
- Check that all observations are included exactly once.
- Compute relative and cumulative frequencies if needed.
- Review whether the chosen grouping reveals the distribution clearly.
Example: Full Categorical Workflow
Suppose \(40\) customers choose among four payment methods:
| Payment Method | Frequency | Relative Frequency | Percentage |
|---|---|---|---|
| Cash | 6 | 0.15 | 15% |
| Credit | 18 | 0.45 | 45% |
| Debit | 12 | 0.30 | 30% |
| Digital Wallet | 4 | 0.10 | 10% |
Checks:
\[ 6+18+12+4=40 \]and
\[ 0.15+0.45+0.30+0.10=1. \]Example: Full Grouped Workflow
Suppose \(25\) observations range from \(4\) to \(27\). Using class width \(5\), one possible grouping is:
| Class | Frequency | Cumulative Frequency |
|---|---|---|
| 0–4 | 2 | 2 |
| 5–9 | 5 | 7 |
| 10–14 | 8 | 15 |
| 15–19 | 6 | 21 |
| 20–24 | 3 | 24 |
| 25–29 | 1 | 25 |
The final cumulative frequency equals the sample size:
\[ \boxed{25}. \]Practice 1: Build a Frequency Table
Construct an ungrouped frequency table for
\[ 1,\ 2,\ 2,\ 3,\ 1,\ 4,\ 2,\ 3,\ 2,\ 1. \]Practice 2: Check the Total
A frequency table has frequencies
\[ 5,\ 8,\ 4,\ 3. \]What is the total number of observations?
Practice 3: Relative Frequency
A category has frequency \(18\) in a sample of \(60\).
Find its relative frequency and percentage.
Practice 4: Cumulative Frequency
Complete the cumulative frequencies for:
| Value | Frequency |
|---|---|
| 1 | 3 |
| 2 | 5 |
| 3 | 4 |
| 4 | 2 |
Practice 5: Class Width
Find the class width for
\[ 10\text{--}19,\quad 20\text{--}29,\quad 30\text{--}39. \]Practice 6: Class Midpoint
Find the midpoint of the class
\[ 40\le x<50. \]Practice 7: Identify the Problem
Explain what is wrong with the classes
\[ 0\le x\le10, \] \[ 10\le x\le20. \]Practice 8: Compare Groups
Group A has \(12\) successes among \(40\) cases. Group B has \(18\) successes among \(90\) cases.
Which group has the higher success proportion?
Practice 9: Missing Responses
A survey has \(80\) participants. Ten do not answer a question, and \(42\) of the remaining participants answer "yes."
Find the percentage answering "yes" among valid responses.
Practice 10: Grouped Table Interpretation
| Score | Frequency |
|---|---|
| 0–9 | 2 |
| 10–19 | 6 |
| 20–29 | 9 |
| 30–39 | 3 |
How many observations are below \(30\) according to this grouping?
Practice Answers
Practice 1
| Value | Frequency |
|---|---|
| 1 | 3 |
| 2 | 4 |
| 3 | 2 |
| 4 | 1 |
Practice 2
\[ 5+8+4+3=\boxed{20}. \]Practice 3
\[ \frac{18}{60}=0.30. \]The relative frequency is \(\boxed{0.30}\), or \(\boxed{30\%}\).
Practice 4
The cumulative frequencies are
\[ \boxed{3,\ 8,\ 12,\ 14}. \]Practice 5
\[ 20-10=\boxed{10}. \]Practice 6
\[ \frac{40+50}{2} = \boxed{45}. \]Practice 7
The classes overlap at \(x=10\), so a value of \(10\) could be counted twice. One endpoint convention must be adjusted.
Practice 8
Group A:
\[ \frac{12}{40}=0.30. \]Group B:
\[ \frac{18}{90}=0.20. \]Group A has the higher success proportion.
Practice 9
There are
\[ 80-10=70 \]valid responses.
\[ \frac{42}{70}=0.60=\boxed{60\%}. \]Practice 10
The classes below \(30\) have frequencies
\[ 2,\quad6,\quad9. \]Therefore,
\[ 2+6+9=\boxed{17}. \]What to Remember
- A frequency is the number of observations in a value, category, or class.
- Frequencies in a complete table should sum to the sample size.
- Relative frequency is \(f/n\).
- Percentage frequency is \(100(f/n)\%\).
- Cumulative frequency is meaningful only for ordered data.
- Ungrouped tables preserve individual observed values.
- Grouped tables combine numerical values into class intervals.
- Classes should not overlap and should not leave unintended gaps.
- Relative frequencies are often more useful than raw counts when comparing groups of different sizes.
- Grouping makes patterns easier to see but sacrifices some exact detail.
Where We Go Next
Frequency tables organize observations numerically, but visual displays can reveal patterns even more quickly.
The next lesson, Categorical and Quantitative Displays, develops graphical methods for representing distributions, including displays appropriate for categorical and numerical variables.