Frequency distribution

Overview

A frequency distribution is a careful way of organizing data so that a learner can see how often each value, category, or class interval occurs. Raw marks such as $48, 47, 57, 56, 71$ are difficult to understand when they are scattered in a list. A frequency distribution turns that list into a table that can be counted, checked, summarized, and later drawn as a histogram, frequency polygon, or cumulative frequency curve.

The main idea is simple: every observation must be placed in one correct place, and the final total frequency must match the number of observations. Most errors in this topic come from rushing the grouping step, using overlapping intervals, or forgetting to check the total.

This chapter builds the bridge from raw data to statistical tables. Later work on Measures of central tendency and Histogram, frequency polygon, and cumulative frequency curve depends on this page because means, medians, modes, histograms, and ogives all need a reliable frequency table first.

Prerequisites

• Ordering whole numbers and decimals from smallest to largest.
• Counting accurately using tallies.
• Understanding intervals such as $20-29$ and $30-39$.
• Basic arithmetic with addition, subtraction, multiplication, and division.
• Approximations, rounding, significant figures, and decimal places for final answers that later use grouped data.
• Sets, subsets, operations with sets, and Venn diagrams of two sets for the idea that each item should belong to one suitable group when categories are separated.

Learning Scope

This chapter covers raw data, ungrouped frequency tables, grouped frequency distributions, class intervals, class limits, class boundaries, class width, class marks, tallying, frequency totals, cumulative frequency, and checking routines.

It does not fully teach graphical presentation, mean, median, or mode. Those topics are linked because they use frequency distributions, but their detailed methods belong to Histogram, frequency polygon, and cumulative frequency curve and Measures of central tendency.

Subtopics

Raw Data And Organized Data

Raw data are values collected before they have been arranged. For example, the marks:

$$ 12,\ 15,\ 11,\ 18,\ 15,\ 12,\ 20,\ 11 $$

are raw data. They are readable, but not yet easy to summarize.

Organized data show how often each value appears:

| Mark | Frequency | | ---: | ---: | | 11 | 2 | | 12 | 2 | | 15 | 2 | | 18 | 1 | | 20 | 1 | | Total | 8 |

The total frequency is $8$, matching the $8$ raw marks. This is the first check.

Frequency

Frequency means the number of times a value or class occurs. If the score $15$ appears $2$ times, its frequency is $2$.

Frequency is not the score itself. In the table above, $20$ is a mark and $1$ is its frequency.

Tallying

A tally is a counting mark used before writing the final frequency. Tallies help prevent lost counts when the raw list is long.

Example:

| Value | Tally | Frequency | | ---: | --- | ---: | | 1 | ||| | 3 | | 2 | |||| | 4 | | 3 | || | 2 |

A good tally routine is:

1. Read one raw value.
2. Put exactly one tally in the correct row.
3. Cross or mark the raw value mentally as counted.
4. Add tallies only after every value has been placed.

Ungrouped Frequency Distribution

An ungrouped frequency distribution lists individual values. It is useful when there are few different values.

For example, for the scores:

$$ 4,\ 5,\ 4,\ 6,\ 7,\ 5,\ 5,\ 6,\ 4,\ 8 $$

the ungrouped table is:

| Score | Frequency | | ---: | ---: | | 4 | 3 | | 5 | 3 | | 6 | 2 | | 7 | 1 | | 8 | 1 | | Total | 10 |

Use ungrouped tables when listing every distinct value is still clear.

Grouped Frequency Distribution

A grouped frequency distribution places values into intervals. It is useful when the data cover many possible values.

For example:

| Class interval | Frequency | | --- | ---: | | 30-39 | 3 | | 40-49 | 8 | | 50-59 | 12 | | 60-69 | 5 | | 70-79 | 2 |

Each class interval covers a range of values. If the data are whole-number marks, $40-49$ includes $40, 41, 42, \ldots, 49$.

Class Limits

Class limits are the visible endpoints of a class interval.

In the class $40-49$:

• Lower class limit: $40$
• Upper class limit: $49$

Class limits are usually what the question gives in a table.

Class Boundaries

Class boundaries remove gaps between classes when the data are treated as continuous.

For whole-number marks:

| Class interval | Class boundaries | | --- | --- | | 40-49 | $39.5-49.5$ | | 50-59 | $49.5-59.5$ |

The upper boundary of one class equals the lower boundary of the next class. This is important for histograms and median calculations in grouped data.

Class Width

Class width is the size of a class interval.

Using boundaries:

$$ \text{class width} = 49.5 - 39.5 = 10 $$

For equal-width classes such as $40-49, 50-59, 60-69$, the class width is $10$.

Class Mark Or Midpoint

The class mark is the midpoint of a class. It represents the whole interval when estimating grouped-data measures.

For $40-49$:

$$ \text{class mark} = \frac{40+49}{2} = 44.5 $$

Using boundaries gives the same result:

$$ \frac{39.5+49.5}{2}=44.5 $$

Choosing Class Intervals

When a question gives the number of classes, class size, and lower limit, use them exactly.

If the lower limit is $32$, class size is $8$, and the number of classes is $8$, the classes are:

| Class number | Class interval | | ---: | --- | | 1 | 32-39 | | 2 | 40-47 | | 3 | 48-55 | | 4 | 56-63 | | 5 | 64-71 | | 6 | 72-79 | | 7 | 80-87 | | 8 | 88-95 |

The next class starts after the previous class ends. There should be no overlap and no missing whole-number values.

Cumulative Frequency

Cumulative frequency is a running total of frequencies. It tells how many observations are up to the end of a class.

| Class interval | Frequency | Cumulative frequency | | --- | ---: | ---: | | 0-9 | 3 | 3 | | 10-19 | 5 | 8 | | 20-29 | 7 | 15 | | 30-39 | 2 | 17 |

The final cumulative frequency must equal the total frequency.

Checking A Frequency Distribution

Use this checking routine before using a distribution:

1. Check that every class has a clear lower and upper end.
2. Check that classes do not overlap.
3. Check that there are no unintended gaps.
4. Add all frequencies.
5. Compare the total frequency with the number of raw observations.
6. Check that the smallest and largest raw values fit inside the class range.

If any check fails, do not continue to mean, median, mode, or graphs until the table is corrected.

Key Terms

• Data: Values or information collected for study.
• Raw data: Data before organization.
• Frequency: The number of times a value or class occurs.
• Frequency distribution: A table showing values or classes and their frequencies.
• Tally: A counting mark used to build frequencies.
• Class interval: A group such as $40-49$.
• Lower class limit: The smallest visible value in a class.
• Upper class limit: The largest visible value in a class.
• Class boundary: A continuous endpoint such as $39.5$ or $49.5$.
• Class width: The size of a class interval.
• Class mark: The midpoint of a class interval.
• Cumulative frequency: A running total of frequencies.
• Total frequency: The sum of all frequencies.

Worked Examples

Example 1: Make An Ungrouped Frequency Table

The scores in a short quiz are:

$$ 2,\ 3,\ 4,\ 2,\ 5,\ 3,\ 3,\ 4,\ 2,\ 5,\ 1,\ 3 $$

Make a frequency table.

First list the values from smallest to largest:

$$ 1,\ 2,\ 3,\ 4,\ 5 $$

Now count each value:

| Score | Tally | Frequency | | ---: | --- | ---: | | 1 | | | 1 | | 2 | ||| | 3 | | 3 | |||| | 4 | | 4 | || | 2 | | 5 | || | 2 | | Total | | 12 |

Check:

$$ 1+3+4+2+2=12 $$

There were $12$ raw scores, so the table is complete.

Example 2: Construct Classes From Given Instructions

Prepare class intervals when the lower limit of the first class is $32$, the class size is $8$, and the number of classes is $8$.

Start at $32$. Since the class size is $8$, the first whole-number class contains:

$$ 32,\ 33,\ 34,\ 35,\ 36,\ 37,\ 38,\ 39 $$

So the first class is $32-39$. Continue by adding $8$ to the lower limit each time:

| Class | Class interval | | ---: | --- | | 1 | 32-39 | | 2 | 40-47 | | 3 | 48-55 | | 4 | 56-63 | | 5 | 64-71 | | 6 | 72-79 | | 7 | 80-87 | | 8 | 88-95 |

Check that the intervals do not overlap. For example, $39$ is in the first class and $40$ begins the second class.

Example 3: Group Raw Marks

The marks are:

$$ 48,\ 47,\ 57,\ 56,\ 71,\ 62,\ 46,\ 45,\ 50,\ 76 $$

Use the intervals $40-47$, $48-55$, $56-63$, $64-71$, and $72-79$.

Place each mark into exactly one interval:

| Class interval | Values in the class | Frequency | | --- | --- | ---: | | 40-47 | 47, 46, 45 | 3 | | 48-55 | 48, 50 | 2 | | 56-63 | 57, 56, 62 | 3 | | 64-71 | 71 | 1 | | 72-79 | 76 | 1 | | Total | | 10 |

Check:

$$ 3+2+3+1+1=10 $$

The total matches the $10$ raw marks.

Example 4: Find Class Boundaries And Midpoints

For the class intervals below, find the boundaries and midpoints.

| Class interval | Boundaries | Midpoint | | --- | --- | ---: | | 20-29 | $19.5-29.5$ | $24.5$ | | 30-39 | $29.5-39.5$ | $34.5$ | | 40-49 | $39.5-49.5$ | $44.5$ |

For $30-39$:

$$ \text{midpoint}=\frac{30+39}{2}=34.5 $$

The boundaries show that the classes touch without a gap.

Example 5: Add Cumulative Frequency

Complete the cumulative frequency column.

| Class interval | Frequency | Cumulative frequency | | --- | ---: | ---: | | 10-19 | 4 | 4 | | 20-29 | 6 | $4+6=10$ | | 30-39 | 9 | $10+9=19$ | | 40-49 | 3 | $19+3=22$ |

Final table:

| Class interval | Frequency | Cumulative frequency | | --- | ---: | ---: | | 10-19 | 4 | 4 | | 20-29 | 6 | 10 | | 30-39 | 9 | 19 | | 40-49 | 3 | 22 |

The total frequency is $22$, and the final cumulative frequency is also $22$.

Common Mistakes

• Mistake: Creating overlapping classes such as $10-20$ and $20-30$. Correction: for whole-number data use classes such as $10-19$ and $20-29$, or use boundaries when working continuously.
• Mistake: Leaving a gap between classes. Correction: check that the next class begins immediately after the previous one ends for discrete data.
• Mistake: Counting a raw value twice. Correction: each observation must be placed in exactly one row.
• Mistake: Forgetting the total frequency. Correction: always add a total row or check the total separately.
• Mistake: Confusing class limit and class boundary. Correction: $40$ and $49$ are limits in $40-49$; $39.5$ and $49.5$ are boundaries.
• Mistake: Using the upper limit as the midpoint. Correction: find the midpoint by averaging the lower and upper limits.
• Mistake: Continuing calculations after a frequency total does not match the raw data count. Correction: repair the table first.
• Mistake: Starting the first class from the smallest data value when the question gives a different lower limit. Correction: follow the given lower limit.

Practice Tasks

Foundation

1. Define frequency.
2. Explain the difference between raw data and organized data.
3. State the total number of observations in: $4, 7, 4, 6, 7, 8, 4$.
4. Make an ungrouped frequency table for: $1, 2, 2, 3, 1, 4, 2, 3$.
5. In the class $50-59$, identify the lower class limit and upper class limit.

Skill-Building

1. Find the class boundaries for $30-39$, $40-49$, and $50-59$.
2. Find the class marks for $10-19$, $20-29$, and $30-39$.
3. Complete a cumulative frequency column for frequencies $2, 5, 8, 4, 1$.
4. Construct $6$ classes of width $5$ starting from lower limit $20$.
5. Group the data $12, 15, 18, 21, 24, 27, 30, 33$ into classes $10-19$, $20-29$, and $30-39$.

Exam-Style

1. The marks of $20$ students are: $32, 35, 38, 41, 44, 46, 48, 51, 52, 53, 56, 58, 61, 64, 66, 69, 70, 73, 75, 78$. Prepare a frequency distribution using class size $5$ starting from $30$.
2. A question gives number of classes $8$, size of each class $8$, and lower limit of the first class $32$. Write all class intervals and explain how you checked them.
3. Given a frequency distribution, add class boundaries, midpoints, and cumulative frequencies.
4. A grouped table has final cumulative frequency $39$, but the question says there were $40$ observations. State what should be checked.

Challenge

1. Design two different grouped frequency distributions for the same $30$ marks: one with class width $5$ and one with class width $10$. Explain how the grouping changes the detail shown.
2. A learner uses classes $0-10$, $10-20$, $20-30$ for whole-number scores. Correct the classes and explain the risk in the original version.
3. Create a short raw data set of $15$ values where the modal class is clear after grouping, then build the grouped frequency distribution.

Generated Question Layer

• Concept questions: Ask learners to identify frequency, class interval, class limit, class boundary, class width, and class mark.
• Construction questions: Generate raw data and require a frequency table with a stated lower limit, class size, and number of classes.
• Checking questions: Give flawed tables with overlaps, gaps, missing totals, or misplaced values and ask learners to correct them.
• Cumulative-frequency questions: Ask learners to complete running totals and identify the total number of observations.
• Bridge questions: Use a completed frequency table as input for later mean, median, histogram, or ogive tasks.
• Reflection questions: Ask learners to explain why grouping loses some detail but makes large data sets easier to handle.

Learner Aid Opportunities

• diagram: annotated frequency table showing value, tally, frequency, class limit, boundary, and midpoint.
• chart: side-by-side raw data, tally table, frequency table, and cumulative frequency table.
• graph: preview showing how class intervals become bars in a histogram.
• animation: values falling into their correct class intervals one at a time.
• interactive: class-builder where learners choose lower limit, class width, and number of classes, then test raw values.
• video: worked construction of a grouped frequency distribution from raw marks.
• LLM tutor: prompts learners to check total frequency, overlaps, gaps, and whether the largest value fits.

Exam-Derived Signals

The official 2022 examination format crosswalk maps the format group Statistics/Circles to this topic, together with Measures of central tendency, Histogram, frequency polygon, and cumulative frequency curve, and Circle angle properties, theorems, tangents, chords, and radians. That format crosswalk is official, but the automatic 2021-2025 question mappings below are unreviewed and must be checked against original papers before being treated as final past-question links.

| Signal source | Status | Frequency distribution signal | | --- | --- | --- | | exam_format_topic_crosswalk_2022.jsonl | official format crosswalk | format-041-spec-11, Statistics/Circles, $1$ item, $7.14\%$ weight, mapped to this topic and related statistics/circle topics. | | topic_frequency_2021_2025.json | unreviewed extraction | $4$ primary mapped records for topic-frequency-distribution: $2$ in 2021 and $2$ in 2023. | | question_map_2021_2025.jsonl | unreviewed extraction | Signals often combine frequency tables with mean, median, or graph tasks, so several records are table-dependent or multi-topic. |

Recent unreviewed extracted records:

| Year | Question ID | Signal | Review note | | ---: | --- | --- | --- | | 2021 | csee_041_2021_p1_q11_a | Prepare a frequency distribution from raw marks using given number of classes, class size, and lower limit. | Unreviewed direct mapping. | | 2021 | csee_041_2021_p1_q11_b | Use the frequency distribution from part (a) to find an actual mean. | Needs manual review because it overlaps with central tendency. | | 2023 | csee_041_2023_p1_q11_a | Use a frequency distribution table to find the median. | Needs manual review; table-dependent and multi-topic. | | 2023 | csee_041_2023_p1_q11_b | Use a frequency distribution table to calculate the mean. | Needs manual review; table-dependent and multi-topic. |

Source And Review Notes

• Official topic registry status: The topic identity, Form IV link, competence, source topic ID, and hub are official through data/curriculum_map.json.
• Official syllabus reference: raw/syllabuses/csee/2023/csee_mathematics_syllabus_2023.pdf.
• Learner expansion status: The explanations, examples, and practice tasks are original learner-facing prose and remain unreviewed.
• Exam signal status: Extracted question mappings from data/question_map_2021_2025.jsonl and counts from data/topic_frequency_2021_2025.json are unreviewed. They are assessment signals only.
• Format crosswalk status: data/exam_format_topic_crosswalk_2022.jsonl is an official format crosswalk and maps Statistics/Circles to this topic.
• No media assets were added. Learner aids are planning markers only.