Histogram, frequency polygon, and cumulative frequency curve

Overview

Statistical graphs turn a frequency distribution into a picture that can be read quickly. A histogram shows how frequencies fill class intervals. A frequency polygon joins points above class marks to show the shape of a distribution. A cumulative frequency curve, also called an ogive, shows running totals and helps estimate positions such as the median, quartiles, and percentiles.

This topic is visual, but it is not guesswork. A good graph begins with a correct table, sensible scale, labelled axes, and accurately plotted values. The graph should preserve the meaning of the data instead of decorating it.

The slow bridge is: raw data become a Frequency distribution, the distribution gives intervals and frequencies, the intervals become graph positions, and the graph supports interpretation or estimation.

Prerequisites

• Frequency distribution for class intervals, class boundaries, class marks, frequencies, and cumulative frequency.
• Measures of central tendency for median, mode, modal class, and graphical estimates.
• Plotting points on axes and reading scales.
• Using rulers and graph paper accurately.
• Approximations, rounding, significant figures, and decimal places for estimated readings from graphs.
• Understanding that grouped data use intervals rather than exact raw values.

Learning Scope

This chapter covers histograms, frequency polygons, cumulative frequency curves, ogives, choosing scales, plotting from class boundaries and class marks, using frequency density when class widths are unequal, estimating mode from a histogram, estimating median from an ogive, and checking graph accuracy.

It does not add finished images or graph assets. Learner aids are recorded only as planning markers. Exact numerical calculation of mean, median, and mode is treated in Measures of central tendency.

Subtopics

Statistical Graphs From Frequency Tables

A graph should be built from a table that has already been checked. Before drawing, confirm:

• The classes are in increasing order.
• The frequencies add to the total number of observations.
• Class boundaries or class marks are known.
• The graph type matches the question.

If the table is wrong, the graph will also be wrong.

Histogram

A histogram uses adjacent bars to represent grouped continuous data. The horizontal axis shows class intervals or class boundaries. The vertical axis shows frequency or frequency density.

For equal class widths, the height of each bar can be the frequency.

For unequal class widths, use:

$$ \text{frequency density}=\frac{\text{frequency}}{\text{class width}} $$

The bars in a histogram touch because the classes represent continuous intervals.

Histogram With Equal Class Widths

Suppose the table is:

| Marks | Frequency | | --- | ---: | | 40-49 | 2 | | 50-59 | 4 | | 60-69 | 7 | | 70-79 | 9 | | 80-89 | 5 | | 90-99 | 3 |

Use class boundaries:

| Marks | Boundaries | Frequency | | --- | --- | ---: | | 40-49 | $39.5-49.5$ | 2 | | 50-59 | $49.5-59.5$ | 4 | | 60-69 | $59.5-69.5$ | 7 | | 70-79 | $69.5-79.5$ | 9 | | 80-89 | $79.5-89.5$ | 5 | | 90-99 | $89.5-99.5$ | 3 |

Each bar has width $10$. The bar heights are $2, 4, 7, 9, 5, 3$.

Histogram With Unequal Class Widths

If class widths are not equal, bar height must be frequency density.

| Class | Frequency | Class width | Frequency density | | --- | ---: | ---: | ---: | | 0-10 | 6 | 10 | 0.6 | | 10-30 | 18 | 20 | 0.9 | | 30-40 | 8 | 10 | 0.8 |

The class $10-30$ has the largest frequency, but its width is also larger. The histogram height uses $0.9$, not $18$.

The area of each bar represents frequency:

$$ \text{frequency}=\text{class width}\times\text{frequency density} $$

Frequency Polygon

A frequency polygon is a line graph drawn from class marks and frequencies. It shows the shape of a distribution.

Steps:

1. Find the class mark for each class.
2. Plot each point $(\text{class mark}, \text{frequency})$.
3. Join the points with straight line segments.
4. Optionally add a point with frequency $0$ before the first class and after the last class to close the shape.

For the class $60-69$:

$$ \text{class mark}=\frac{60+69}{2}=64.5 $$

The point is $(64.5, 7)$ if the frequency is $7$.

Drawing A Frequency Polygon From A Histogram

When a histogram has equal class widths, a frequency polygon can be drawn by marking the midpoint at the top of each bar and joining those midpoints.

This method works because each midpoint has horizontal position equal to the class mark and vertical position equal to the class frequency.

For unequal class widths, be careful: if the histogram uses frequency density, the polygon drawn on it follows density heights, not raw frequencies.

Cumulative Frequency

Cumulative frequency is a running total.

| Class | Frequency | Cumulative frequency | | --- | ---: | ---: | | 40-49 | 2 | 2 | | 50-59 | 4 | 6 | | 60-69 | 7 | 13 | | 70-79 | 9 | 22 | | 80-89 | 5 | 27 | | 90-99 | 3 | 30 |

The final cumulative frequency is $30$, which is the total number of observations.

Cumulative Frequency Curve Or Ogive

A cumulative frequency curve, or ogive, plots cumulative frequency against upper class boundaries.

For a "less than" ogive, plot:

$$ (\text{upper boundary}, \text{cumulative frequency}) $$

It is also common to start with the lower boundary of the first class at cumulative frequency $0$.

Using the table above, the plotted points are:

| Boundary point | Cumulative frequency | | ---: | ---: | | 39.5 | 0 | | 49.5 | 2 | | 59.5 | 6 | | 69.5 | 13 | | 79.5 | 22 | | 89.5 | 27 | | 99.5 | 30 |

The points are joined with a smooth increasing curve.

Estimating The Median From An Ogive

To estimate the median from an ogive:

1. Find the total frequency $N$.
2. Calculate $\frac{N}{2}$.
3. Locate $\frac{N}{2}$ on the cumulative frequency axis.
4. Draw a horizontal line to the curve.
5. Draw a vertical line down to the data axis.
6. Read the median estimate.

For $N=30$:

$$ \frac{N}{2}=15 $$

The median is read at cumulative frequency $15$.

Estimating Quartiles From An Ogive

An ogive can also estimate quartiles:

$$ Q_1 \text{ is at } \frac{N}{4} $$

$$ Q_3 \text{ is at } \frac{3N}{4} $$

For $N=40$, use cumulative frequencies $10$ and $30$ to estimate $Q_1$ and $Q_3$.

Estimating Mode From A Histogram

The modal class is the tallest bar when class widths are equal. A graphical estimate of the mode can be found by drawing diagonal guide lines from the top corners of the modal bar to the top corners of the adjacent bars. The vertical line from their intersection to the horizontal axis gives an estimated mode.

This estimate depends on drawing accuracy. A formula-based grouped mode may be more reliable when exact computation is required.

Choosing Scales And Labelling Axes

A graph is only useful if the scale is readable.

Good scale habits:

• Use equal spacing for equal values.
• Start from a sensible value, often $0$ on the frequency axis.
• Label both axes with quantities.
• Mark class boundaries, class marks, or cumulative frequencies clearly.
• Use enough graph space so points are not crowded.

Bad scales can make a graph look correct while giving wrong readings.

Key Terms

• Histogram: A graph with adjacent bars representing grouped continuous data.
• Frequency polygon: A line graph joining points plotted at class marks and frequencies.
• Cumulative frequency curve: A curve showing running totals against class boundaries.
• Ogive: Another name for a cumulative frequency curve.
• Class boundary: A continuous endpoint of a class interval.
• Class mark: The midpoint of a class interval.
• Frequency density: Frequency divided by class width.
• Modal class: The class with the greatest frequency or greatest density, depending on the histogram scale.
• Median estimate: A value read from an ogive at $\frac{N}{2}$.
• Upper class boundary: The boundary used for plotting a less-than cumulative frequency curve.
• Scale: The value spacing used on an axis.

Worked Examples

Example 1: Prepare Histogram Values

Prepare the values needed to draw a histogram for:

| Marks | Frequency | | --- | ---: | | 40-49 | 2 | | 50-59 | 4 | | 60-69 | 7 | | 70-79 | 9 | | 80-89 | 5 | | 90-99 | 3 |

The classes have equal width, so use frequency as bar height.

| Marks | Boundaries | Bar height | | --- | --- | ---: | | 40-49 | $39.5-49.5$ | 2 | | 50-59 | $49.5-59.5$ | 4 | | 60-69 | $59.5-69.5$ | 7 | | 70-79 | $69.5-79.5$ | 9 | | 80-89 | $79.5-89.5$ | 5 | | 90-99 | $89.5-99.5$ | 3 |

Draw adjacent bars over each boundary interval.

Check: the tallest bar is $70-79$, so the modal class is $70-79$.

Example 2: Frequency Density For Unequal Classes

Prepare histogram heights.

| Time in minutes | Frequency | | --- | ---: | | 0-10 | 5 | | 10-20 | 8 | | 20-40 | 14 | | 40-70 | 9 |

The class widths are not all equal, so use frequency density.

| Time | Frequency | Width | Frequency density | | --- | ---: | ---: | ---: | | 0-10 | 5 | 10 | 0.5 | | 10-20 | 8 | 10 | 0.8 | | 20-40 | 14 | 20 | 0.7 | | 40-70 | 9 | 30 | 0.3 |

The histogram heights are $0.5, 0.8, 0.7, 0.3$.

Check one bar:

$$ 20 \times 0.7 = 14 $$

The area of the $20-40$ bar gives frequency $14$.

Example 3: Frequency Polygon Points

Use the table:

| Marks | Frequency | | --- | ---: | | 40-49 | 2 | | 50-59 | 4 | | 60-69 | 7 | | 70-79 | 9 | | 80-89 | 5 | | 90-99 | 3 |

Find class marks:

| Marks | Class mark | Frequency | Polygon point | | --- | ---: | ---: | --- | | 40-49 | 44.5 | 2 | $(44.5,2)$ | | 50-59 | 54.5 | 4 | $(54.5,4)$ | | 60-69 | 64.5 | 7 | $(64.5,7)$ | | 70-79 | 74.5 | 9 | $(74.5,9)$ | | 80-89 | 84.5 | 5 | $(84.5,5)$ | | 90-99 | 94.5 | 3 | $(94.5,3)$ |

Plot these points and join them with straight line segments.

Example 4: Ogive Plotting Table

Prepare points for a cumulative frequency curve.

| Marks | Frequency | | --- | ---: | | 40-49 | 2 | | 50-59 | 4 | | 60-69 | 7 | | 70-79 | 9 | | 80-89 | 5 | | 90-99 | 3 |

Add cumulative frequency:

| Marks | Upper boundary | Frequency | Cumulative frequency | Ogive point | | --- | ---: | ---: | ---: | --- | | start | 39.5 | | 0 | $(39.5,0)$ | | 40-49 | 49.5 | 2 | 2 | $(49.5,2)$ | | 50-59 | 59.5 | 4 | 6 | $(59.5,6)$ | | 60-69 | 69.5 | 7 | 13 | $(69.5,13)$ | | 70-79 | 79.5 | 9 | 22 | $(79.5,22)$ | | 80-89 | 89.5 | 5 | 27 | $(89.5,27)$ | | 90-99 | 99.5 | 3 | 30 | $(99.5,30)$ |

Plot these points and draw a smooth increasing curve.

Check: the last point has cumulative frequency $30$, the total frequency.

Example 5: Estimate Median Position From An Ogive

A cumulative frequency curve has total frequency $66$. Which cumulative-frequency value should be used to estimate the median?

Use:

$$ \frac{N}{2}=\frac{66}{2}=33 $$

On the graph, locate $33$ on the cumulative frequency axis, move horizontally to the curve, then move vertically down to the score axis.

The score read from the axis is the estimated median.

Example 6: Choose The Correct Graph

A table gives class intervals and frequencies for exam marks. The question asks:

1. "Draw a histogram."
2. "Draw a cumulative frequency curve."
3. "Use the graph to estimate the median."

For part 1, use class boundaries on the horizontal axis and frequency or density on the vertical axis.

For part 2, use upper class boundaries and cumulative frequencies.

For part 3, use the ogive, not the histogram, because the median is a position in the cumulative total.

Common Mistakes

• Mistake: Leaving gaps between histogram bars. Correction: histogram bars touch because class intervals are continuous.
• Mistake: Using class limits instead of boundaries on the horizontal axis. Correction: convert $40-49$ to $39.5-49.5$ when needed.
• Mistake: Using frequency instead of frequency density for unequal class widths. Correction: calculate frequency density first.
• Mistake: Plotting a frequency polygon at class limits. Correction: plot at class marks.
• Mistake: Plotting an ogive against class marks. Correction: a less-than ogive uses upper class boundaries.
• Mistake: Forgetting the starting point of an ogive. Correction: begin at the lower boundary of the first class with cumulative frequency $0$.
• Mistake: Drawing a cumulative frequency curve that goes down. Correction: cumulative frequency must never decrease.
• Mistake: Reading the median from the wrong axis. Correction: start at $\frac{N}{2}$ on the cumulative frequency axis, then read the data value from the horizontal axis.
• Mistake: Using an uneven scale. Correction: equal distances on an axis must represent equal numerical increases.
• Mistake: Treating a graphical estimate as exact. Correction: say "estimate" when reading from a graph.

Practice Tasks

Foundation

1. Define histogram, frequency polygon, and cumulative frequency curve.
2. Explain why histogram bars touch.
3. Find the class mark for $30-39$.
4. Find the class boundaries for $50-59$.
5. Complete cumulative frequencies for frequencies $3, 5, 7, 4, 1$.

Skill-Building

1. Prepare a histogram table with boundaries for classes $10-19, 20-29, 30-39$ and frequencies $4, 8, 5$.
2. Prepare frequency polygon points for classes $0-9, 10-19, 20-29, 30-39$ with frequencies $2, 6, 9, 3$.
3. Prepare ogive points for classes $40-49, 50-59, 60-69, 70-79$ with frequencies $5, 7, 10, 8$.
4. For classes $0-5, 5-15, 15-30$ with frequencies $4, 10, 9$, calculate frequency densities.
5. State the cumulative frequency value used to estimate the median when $N=48$.

Exam-Style

1. A frequency table has classes $40-49, 50-59, 60-69, 70-79, 80-89, 90-99$ and frequencies $2, 4, 7, 9, 5, 3$. Draw a histogram and use it to estimate the modal class.
2. Use the same table to prepare a frequency polygon.
3. Use the same table to draw a cumulative frequency curve and state the value of $N/2$ used for the median estimate.
4. A table gives cumulative frequencies $10, 22, 43, 49, 58, 62, 66$ for scores $65-69, 70-74, 75-79, 80-84, 85-89, 90-94, 95-99$. Prepare the ogive plotting points.
5. Explain why the first cumulative frequency in the previous task is not the first frequency of the distribution.

Challenge

1. Create a grouped table with unequal class widths and explain why frequency density is needed.
2. Draw a frequency polygon from a histogram and compare it with a polygon drawn directly from class marks and frequencies.
3. A learner draws an ogive that decreases between two points. Diagnose the error and correct the cumulative frequency table.
4. Explain how a histogram can help estimate mode but not median as directly as an ogive.

Generated Question Layer

• Graph-preparation questions: Ask learners to convert classes into boundaries, class marks, cumulative frequencies, and densities.
• Drawing questions: Generate tables suitable for histograms, frequency polygons, and ogives.
• Interpretation questions: Ask learners to identify modal class, estimate median position, or compare shapes.
• Error-analysis questions: Provide graphs or plotting tables with wrong axes, missing boundaries, gaps between bars, or decreasing cumulative frequency.
• Scale questions: Ask learners to choose a suitable scale and justify it.
• Bridge questions: Connect histogram mode estimates to grouped-mode calculation and ogive median estimates to grouped-median calculation.

Learner Aid Opportunities

• diagram: labelled axes for histogram, frequency polygon, and ogive on separate panels.
• chart: comparison table showing graph type, horizontal axis, vertical axis, plotted values, and common use.
• graph: sample plotting table rendered as points before the final graph is drawn.
• animation: cumulative frequency points appearing one by one and forming an ogive.
• interactive: learners choose graph type and receive prompts for boundaries, class marks, or cumulative frequencies.
• video: full graph-paper walkthrough for histogram, frequency polygon, and ogive from the same table.
• LLM tutor: checks whether the learner is using boundaries, class marks, or upper boundaries for the selected graph.

Exam-Derived Signals

The official 2022 examination format crosswalk maps the format group Statistics/Circles to this topic, together with Frequency distribution, Measures of central tendency, and Circle angle properties, theorems, tangents, chords, and radians. That format crosswalk is official. The extracted question records below are unreviewed and table-dependent, so they must be verified against original papers before becoming final past-question links.

| Signal source | Status | Graphing 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 | $2$ primary mapped records for topic-histogram-frequency-polygon-and-cumulative-frequency-curve: $1$ in 2023 and $1$ in 2024. | | question_map_2021_2025.jsonl | unreviewed extraction | Signals involve drawing a histogram to estimate mode and drawing a cumulative frequency curve from a grouped table. |

Recent unreviewed extracted records:

| Year | Question ID | Signal | Review note | | ---: | --- | --- | --- | | 2023 | csee_041_2023_p1_q11_c | Draw a histogram from a frequency distribution table and use it to estimate the mode. | Needs manual review; table-dependent and multi-topic. | | 2024 | csee_041_2024_p1_q11_a_ii | Draw the cumulative frequency curve, or ogive, of grouped scores. | Needs manual review; table-dependent. |

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. Both primary graphing records are table-dependent.
• 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.