Construct a histogram from a frequency table

A table records library visits for 18 students in equal-width whole-number classes. What frequency height belongs to 10–14 visits?

Change the second class frequency. Watch that actual column change while the other three stay fixed. The zero-frequency third class still occupies space

When the second class frequency reaches zero, what happens to that interval?

Construct the histogram

A histogram displays numerical data in adjoining intervals. In this lesson all intervals have equal width

1. 1Horizontal axis Use a numerical scale with equal widths and put each column over its own interval
2. 2Vertical axis For these equal-width classes, use frequency or percentage frequency as labelled. Do not use cumulative totals as column heights
3. 3Empty classes A zero-frequency class has height zero but keeps its place and width on the axis
4. 4Check the result Label axes and units, start the frequency scale at zero, and compare every column with the table

The class frequencies are 4, 7, 5 and 2. How many observations must the histogram represent?

Add the four class frequencies to find the total

Why do positive-height neighbouring columns touch?

First six waits (minutes)

\(\displaystyle 0.4,1.8,2.0,2.3,3.1,4.6\)

Next six waits (minutes)

\(\displaystyle 5.9,6.0,6.2,8.4,10.1,11.8\)

These twelve waits use width-2 classes starting at 0, with frequencies 2, 3, 2, 2, 1, 2. Predict the first column height if the class width becomes 4 minutes

Change the class width from 2 to 4 minutes. The graph regroups the actual waits. Watch the column widths and heights, then check that the total remains twelve

Why are some columns taller after regrouping?

Class count

\(\displaystyle 4\)

Total count

\(\displaystyle 12\)

In the width-4 histogram, the 4 ≤ x < 8 class has frequency 4. What is its percentage-frequency height, to one decimal place?

Divide 4 by 12, multiply by 100 and round once

Two groups use different class widths and different sample sizes. Is comparing the tallest frequency columns alone a fair comparison?

Class frequency

\(\displaystyle 9\)

Total frequency

\(\displaystyle 40\)

For an equal-width percentage histogram, a class contains 9 of 40 observations. What height should its column have?

Divide the class frequency by the total frequency, then multiply by 100

The second column in this frequency histogram is empty. Which statement is correct?

Tallest column

\(\displaystyle 13\)

Tick step

\(\displaystyle 5\)

The tallest column has frequency 13. The frequency axis is marked every 5 units. What is the first marked axis value strictly above the tallest column?

Count in steps of 5 from zero until you pass 13

Tallest column

\(\displaystyle 15\)

Tick step

\(\displaystyle 5\)

The tallest column now has frequency 15, with marks every 5 units. What is the first marked axis value strictly above it?

Because the question says strictly above, move to the next mark after 15

A class includes values from 10 up to but not including 20. Which edges should its histogram column have?

An ungrouped histogram counts integer scores 2, 3 and 4. Where should the column for score 3 be centred?

Can you compare frequencies by column height if class widths are unequal?

Draw and check a complete histogram

On paper, draw a frequency histogram with intervals from 0 to less than 10, from 10 to less than 20, from 20 to less than 30, and from 30 to less than 40, with counts 4, 6, 0, 8. Label time in minutes and frequency

1. 1Construction checklist Use boundaries 0, 10, 20, 30, 40; heights 4, 6, 0, 8; no arbitrary gaps; retain the empty third class
2. 2Self or teacher review Compare your drawing with the reference example after finishing. The app checks selected heights and graph recognition; it does not automatically grade your hand-drawn graph
3. 3Choose and use technology Also create the complete histogram using a tool that can represent the supplied class frequencies. Explain your function choice and record the input, interval labels or boundaries, frequency settings, gap settings and axes. Check whether your chosen function expects raw observations or already grouped counts
4. 4Verify independently Match each height to its own class count and check the total by hand. Explain why entering 4, 6, 0, 8 as four raw times into a histogram function would represent a different dataset. Save your drawing and tool record before opening the next model; the complete drawing and external tool use are not automatically marked

Model response: histogram technology and checks

The retained reference example shows the complete correct histogram. Use it with the following input, scale and verification record; other tools are valid when they reproduce the stated class frequencies

1. 1One reproducible workflow Enter interval labels 0 ≤ x < 10, 10 ≤ x < 20, 20 ≤ x < 30, 30 ≤ x < 40 and counts 4, 6, 0, 8 in two columns. A column-chart function can represent these equal-width intervals when the columns are in numerical order and the gap is set to zero. Retain the zero-height third interval and label every interval. Use the title “Recorded times (18 observations)”, horizontal axis Time (minutes), and vertical axis Frequency. A vertical scale from 0 to 10 with equal steps of 2 is one suitable choice
2. 2Why this function and input The input is a frequency table, so each supplied count must become the height of its own width-10 interval. A histogram function that accepts class boundaries and frequencies is also suitable. A raw-data histogram of the four numbers 4, 6, 0, 8 would instead show only four observations, including an invented zero-minute observation. The given table represents 18 observations and does not reveal their exact individual times
3. 3Independent checks Heights are 4, 6, 0, 8 and sum to 18. Boundaries are 0, 10, 20, 30, 40 with equal numerical widths; the empty third interval retains its place. Check each class separately: cumulative heights 4, 10, 10, 18 would not represent this frequency table. Verify that no Total row has been plotted and that the frequency baseline is zero
4. 4Self-review checklist Check the complete diagram against the retained reference: title, variable and units, adjoining equal-width intervals, correct empty class, all four heights, readable axes, a recorded tool/function choice and rationale, correct input interpretation and an independent total. The drawing and tool operation remain self or teacher reviewed

Which check detects a cumulative-frequency graph accidentally drawn instead of a histogram?