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Histograms with unequal class widths

Frequency density · Statistics · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Class widths
  2. Frequency density
  3. Draw the bars
  4. Check

The steps for Frequency density: 0-10: 5, 10-30: 12, 30-60: 9. Every solution names its steps, so you always know what comes next.

Worked examples

Frequency density: 0-10: 5, 10-30: 12, 30-60: 9 Easy

The question

Question: draw the histogram for 0-10: 5, 10-30: 12, 30-60: 9.

frequency density: 0-10: 5, 10-30: 12, 30-60: 9

The classes are not the same width.

So the bars cannot just be as high as the frequency.

1 Class widths

A table: each class and its frequency.

The width of a class is its end minus its start: 10, 20, 30.

2 Frequency density

Frequency density is the frequency divided by the width: 5 ÷ 10 = 0.5, 12 ÷ 20 = 0.6, 9 ÷ 30 = 0.3.

The rule for the height of each bar.

frequency density = frequencyclass width

Careful!

With different widths, a wide bar as high as its frequency would look too big.

Its area must be the frequency.

Careful: bar height = frequency density, not frequency

3 Draw the bars

Across: the classes.

Up: the frequency density.

Each bar goes across its whole class and up to its density: 0.5 for the first.

Now the area of each bar shows its frequency.

4 Check

Let's check: width times height gives back each frequency: 5, 12, 9.

So the bars are right!

10 × 0.5 = 5, 20 × 0.6 = 12, 30 × 0.3 = 9

Let's look back.

First: class widths.

Then: frequency density.

Then: draw the bars.

Last: check.

Answer
frequency density = frequencyclass width

▶ Watch it solved

Frequency density: 10-20: 7, 20-25: 9, 25-30: 12, 30-50: 8 Medium

The question

Question: draw the histogram for 10-20: 7, 20-25: 9, 25-30: 12, 30-50: 8.

frequency density: 10-20: 7, 20-25: 9, 25-30: 12, 30-50: 8

The classes are not the same width.

So the bars cannot just be as high as the frequency.

1 Class widths

A table: each class and its frequency.

The width of a class is its end minus its start: 10, 5, 5, 20.

2 Frequency density

Frequency density is the frequency divided by the width: 7 ÷ 10 = 0.7, 9 ÷ 5 = 1.8, 12 ÷ 5 = 2.4, 8 ÷ 20 = 0.4.

The rule for the height of each bar.

frequency density = frequencyclass width

Careful!

With different widths, a wide bar as high as its frequency would look too big.

Its area must be the frequency.

Careful: bar height = frequency density, not frequency

3 Draw the bars

Across: the classes.

Up: the frequency density.

Each bar goes across its whole class and up to its density: 0.7 for the first.

Now the area of each bar shows its frequency.

4 Check

Let's check: width times height gives back each frequency: 7, 9, 12, 8.

So the bars are right!

10 × 0.7 = 7, 5 × 1.8 = 9, 5 × 2.4 = 12, 20 × 0.4 = 8

Let's look back.

First: class widths.

Then: frequency density.

Then: draw the bars.

Last: check.

Answer
frequency density = frequencyclass width

▶ Watch it solved

Frequency density: 0-30: 12, 30-40: 14, 40-45: 9, 45-60: 6 Harder

The question

Question: draw the histogram for 0-30: 12, 30-40: 14, 40-45: 9, 45-60: 6.

frequency density: 0-30: 12, 30-40: 14, 40-45: 9, 45-60: 6

The classes are not the same width.

So the bars cannot just be as high as the frequency.

1 Class widths

A table: each class and its frequency.

The width of a class is its end minus its start: 30, 10, 5, 15.

2 Frequency density

Frequency density is the frequency divided by the width: 12 ÷ 30 = 0.4, 14 ÷ 10 = 1.4, 9 ÷ 5 = 1.8, 6 ÷ 15 = 0.4.

The rule for the height of each bar.

frequency density = frequencyclass width

Careful!

With different widths, a wide bar as high as its frequency would look too big.

Its area must be the frequency.

Careful: bar height = frequency density, not frequency

3 Draw the bars

Across: the classes.

Up: the frequency density.

Each bar goes across its whole class and up to its density: 0.4 for the first.

Now the area of each bar shows its frequency.

4 Check

Let's check: width times height gives back each frequency: 12, 14, 9, 6.

So the bars are right!

30 × 0.4 = 12, 10 × 1.4 = 14, 5 × 1.8 = 9, 15 × 0.4 = 6

Let's look back.

First: class widths.

Then: frequency density.

Then: draw the bars.

Last: check.

Answer
frequency density = frequencyclass width

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. frequency density: 0-10: 5, 10-30: 12, 30-60: 9
    Answer
    frequency density = frequencyclass width
  2. frequency density: 0-10: 8, 10-20: 6, 20-40: 10
    Answer
    frequency density = frequencyclass width
  3. frequency density: 0-5: 4, 5-10: 6, 10-20: 8
    Answer
    frequency density = frequencyclass width
  4. frequency density: 0-20: 10, 20-30: 15, 30-50: 6
    Answer
    frequency density = frequencyclass width
  5. histogram: class 20-40 has frequency density 1.5, find the frequency
    Answer
    Careful: not frequency = 1.5 ✗
  6. histogram: class 10-15 has frequency density 4, find the frequency
    Answer
    Careful: not frequency = 4 ✗
  7. histogram: class 0-25 has frequency density 0.8, find the frequency
    Answer
    Careful: not frequency = 0.8 ✗

Medium

  1. frequency density: 10-20: 7, 20-25: 9, 25-30: 12, 30-50: 8
    Answer
    frequency density = frequencyclass width
  2. frequency density: 0-10: 3, 10-15: 6, 15-20: 9, 20-40: 10
    Answer
    frequency density = frequencyclass width
  3. frequency density: 100-120: 6, 120-130: 9, 130-150: 4
    Answer
    frequency density = frequencyclass width
  4. histogram: class 50-80 has frequency density 0.4, find the frequency
    Answer
    Careful: not frequency = 0.4 ✗
  5. histogram: class 15-20 has frequency density 2.4, find the frequency
    Answer
    Careful: not frequency = 2.4 ✗
  6. histogram: class 30-60 has frequency density 1.2, find the frequency
    Answer
    Careful: not frequency = 1.2 ✗
  7. frequency density: 0-4: 6, 4-6: 8, 6-10: 10, 10-20: 5
    Answer
    frequency density = frequencyclass width

Harder

  1. frequency density: 0-30: 12, 30-40: 14, 40-45: 9, 45-60: 6
    Answer
    frequency density = frequencyclass width
  2. frequency density: 140-150: 5, 150-155: 8, 155-160: 11, 160-180: 6
    Answer
    frequency density = frequencyclass width
  3. frequency density: 0-2: 3, 2-5: 9, 5-10: 10, 10-20: 4, 20-30: 2
    Answer
    frequency density = frequencyclass width
  4. histogram: class 40-100 has frequency density 0.25, find the frequency
    Answer
    Careful: not frequency = 0.25 ✗
  5. frequency density: 20-30: 15, 30-35: 12, 35-40: 10, 40-60: 8
    Answer
    frequency density = frequencyclass width
  6. frequency density: 0-50: 20, 50-75: 25, 75-100: 15, 100-200: 10
    Answer
    frequency density = frequencyclass width