Learning Objectives
  • Interpret grouped continuous data and class intervals.
  • Calculate class width and frequency density.
  • Construct histograms with unequal class widths.
  • Find frequencies from a histogram using bar areas.
  • Compare distributions and avoid confusing histograms with bar charts.
Key Terms
Histogram
A graph for grouped continuous data in which bar area represents frequency.
Class interval
A range used to group continuous values.
Class boundary
The exact endpoint separating adjacent continuous classes.
Class width
Upper class boundary minus lower class boundary.
Frequency density
Frequency divided by class width.
Continuous data
Data that can take any value within an interval.
Bar area
In a histogram, class width multiplied by frequency density, proportional to frequency.
8th Edition Chapter Map
  • Continuous classes and boundaries
  • Frequency density
  • Constructing histograms
  • Recovering frequencies from areas
  • Interpreting and comparing histograms
Histogram Or Bar Chart?

A histogram is used for grouped continuous data. Adjacent bars touch because there are no gaps in the measurement scale. The horizontal axis shows class intervals, and the area of each bar represents frequency. A bar chart is generally used for discrete or categorical data; its bars are separated and height directly represents frequency.

Feature Histogram Bar chart
Data type Grouped continuous Categorical or discrete
Gaps between bars No Usually yes
Meaning of height Frequency density when widths vary Frequency or another stated value
Meaning of area Frequency Usually no special meaning
Class Boundaries

Measurements written as 10\le x\lt 20 already provide boundaries 10 and 20. Data rounded to the nearest unit may need conversion: a class 10–19 for whole-number ages has continuous boundaries 9.5 and 19.5. Cambridge questions normally make the required interpretation clear.

Class Width
\text{class width}=\text{upper boundary}-\text{lower boundary}

For 15\le x\lt 27, the class width is 27-15=12, not 11.

Frequency Density
\text{frequency density}=\frac{\text{frequency}}{\text{class width}}
Worked Example: Complete A Density Table

Question: For the class 20\le x\lt 35, the frequency is 45. Find the frequency density.

  1. Class width =35-20=15.
  2. Density =45/15=3.

Answer: 3.

Constructing A Histogram

Calculate the density for every class, choose a vertical scale labelled “frequency density,” and draw touching rectangles with widths equal to the class intervals. The graph must not use raw frequency as height when class widths differ.

Class interval Frequency Width Frequency density
0\le x\lt 10 15 10 1.5
10\le x\lt 15 20 5 4
15\le x\lt 30 30 15 2
30\le x\lt 50 20 20 1

The second class has the highest bar because its values are most concentrated per unit of class width, even though another class has a larger raw frequency.

Finding Frequency From A Histogram
\text{frequency}=\text{frequency density}\times\text{class width}
Worked Example: Recover A Frequency

Question: A histogram bar covers 40\le x\lt 55 and has frequency density 2.8. Find the frequency.

  1. Width =55-40=15.
  2. Frequency =2.8\times15=42.

Answer: 42.

When The Vertical Scale Is Not Frequency Density

Sometimes a histogram gives a vertical scale in arbitrary units or provides one known class frequency. Use the known bar area to determine the scale factor, then apply the same factor to other bar areas. The key principle remains that frequency is proportional to bar area.

Worked Example: Area Scale

Question: On a histogram, a class of width 8 and height 6 represents frequency 24. Another class has width 5 and height 9. Find its frequency.

  1. First bar area =8\times6=48 square units represents 24, so 1 square unit represents 0.5 frequency.
  2. Second bar area =5\times9=45.
  3. Frequency =45\times0.5=22.5.
  4. If frequencies must be whole, re-check the graph scale or supplied values; in an examination, data will normally give a sensible result.

Answer: 22.5 according to the stated proportional scale.

Estimating Frequencies Within Part Of A Class

If values are assumed evenly distributed within a class, use the fraction of the class width. For example, in 20\le x\lt 30 with frequency 50, an estimate for 20\le x\lt 24 is (4/10)\times50=20. State that this is an estimate.

Interpreting Shape

A histogram can show where data are concentrated and whether a distribution is spread out or skewed. Compare areas, class widths and densities rather than looking only at heights. A tall narrow bar may contain fewer observations than a shorter wide bar.

Examination Guidance
  • Label the vertical axis “frequency density” when that is what is plotted.
  • Calculate class width from boundaries, not from the number of integers displayed.
  • Use a ruler and draw adjacent rectangles accurately.
  • Remember: histogram frequency is represented by area.
Common Mistakes
  • Using frequency as bar height when class widths are unequal.
  • Leaving gaps between histogram bars.
  • Comparing bar heights alone to compare frequencies.
  • Using the midpoint instead of class width in the density calculation.
Knowledge Check And Practice

1. A class 12\le x\lt 20 has frequency 32. Find its density.

Answer: Width 8, so density 4.

2. A class has width 6 and density 3.5. Find its frequency.

Answer: 21.

3. Why do histogram bars touch?

Answer: The data are continuous and the class intervals are adjacent.

4. A bar of width 10 and density 2.4 represents what frequency?

Answer: 24.

5. State the main difference between bar-chart height and histogram height.

Answer: In a histogram with unequal widths, height is frequency density and area represents frequency; in a bar chart, height usually represents frequency directly.
Extended Worked Practice
Comparing Two Histogram Bars

Question: Class A has width 5 and frequency density 8. Class B has width 12 and density 4. Which class contains more observations?

Frequency A =5\times8=40. Frequency B =12\times4=48. Answer: Class B, even though its bar is shorter.

Converting Rounded Classes To Boundaries

Question: Masses are recorded to the nearest kilogram in classes 40-49, 50-59 and 60-69. State the continuous boundaries of the middle class and its width.

The middle class is 49.5\le m<59.5. Its width is 59.5-49.5=10 kg.

Estimating Part Of A Class

Question: The class 20\le x<35 has frequency 60. Assuming values are evenly distributed, estimate how many satisfy 20\le x<26.

  1. Required width is 26-20=6.
  2. Whole class width is 35-20=15.
  3. Estimate =\frac{6}{15}\times60=24.

Answer: 24, stated as an estimate.

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