Histograms | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Frequency Density

What is frequency density?

• Frequency density is calculated using the formula:
  • Frequency density = frequency / class width
• It is used with grouped data (class intervals).
• Frequency density is particularly useful for class intervals of unequal width.
• It measures how spread out data within a class interval is, relative to its size.
• Example:
  • 10 data values over a class interval of 20:
    • Frequency density = 10 / 20 = 1/2
  • 20 data values over a class interval of 100:
    • Frequency density = 20 / 100 = 1/5
• Since 1/2 is greater than 1/5, the data in the first interval is more densely spread (closer together) than in the second interval, even though the second interval has more data values.

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How do I calculate frequency density?

• Questions typically present grouped data in a table.
• Add two extra columns to this table:
  • One column for calculating and recording the class width of each interval.
  • A second column for calculating the frequency density for each group (row).

Drawing Histograms

What is a Histogram? 

• Histograms are not just really hard bar charts; they serve a distinct purpose.
• Unlike bar charts which handle discrete and non-numerical data, histograms are designed for continuous data, often organized in uneven class intervals.
• In a bar chart, frequency is determined by height or length, while in a histogram, it's the area of a bar that signifies the frequency.
• The frequency of a class interval is directly linked to the area of the bar representing that interval.
• Merely glancing at a histogram doesn't reveal much; one needs to perform basic calculations for meaningful conclusions and comparisons.

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How do I draw a histogram?

• To draw a histogram, first calculate the frequency densities for each class interval (group).
  • Most questions will require you to complete an incomplete histogram rather than starting from scratch.
• Since frequency is proportional to frequency density:
  
• In most cases, k = 1, so the proportionality can be ignored.
• Once the frequency densities are calculated:
  • Draw bars (rectangles) with widths measured on the horizontal (x) axis.
  • The height of each bar represents the class's frequency density and is measured on the vertical (y) axis.
  • As the data is continuous, the bars will be touching.

Interpreting Histograms

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How do I interpret a histogram?

• The frequency density (y-) axis does not indicate frequency directly.
• The area of the bar is proportional to the frequency.
• Frequency is determined by the area of the bar using:
  • Frequency = area
• To estimate the frequency of part of a bar/class interval within a histogram:
  • Find the area of the required part of the interval.
  • Use the area to calculate the frequency as described above.