Representing Data: Histograms and Frequency Polygons

What you'll learn

This revision guide covers how to construct and interpret histograms and frequency polygons for grouped continuous data. You'll learn to calculate frequency density, draw accurate diagrams with unequal class widths, and extract information from these visual representations. These techniques are essential for representing large datasets effectively in your AQA GCSE Statistics exam.

Key terms and definitions

Histogram — a diagram showing the distribution of continuous data where the area of each bar is proportional to the frequency, with no gaps between bars.

Frequency density — the frequency per unit of class width, calculated using the formula: frequency density = frequency ÷ class width.

Class width — the range covered by each class interval, calculated by subtracting the lower class boundary from the upper class boundary.

Class interval — the range of values grouped together in a frequency table, such as 10 ≤ h < 20.

Frequency polygon — a line graph formed by plotting frequency or frequency density against the midpoint of each class interval and joining the points with straight lines.

Midpoint — the central value of a class interval, calculated by adding the lower and upper boundaries and dividing by 2.

Continuous data — numerical data that can take any value within a range, such as height, time or mass, measured rather than counted.

Modal class — the class interval with the highest frequency (or the tallest bar in a histogram when class widths are equal).

Core concepts

Understanding histograms

Histograms differ fundamentally from bar charts. In a histogram:

• The horizontal axis shows a continuous scale with no gaps
• Bars touch each other because the data is continuous
• The area of each bar represents frequency, not the height
• The vertical axis shows frequency density, not frequency
• Unequal class widths are common and require frequency density calculations

When all class widths are equal, the heights of the bars are proportional to frequency, making the histogram appear similar to a bar chart. However, when class widths vary, you must calculate frequency density to ensure areas remain proportional to frequencies.

Why use frequency density?

Without frequency density, classes with wider intervals would appear more prominent simply because they cover more values, creating a misleading visual representation. Frequency density standardises the display so that area accurately represents frequency regardless of class width.

Calculating frequency density

The key formula you must know is:

Frequency density = Frequency ÷ Class width

Follow this process:

1. Identify the class boundaries (the actual limits of each class)
2. Calculate the class width for each interval
3. Divide the frequency by the class width
4. Record the frequency density value

Example calculation:

For the class 20 ≤ t < 35 with frequency 45:

• Class width = 35 - 20 = 15
• Frequency density = 45 ÷ 15 = 3

When inequalities include different symbols (like 10 < m ≤ 25), treat them as continuous. The class boundaries are still 10 and 25.

Drawing histograms

To construct an accurate histogram:

1. Draw and label both axes clearly
   • Horizontal axis: the variable name and units
   • Vertical axis: "Frequency density"
2. Use a consistent scale with equal intervals
3. Mark class boundaries on the horizontal axis (not midpoints)
4. Calculate frequency density for each class
5. Draw each bar with the correct height and width
6. Ensure bars touch with no gaps
7. Use a ruler for straight, neat lines

Scale considerations:

Choose scales that:

• Make efficient use of the graph paper
• Allow all data to fit comfortably
• Use simple intervals (1, 2, 5, 10, etc.)
• Are clearly numbered at regular intervals

Interpreting histograms

From a histogram, you can:

Find frequencies:

• Frequency = Frequency density × Class width
• Measure the height (frequency density) and width of any bar
• Multiply these together to find how many values fall in that interval

Identify the modal class:

• When class widths are equal: the tallest bar
• When class widths vary: the bar with the largest area (not necessarily the tallest)

Estimate values:

• Total frequency: sum the frequencies of all classes
• Proportion in a range: add relevant frequencies and divide by total
• Median class: find the class containing the middle value

Constructing frequency polygons

A frequency polygon shows the shape of a distribution using straight line segments. You can create one from a frequency table or from a histogram.

From a frequency table:

1. Calculate the midpoint of each class interval
2. Plot frequency (or frequency density) against each midpoint
3. Join consecutive points with straight lines
4. Extend the polygon to the horizontal axis at both ends by plotting zero frequency at the midpoint of imaginary classes before and after the data

From a histogram:

1. Mark the midpoint of the top of each bar
2. Join these midpoints with straight lines
3. Complete the polygon by connecting to the horizontal axis

Key features:

• Use the same axes as the histogram
• Points are plotted at class midpoints, not boundaries
• The area under a frequency polygon approximates the total frequency
• Multiple frequency polygons can be drawn on one set of axes for comparison

Comparing distributions using frequency polygons

Frequency polygons are particularly useful for comparing two or more datasets because:

• Multiple polygons can appear on the same diagram without obscuring each other
• The shape, spread and central tendency of distributions are clearly visible
• Differences in modal classes and ranges are easy to identify

When comparing, always comment on:

• Shape: symmetric, skewed left/right, uniform
• Central tendency: which dataset has higher typical values
• Spread: which dataset is more variable or consistent
• Modal classes: where the peaks occur

Worked examples

Example 1: Drawing a histogram with unequal class widths

The table shows the waiting times at a doctor's surgery.

Draw a histogram to represent this data.

Solution:

First, create a frequency density table:

Then draw the histogram:

• Horizontal axis: Waiting time (minutes), scale 0 to 60
• Vertical axis: Frequency density, scale 0 to 4
• Draw bars with heights matching frequency density values
• Ensure bars touch at boundaries 5, 10, 20, 40, 60

Mark scheme points: (1 mark) Correct frequency density calculations; (1 mark) appropriate scales with labels; (1 mark) accurate bars with correct heights and widths.

Example 2: Finding frequency from a histogram

A histogram shows the heights of plants in a greenhouse. The bar representing heights from 15 cm to 25 cm has a frequency density of 4.5 and the bar from 25 cm to 30 cm has a frequency density of 6.

(a) Calculate the frequency of plants in the 15 ≤ h < 25 interval. (b) Calculate the frequency of plants in the 25 ≤ h < 30 interval. (c) How many more plants are in the first interval than the second?

Solution:

(a) Class width = 25 - 15 = 10 cm Frequency = frequency density × class width = 4.5 × 10 = 45 plants

(b) Class width = 30 - 25 = 5 cm Frequency = 6 × 5 = 30 plants

(c) Difference = 45 - 30 = 15 plants There are 15 more plants in the 15 ≤ h < 25 interval.

Mark scheme points: (1 mark) each for parts (a) and (b) showing correct calculation; (1 mark) for part (c) with correct comparison.

Example 3: Drawing a frequency polygon

The table shows the ages of people attending a community event.

Draw a frequency polygon for this data.

Solution:

First, calculate midpoints:

Plot points: (5, 24), (15, 36), (25, 42), (35, 28), (45, 15)

Join with straight lines and close the polygon by connecting to the horizontal axis at the midpoints before 0 (at -5) and after 50 (at 55) with zero frequency.

Mark scheme points: (1 mark) correct midpoint calculations; (1 mark) accurate plotting of points; (1 mark) points joined correctly with straight lines.

Common mistakes and how to avoid them

• Confusing frequency with frequency density — Always check the vertical axis label. If drawing a histogram with unequal class widths, you must use frequency density. Remember: area represents frequency, not height.

• Incorrect class width calculations — Class width is upper boundary minus lower boundary, even when the inequalities use different symbols (< or ≤). For 10 < m ≤ 20, the class width is still 20 - 10 = 10.

• Plotting frequency polygons at boundaries instead of midpoints — Frequency polygons use the centre of each class interval, not the edges. Calculate the midpoint by adding the two boundaries and dividing by 2.

• Leaving gaps between histogram bars — Histograms represent continuous data, so bars must touch. Gaps suggest discrete categories, which is incorrect for continuous measurements.

• Not closing frequency polygons properly — Extend the polygon down to the horizontal axis at both ends. Plot points at the midpoints of imaginary classes before and after your data with zero frequency.

• Misidentifying the modal class with unequal widths — When class widths vary, the modal class has the largest area (frequency), which may not be the tallest bar. Calculate actual frequencies if needed.

Exam technique for "Representing Data: Histograms and Frequency Polygons"

• Command words matter: "Draw" means construct accurately with a ruler and appropriate scale (2-3 marks). "Sketch" allows a rougher diagram showing the shape (1-2 marks). "Calculate" requires showing your working using the frequency density formula.

• Show all working for frequency density calculations — Write out the formula, substitute values, and show the division. Even if your final answer is wrong, you can earn method marks for correct working.

• Use the scales provided on exam paper — If grid paper is given, use it efficiently. If you must choose scales, avoid awkward intervals like 3s or 7s. Label axes clearly with variable names and units.

• Read the table carefully for class boundaries — Check whether inequalities use < or ≤, though for continuous data this rarely affects class width. Note any unusual groupings like "up to 10" or "at least 20 but less than 30."

Quick revision summary

Histograms display continuous data with bars touching; the area of each bar represents frequency. When class widths vary, calculate frequency density (frequency ÷ class width) for the vertical axis. Frequency polygons plot frequency or frequency density against class midpoints, joined by straight lines, useful for comparing distributions. To find frequency from a histogram, multiply frequency density by class width. The modal class has the highest frequency (largest area when widths vary). Always label axes clearly, use rulers, and show full working in calculations.