Cambridge IGCSE Statistics 3: Frequency distributions

Cambridge IGCSE Statistics 0479 Topic 3 develops grouped and ungrouped frequency distributions, class notation, histograms, frequency polygons and cumulative frequency representations. The central principle is that a display must preserve the meaning of frequency even when class widths differ.

1. Ungrouped and grouped frequency distributions

An ungrouped frequency distribution records every distinct value and its frequency. It preserves exact values and supports exact calculations, but a long table can conceal the overall pattern.

A grouped distribution combines values into non-overlapping classes, such as 30 ≤ x < 40. Grouping makes shape, concentration and spread easier to see. Its cost is information loss: the original values within each class are unknown, so later calculations based on class midpoints are estimates.

Classes must cover the relevant range without gaps or overlaps. State the convention clearly. The class 30 ≤ x < 40 includes 30 but excludes 40, which enters the next class.

2. Class limits and boundaries

For discrete data, class limits are the smallest and largest recorded values assigned to a class. If ages in completed years are grouped as 10 to 14, the lower and upper class limits are 10 and 14. Assuming measurement to the nearest year, the corresponding continuous class boundaries are 9.5 and 14.5.

For continuous data, classes are defined by boundaries directly. A mass class 50 ≤ m < 60 has boundaries 50 and 60. There is no artificial gap between it and 60 ≤ m < 70.

The class midpoint is (lower boundary + upper boundary) ÷ 2.

The class width is upper boundary minus lower boundary. Use boundaries, not displayed integer limits, when determining histogram widths.

3. Histograms and frequency density

A histogram represents continuous grouped data with touching rectangles. The area of each rectangle represents frequency. With equal class widths, frequency and frequency density produce the same relative pattern, but with unequal widths the vertical axis must be frequency density:

frequency density = frequency ÷ class width

Therefore:

frequency = frequency density × class width

Label both axes with the variable and units. The horizontal scale follows class boundaries. The vertical scale is frequency density, including compound units when appropriate, such as people per minute.

A taller bar does not necessarily contain more observations. Compare areas when widths differ. To find a frequency from a published histogram, read the density and multiply by the relevant width. If the vertical scale is absent but one class frequency is known, use that class to establish the scale factor.

4. Interpreting histogram shape

Describe where observations are concentrated, how widely they spread, whether the shape is roughly symmetric or skewed, and whether there are gaps or multiple peaks. Avoid causal claims. A histogram can show an association between measurement range and frequency, not explain why the pattern arose.

Grouping affects appearance. Broad classes can conceal clusters and gaps; narrow classes can make a modest sample look irregular. Comparisons are strongest when the same class boundaries and density scale are used.

5. Frequency polygons

For Cambridge 0479, frequency polygons use equal-width groups. Plot each class midpoint against its frequency, then join adjacent points with straight lines. A polygon emphasises overall shape and allows two distributions to be compared on the same axes with less visual obstruction than overlapping histogram bars.

Use a key for paired distributions. Compare location, spread, peaks and shape, while remembering that joined segments are a visual guide rather than evidence of values at every point between midpoints.

6. Cumulative frequency distributions

Cumulative frequency is a running total. An upper cumulative distribution records the number of observations below each upper class boundary. Starting from ordinary frequencies, add successively. To reverse the process, subtract consecutive cumulative totals.

For example, cumulative frequencies 5, 13, 25 and 30 correspond to class frequencies 5, 8, 12 and 5. The final cumulative frequency must equal the total sample size.

A lower cumulative distribution can instead accumulate from the upper end. Read the wording and boundary direction rather than assuming every cumulative table has the same orientation.

7. Cumulative frequency polygons and curves

For a conventional “less than” graph, plot cumulative frequency at each upper class boundary and include zero cumulative frequency at the first lower boundary. Join points with straight segments for a polygon or a smooth increasing curve where requested.

The graph must never fall because a running total cannot decrease. Read an estimated frequency below a value from the vertical coordinate. Estimate the number between two values by subtracting their cumulative frequencies. Quartiles, percentiles and the median are developed further in Topics 4 and 5.

Interpolation within a class assumes observations are spread sufficiently evenly for an estimate. It does not recover the exact unknown raw values.

Worked example: unequal-width histogram

A journey-time distribution has classes 0 to under 5 minutes with frequency 12, 5 to under 15 with frequency 30, and 15 to under 35 with frequency 28. Their widths are 5, 10 and 20, so the frequency densities are 12 ÷ 5 = 2.4, 30 ÷ 10 = 3 and 28 ÷ 20 = 1.4 people per minute. The middle bar is tallest and also has the largest area, but the first bar is taller than the third despite containing fewer than half as many observations. If a fourth bar from 35 to under 45 has density 0.8, its frequency is 0.8 × 10 = 8. The total frequency is therefore 78. This calculation shows why height alone is not frequency when class widths differ.

Common misconceptions and how to correct them

• Treating grouped data as exact raw data. Grouping loses within-class positions.
• Allowing classes to overlap. Use an explicit inclusive and exclusive boundary convention.
• Leaving gaps between continuous classes. Adjacent boundaries must meet.
• Using integer class limits as histogram widths. Convert discrete limits to boundaries when required.
• Calculating midpoint from two class widths. Average the lower and upper boundaries.
• Drawing gaps between histogram bars. Continuous classes use touching rectangles.
• Putting frequency on an unequal-width histogram axis. Use frequency density.
• Reading bar height as frequency. Frequency is represented by area.
• Assuming the tallest bar always has the greatest frequency. Compare density multiplied by width.
• Giving frequency density no scale or label. State the quantity and units.
• Using unequal groups for the required frequency polygon. Cambridge specifies equal class widths here.
• Plotting frequency polygons at class boundaries. Plot at midpoints.
• Treating joined polygon segments as observed continuous data. They summarise grouped frequencies.
• Adding cumulative frequencies again when reversing a table. Subtract successive totals.
• Omitting the zero point from a less-than cumulative graph. Begin at the first lower boundary.
• Drawing a decreasing cumulative curve. Running totals cannot fall.
• Reading a between-values frequency directly once. Subtract two cumulative readings.
• Presenting interpolation as exact. It is an estimate within a grouped interval.

Assessment guidance

Write class inequalities or boundaries clearly before drawing. In a histogram question, show class widths and frequency-density calculations, label the density axis, and use boundaries on the horizontal axis. Interpret unequal bars through area, not height alone. For a frequency polygon, use equal-width class midpoints and distinguish paired distributions with a key. In cumulative-frequency work, show the running totals, check the final total, plot upper boundaries for a less-than graph and include the zero starting point. When reading between two values, subtract cumulative readings. State that any within-class interpolation is an estimate caused by grouping.

Retrieval practice

Convert a discrete class stated with limits into boundaries, midpoint and width. Build density values for four unequal classes and reconstruct frequencies from a histogram. Plot two equal-width frequency polygons and compare their location and spread. Convert an ordinary frequency table into cumulative totals and back again, then use a cumulative graph to estimate counts below, above and between chosen values.

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