Cumulative frequency diagrams
Section: Statistics | Syllabus: Cambridge IGCSE Mathematics (0580)
Constructing a Cumulative Frequency Table and Diagram
Extended Only
- Cumulative frequency is a running total of the frequencies, built up class by class from a grouped frequency table
- Each cumulative frequency is plotted against the upper boundary of its class interval (not the midpoint), since it represents "at most this many items are less than or equal to this boundary"
- The plotted points are joined with a smooth curve (often called an ogive), starting from a point at the lower boundary of the first class, where the cumulative frequency is 0
Worked Example: Constructing a Cumulative Frequency Table
- Question: The table shows the mass, m grams, of 60 apples. Construct a cumulative frequency table for this data.
Mass, m (g) 80 90 100 110 120 Frequency 6 14 22 12 6 - Step 1: Add each frequency to the running total of all the frequencies before it
- Answer:
Mass (g) 90 100 110 120 130 Cumulative frequency 6 20 42 54 60
Common Mistakes
MistakePlotting each cumulative frequency against the midpoint of its class interval instead of the upper boundary
Fixa cumulative frequency always represents a total "up to and including" a boundary value, so it must be plotted at the upper boundary, not the midpoint
MistakeWriting each class's own frequency in the cumulative frequency table instead of the running total
Fixeach cumulative frequency value must include every frequency up to that point, not just the frequency of that one class
Estimating the Median, Quartiles and Interquartile Range
Extended Only
- On a cumulative frequency diagram with n items: the median is read at cumulative frequency (n/2); the lower quartile (LQ) at (n/4); the upper quartile (UQ) at (3n/4)
- Interquartile range (IQR) = UQ - LQ; this measures the spread of the middle 50% of the data and is less affected by extreme values than the range
- Readings taken from a graph are estimates - answers within a small tolerance of the exact curve are generally accepted
Worked Example: Estimating the Median and Quartiles
- Question: Using the apple-mass cumulative frequency data above (n=60), estimate the median mass and the interquartile range.
- Step 1 (median): Read at cumulative frequency (60/2)=30, which falls between (100,20) and (110,42) - interpolating gives a median of approximately 105 g
- Step 2 (LQ): Read at cumulative frequency (60/4)=15, between (90,6) and (100,20) - approximately 96 g
- Step 3 (UQ): Read at cumulative frequency (3(60)/4)=45, between (110,42) and (120,54) - approximately 112.5 g
- Step 4 (IQR): 112.5-96=16.5
- Answer: median 105 g; interquartile range 16.5 g
Common Mistakes
MistakeReading the median at n instead of (n/2), or the quartiles at the wrong fraction of n
Fixalways calculate the correct cumulative frequency position first (n/2, n/4, or 3n/4) before reading across to the curve
MistakeSubtracting LQ and UQ the wrong way round, or subtracting the minimum and maximum instead of the quartiles, when finding the IQR
Fixthe interquartile range is always upper quartile minus lower quartile (UQ − LQ), not the full range of the whole data set
Estimating Percentiles
Extended Only
- A percentile splits the data into 100 equal parts; the k-th percentile is read at cumulative frequency (k/100)× n
- Percentiles describe a position within a distribution, e.g. "the top 10%" of a data set corresponds to the 90th percentile
- Reading a percentile uses exactly the same method as reading the median or a quartile, just at a different cumulative frequency position
Worked Example: Estimating a Percentile
- Question: The top 10% of the 60 apples (by mass) are classified as "large". Use the cumulative frequency table above to estimate the minimum mass for an apple to be classified as large.
- Step 1: "Top 10%" corresponds to the 90th percentile, at cumulative frequency (90/100)60=54
- Step 2: From the table, a cumulative frequency of exactly 54 corresponds to a mass of 120 g - no interpolation is needed since this lands exactly on a plotted point
- Answer: the minimum mass for an apple to be classified as large is approximately 120 g
Common Mistakes
MistakeUsing k directly as the cumulative frequency position instead of (k/100)× n
Fixa percentile position always depends on the total number of items, n - it is not simply the percentile number itself
MistakeConfusing "the top 10%" with the 10th percentile instead of the 90th percentile
Fix"the top 10%" means the highest values, which lie above the 90th percentile - the bottom 10% is what the 10th percentile describes
Interactive revision notes, videos and practice questions load below.