Cumulative Frequency and Quartiles

Build a running total

Cumulative frequency is the running total of frequencies. Start with the first frequency and keep adding the next one. The final total must equal the number of observations; a cumulative total cannot decrease.

Worked example — interpret a cumulative total

At the upper boundary 20, the cumulative frequency is 4 + 12 = 16: 16 journeys took less than 20 minutes. To recover the 20–40 class frequency, subtract successive totals: 32 − 16 = 16.

If a table instead uses classes ending with ≤, its running totals mean “at most” the boundary. Match the wording to the given inequalities.

Plot at upper class boundaries

Put the measured variable on the horizontal axis and cumulative frequency on the vertical axis. Plot each running total at its upper class boundary, not at a midpoint. Add the lowest boundary with cumulative frequency zero.

Worked example — construct the graph

Plot (0,0), (10,4), (20,16), (40,32), (60,40). Here straight segments explicitly use linear interpolation within each class. A question may instead ask for a smooth increasing curve; draw it through the points without making it turn downwards.

The table fixes the plotted boundary totals. Readings between them are estimates because the exact observations inside each class are unknown. This graph is not a frequency polygon: both its plotting positions and its vertical quantities are different.

Read the median and quartiles

Quartiles divide ordered data into quarters. Q1 is the lower quartile, the median is the halfway value and Q3 is the upper quartile. For a cumulative-frequency graph with total n, read across from cumulative frequencies n/4, n/2 and 3n/4, then down to the value axis.

Worked example — read three estimates for 40 journeys

Use levels 10, 20 and 30. The graph gives Q1 ≈ 15 min, median ≈ 25 min, Q3 ≈ 37.5 min. These are times, not the cumulative counts 10, 20 and 30.

For the median, cumulative frequency rises from 16 to 32 between 20 and 40 minutes. Level 20 is 4 of the 16 observations into that class: 20 + (4/16) × 20 = 25 min. This calculation matches a straight-line graph reading.

Interquartile range (IQR) = Q3 − Q1. Here IQR ≈ 37.5 − 15 = 22.5 min. It describes the spread of the middle half of the data and is less affected by extreme values than the full range.

Estimate counts and proportions

Worked example — read above a threshold

At 30 minutes the cumulative frequency is about 24. The estimated count taking 30 minutes or longer is 40 − 24 = 16, which is 40% of the sample.

Worked example — read an interval

At 15 minutes the running total is about 10; at 45 minutes it is about 34. The estimated count from 15 up to 45 minutes is 34 − 10 = 24, or 60%.

Use subtraction for an interval and subtract from the total for values above a threshold. For continuously measured data, a graph cannot normally resolve an exact count at one particular value; do not claim more precision than the grouped evidence supports.

Distinguish grouped estimates from raw-data quartiles

With a small ordered list, quartiles can be found directly. A common school convention is to take the median of each half, excluding the overall median when the count is odd. Follow any convention or positions supplied in the question; conventions can differ for small datasets.

Worked example — use an eight-value list

Times are 4, 6, 8, 10, 12, 14, 18, 24 minutes. Median = (10 + 12)/2 = 11. Lower half 4,6,8,10 gives Q1 = (6 + 8)/2 = 7. Upper half 12,14,18,24 gives Q3 = (14 + 18)/2 = 16.

IQR = 16 − 7 = 9 min; range = 24 − 4 = 20 min. These exact raw-data calculations are different from reading quartile levels from grouped data.

For a cumulative-frequency graph, use one quarter, one half and three quarters of its total count. Do not replace those graph levels with raw-list middle-position rules. The next lesson uses quartiles to draw and compare box plots.

Think Like a Mathematician

A steep part of a cumulative-frequency graph means many observations occur in a short interval of values. It does not mean the cumulative frequency is falling or that the measured quantity is changing rapidly over time.

Common Mistakes

Midpoint plotting: cumulative frequency uses upper boundaries.

Count versus value: n/2 locates the median; read its value from the other axis.

Forgetting the total: counts above a threshold need total minus the cumulative count.

False exactness: grouped-data readings depend on interpolation.

Practice

1. Frequencies are 5, 9, 12 and 4. Give the cumulative frequencies.

2. Classes 0–5, 5–15, 15–25 have frequencies 4, 10, 6. Give the cumulative-graph coordinates, including the start.

3. For a sample of 80, which cumulative-frequency levels locate Q1, median and Q3?

4. In the journey graph, estimate Q1, the median, Q3 and IQR.

5. Use the journey graph to estimate the number taking at least 50 minutes.

6. Use that graph to estimate the percentage from 10 up to 40 minutes.

7. Using medians of halves, find the median, Q1 and Q3 for 2,4,6,8,10,12,14,16.

8. Why can a correct cumulative-frequency graph not slope downwards?

Frequently Asked Questions

Must I calculate interpolation algebraically?

You should be able to read and estimate accurately from the graph. A proportional calculation is a useful check when straight-line interpolation is specified or used.

Does the median split every small dataset into two perfectly distinct halves?

It marks the centre of the ordered data. Repeated values and different finite-sample conventions require care; grouped graphs provide approximate percentile readings, not exact counts at every boundary.

Continue Learning

Previous: Histograms and Frequency Density

Next: Box Plots and Comparing Distributions

Statistics topic overview