Did you know that two school clubs can finish a chart at 150 and 110, yet the club at 110 may still have far more visitors? The labels can be perfectly accurate. It is the meaning of the vertical axis—not the arithmetic—that readers sometimes miss. That is a useful education problem: we should learn to ask what a number represents before using it to make a claim.
An index chart rebased to 100 expresses each series relative to its own value in a chosen reference period. The base becomes 100; a later value of 150 means 50% above that series’ baseline, not 150 people or 150%. To read it correctly, find the base period, calculate current ÷ base × 100, and keep relative change separate from actual size. Two lines that start at 100 do not necessarily begin with equal real-world quantities.
A quick diagnostic: what did 100 stand for?
Put two questions to a learner before showing any formula: ‘If a club had 40 visitors in the base month and its index is now 150, how many visitors does it have now?’ Then ask: ‘If another club began with 400 visitors and reaches an index of 110, which club is bigger now?’ A student who says the first club is bigger because 150 is taller than 110 has confused relative growth with the actual count. That is a diagnostic clue, not evidence that they lack multiplication skills.
The answer to the first question is 40 × 1.5 = 60 visitors. For the second club it is 400 × 1.1 = 440 visitors. The smaller club has grown faster in percentage terms; the bigger club still records many more visitors. An index preserves a comparison with a selected baseline, while deliberately hiding the original units unless they are supplied elsewhere.
Worked example: two fictional reading corners
Every person, club and figure in this example is invented for teaching. Imagine an after-school project comparing two reading corners. In September, Corner A records 40 visits and Corner B records 400. The same counting method is applied in October: A records 60 visits, B records 440. Each visit is counted once on entry; a repeat visit on another day counts again. These are counts of visits, not necessarily unique children.
- Corner A, September base = 40 visits; October = 60 visits. Its October index is 60 ÷ 40 × 100 = 150.
- Corner B, September base = 400 visits; October = 440 visits. Its October index is 440 ÷ 400 × 100 = 110.
- Corner A rose by (60 − 40) ÷ 40 × 100 = 50%, which is an increase of 20 visits.
- Corner B rose by (440 − 400) ÷ 400 × 100 = 10%, which is an increase of 40 visits.
Now compare three different questions. ‘Which grew faster relative to its starting point?’ A, because 50% is greater than 10%. ‘Which added more visits?’ B, because 40 is greater than 20. ‘Which had more visits in October?’ B, because 440 is greater than 60. All three answers can be correct together. The apparent contradiction disappears once we stop using ‘bigger’ as an imprecise substitute for three distinct measures.
A sensible graph legend would read ‘September = 100 for each reading corner; index of monthly visits’. A separate table or note should reveal the base-month visit counts if the reader needs absolute comparisons. A vertical label saying only ‘Visitors’ would be misleading for these indexed values; a stronger label is ‘Visit index (September = 100)’.
Extend the example to November
Suppose November records are 52 visits for A and 480 for B, using the same fictional rules. Their indices become 52 ÷ 40 × 100 = 130 and 480 ÷ 400 × 100 = 120. Between October and November, A actually falls from 60 to 52 visits (a fall of 8), while its index falls from 150 to 130 (20 index points). That 20-point drop is not the same as a 20% drop from October: the percentage fall from October is 8 ÷ 60 × 100 ≈ 13.3%.
Meanwhile B increases from 440 to 480 visits, a gain of 40 visits; its index rises from 110 to 120, or 10 points. A November index of 120 is 20% above its September base, while its change from October is (120 − 110) ÷ 110 × 100 ≈ 9.1%. A base-relative percentage and a month-to-month percentage answer different questions even when they describe the same time series.
The mechanism: why rebasing works—and what it erases
For a positive reference value, the transformation is index at time t = (value at time t ÷ value in the reference period) × 100. The units cancel. It becomes possible to compare how far different series move relative to their own starting levels. This can be useful for a school project about activity, library demand or an experiment measured on comparable dates.
The same operation also removes information. A base of 20 and a base of 2,000 both plot at 100. A chart of indexed values can therefore support a claim about relative movement without supporting a claim that the series have the same size. We should not reverse that limitation by declaring indexed charts useless: they answer a genuine, different question well.
Official statistical organisations use index reference periods to put values on a comparable index scale. For a technical explanation of re-referencing a series, see the UK Office for National Statistics explanation of rebasing. Our school example is a simple fixed-base index, not a claim about how the ONS produces every real statistical index; some published indices also incorporate weights and chaining.
Five misconceptions worth catching early
- ‘150 means 150 visitors.’ Not unless the chart explicitly reports a count. Here 150 is an index number.
- ‘A line ending at 150 is larger than one ending at 110.’ It shows stronger growth from its own base, not a larger absolute total.
- ‘An increase of 20 index points means 20% growth since last month.’ Index points use the reference-period scale. Recompute the change using the previous month if that is the question.
- ‘Rebasing changes the history.’ Re-expressing values on a different positive reference scale changes the displayed index numbers, not the underlying visit counts.
- ‘I can index a zero baseline by dividing by zero.’ You cannot use this ratio when the chosen baseline is zero. Choose a meaningful alternative measure or explain why an index is not suitable.
From guided practice to independent judgement
Guided practice: keep units visible
Start with the raw pairs 25 and 35 visits for Corner C. Ask the learner to calculate 35 ÷ 25 × 100 = 140, then say aloud ‘40% above the base’, not ‘140 visits’. Repeat with 100 and 120 visits for Corner D: its index is 120. The learner should be able to say C grew faster but D had more visits in the final month.
Shared practice: choose the correct claim
Give this sentence to two learners: ‘The reading-corner chart proves Corner C was the most popular.’ One learner checks the vertical label; the other searches for the baseline counts. Together they should rewrite it as: ‘Corner C showed the larger proportional rise from the selected base, but the indexed graph alone does not show which corner had the most visits.’ This strengthens claim-evidence discipline alongside arithmetic.
Independent check: a new base changes the plotted numbers
Without looking back, use Corner A’s October count of 60 as a new base of 100. Its November count is 52; the November index relative to October is 52 ÷ 60 × 100 ≈ 86.7. The actual November count remains 52. A learner who can explain why the original September-based November index was 130 has understood both the formula and the interpretation.
A four-week transfer routine
- Week 1: Label each chart as raw count, percentage, index or rate before answering the question.
- Week 2: Convert three pairs of base/current values into indices, then independently reconstruct the actual values from a supplied baseline.
- Week 3: Write two short conclusions from the same chart: one about proportional change and another about absolute values from a separate table.
- Week 4: Find an unfamiliar index chart, identify its reference period and state one justified conclusion and one conclusion the chart cannot support.
Parents can ask their child for a plain-English explanation of the base rather than merely checking a formula. Teachers can deliberately pair an index plot with its raw-data table. Learners can track progress by counting how often they state the reference period, give units correctly and distinguish an index-point change from a percentage change. Improvement means fewer interpretation errors on fresh examples, not just a higher score on a rehearsed worksheet.
Limits and responsible use
An index comparison assumes that the underlying measures are defined consistently. If Corner B changed from counting all visits to counting unique visitors, the series would contain a measurement break. If the two corners operate for different numbers of days each month, daily averages may be more informative than monthly totals. A small group starting from a very low base can show a large percentage increase after only a few extra visits. These limitations do not invalidate the arithmetic; they limit how much the arithmetic establishes.
Frequently asked questions
Can an index be below 100?
Yes. With a positive base, an index of 80 represents a value 20% lower than that reference value. An index of 100 means the value matches the base value, not that the project has met every target.
If both series start at 100, were they equal in the first month?
Only on the indexed scale. Their original counts could be very different, as the fictional 40-visit and 400-visit baselines demonstrate.
Why not compare the slopes visually and stop there?
The apparent steepness depends on the axis scale and chart proportions. Compare clearly identified values and the periods covered; check whether the claim concerns relative growth, actual count or actual increase.
Should a student always convert indexed data back to counts?
No. Sometimes the actual baseline counts are unavailable, and the task asks only about proportional movement. Answer that question responsibly, and state what cannot be reconstructed.
Where this learning connects
For a related check on visual comparisons, read how to compare grouped bar charts when category order changes and how to describe a school graph accessibly. The wider Core Aim of Education learning hub connects careful interpretation to transferable study skills. Our immutable Secondary 1 Mathematics tutorial reference illustrates diagnosis, explanation and progressive independent checks in teaching.
The core aim is not to make the tallest line win. It is to help a learner choose the question, read the units, test the calculation and communicate only what the data can establish. That habit travels far beyond one school graph.