Index numbers express values relative to a chosen reference, often set equal to 100. They make changes easier to compare when the original quantities use different scales. If a quantity rises from an index of 100 to 120, it has increased by 20% relative to that reference. The number 120 is not necessarily a price, a percentage share or a physical amount.
Understanding index numbers helps readers interpret graphs of production, prices, population and many other subjects. It also explains why changing the reference period can change every displayed number without changing the underlying history. This guide uses original worked examples to show how indices are constructed, what they reveal and which comparisons they cannot support.
Key takeaways
- An index of 100 usually marks a reference value, not a maximum or an average score.
- A simple index equals the current value divided by the reference value, multiplied by 100.
- Index-point changes and percentage changes are different.
- Re-referencing rescales a series without changing its proportional movements.
- Weights, definitions and methods determine what a composite index means.
What is an index number?
A simple index converts an observed quantity into a ratio against a reference. Suppose a school library lends 800 books in one term. Set that term to 100. If it lends 1,000 books in a later term, the new index is 1,000 divided by 800, multiplied by 100: 125.
The result tells us that lending is 25% above the reference term. It does not say that 125 books were borrowed. The original count and the indexed count are different ways of describing the same observation.
Indices are useful when the relative change matters more than the original unit. A graph can show the growth of library loans and museum visits together even though their starting totals differ. However, that convenience comes at a cost: the graph no longer directly displays their absolute sizes.
Why use 100?
Using 100 makes proportional comparisons convenient. An index of 110 is 10% above the reference, while 90 is 10% below it. Other reference values are possible, but 100 is a familiar convention rather than a law of nature.
Earth Site’s introduction to percentages and proportional calculation explains the arithmetic. The important habit is to identify the denominator: relative to what has the change been measured?
How to calculate a simple index
Use the relationship: index = observed value ÷ reference value × 100. The reference must be appropriate and clearly identified. It might be a particular year, a month or an average over a period.
In an invented example, a workshop makes 400 units in Year A, 460 in Year B and 500 in Year C. With Year A equal to 100, the indices are 100, 115 and 125. The calculation preserves the relative movements while removing the original unit from the displayed scale.
The index cannot fix problems in the underlying data. If the workshop changes what it calls a unit, expands its reporting coverage or begins counting unfinished products, the apparent trend may no longer compare like with like. Indexing is a transformation, not a quality guarantee.
What if the reference value is zero?
The simple formula cannot use a zero reference because division by zero is undefined. A different reference or a different way of presenting change is needed. Very small reference values can also generate strikingly large percentage changes that require context.
For instance, increasing an invented count from one to five produces an index of 500. That is a fivefold level, but only four additional items. Relative and absolute changes answer different questions, and both may be useful.
Index points are not percentage changes
If an index rises from 120 to 132, the increase is 12 index points. The percentage increase is 12 divided by 120, multiplied by 100, which equals 10%. Calling it a 12% increase would be wrong.
The confusion arises because, at a reference value of 100, index-point and percentage changes happen to have the same numerical size. A move from 100 to 112 is both 12 points and 12%. That coincidence does not continue from every starting value.
To compare any two periods, use: percentage change = later index ÷ earlier index × 100 − 100. The common reference scale cancels, provided both figures belong to a compatible series.
A fall does not undo an equal percentage rise
Start with 100 and increase it by 20%, reaching 120. A subsequent 20% fall takes away 24, leaving 96. The final value is below the starting point because the second percentage uses a different denominator.
To return from 120 to 100 requires a fall of about 16.7%, not 20%. This is not a peculiarity of indices; it is ordinary percentage arithmetic made visible through a reference scale.
What does re-referencing do?
Re-referencing changes the period set equal to 100. To make a chosen period the new reference, divide every old index value by the old value in that period and multiply by 100.
Return to the workshop series of 100, 115 and 125. If Year C becomes the reference, the new values are 80, 92 and 100. The workshop has not produced fewer goods because the earlier index now reads 80. Only the scale has changed.
The movement from Year A to Year B remains 15%: 92 divided by 80 equals 1.15. Re-referencing preserves proportional change, apart from possible effects of rounding in published figures.
Why the word rebasing needs care
Statistical agencies sometimes use rebasing for a wider process that can update weights, price references or methods as well as the index reference period. A pure rescaling and a methodological revision are not the same operation.
Read the accompanying explanation before concluding that all differences between two versions are cosmetic. If only the reference changes, ratios remain the same. If coverage or weights change, the measured history may change too, depending on the method.
Comparing two indexed series
Imagine two museums. Museum A receives 10,000 visits in the reference year and 12,000 later. Museum B receives 100,000 and then 110,000. Their later indices are 120 and 110 respectively.
Museum A has grown faster proportionally: 20% compared with 10%. Museum B still has many more visits and has gained more visits in absolute terms: 10,000 rather than 2,000. An indexed graph can show the first comparison clearly while hiding the second.
A headline saying A is now bigger because its index is higher would confuse growth with size. The correct statement identifies which quantity is being compared.
Why the chosen starting date matters
If the reference period is unusually high or low, subsequent movements can look especially dramatic or modest. Choosing a common starting point is useful, but it does not make the choice neutral for every analytical purpose.
Earth Site’s guide to reading graphs and identifying misleading presentation explains why dates and axes deserve scrutiny. Try a longer period and inspect the original levels before accepting a story based on two indexed lines.
How composite indices combine different things
Many official indices combine numerous items rather than tracking one quantity. A price index may combine changes in many goods and services. Its construction requires a method for giving those changes appropriate influence.
Weights express that influence. In a simple hypothetical fixed-weight example, suppose one component has a weight of 75% and rises by 4%, while another has a weight of 25% and rises by 12%. The weighted change is 0.75 × 4 plus 0.25 × 12, or 6%.
An unweighted average would give 8%, a different answer. Neither weighting can be judged without knowing the question. If the aim is to represent expenditure, spending shares are relevant. If the aim is something else, another design may be appropriate.
Different indices can both be valid
Two indices may cover different populations, products or concepts. Their disagreement does not automatically show that one is dishonest. It may show that they measure different things or use methods suited to different purposes.
Earth Site’s explanation of consumer price indices and their shopping-basket framework gives a practical application. This article focuses on interpreting index scales, rather than repeating the construction of particular inflation measures.
Why weights may need updating
What people buy and what businesses produce change over time. An index using an old pattern indefinitely may become less representative of the question it is meant to answer. Updating weights can help keep the measure relevant.
Chain-linking connects segments of an index calculated using updated information. It requires a defined method so that the join does not create an artificial jump mistaken for real change. Users should consult documentation rather than splice unrelated series together themselves.
A useful analogy is maintaining a long-running school survey. If the school adds new year groups, the analyst must decide how to compare the enlarged population with earlier years. Merely carrying forward an old label cannot resolve the change in coverage.
Precision and rounding
Published indices are often rounded. Recalculating a percentage change from rounded values may produce a slightly different result from the agency’s calculation using more precise underlying figures. That is not necessarily an inconsistency.
Do not invent precision by reporting many decimal places from limited inputs. If a source publishes an index to one decimal place, a long string of calculator digits does not represent newly acquired information.
A practical method for checking an index claim
First identify the concept: prices, quantities, values or something else. Next locate the reference period and read what 100 means. Then check whether the claim concerns an index-point difference, a percentage movement or a comparison of underlying levels.
After that, inspect the scope and method. Are the same places, items and populations included throughout? Were weights updated? Is the series seasonally adjusted? Those details can change the meaning of a short-term movement.
Finally, reproduce the key arithmetic. If the claim says a series rose by 25% while its index moved from 160 to 180, the calculation gives 12.5%, not 25%. A simple check can reveal an error that a persuasive graph conceals.
An exercise with a complete answer
A fictional reading programme has indices of 100, 120 and 150 across three terms. Growth between the second and third terms is 25%, because 150 divided by 120 is 1.25. Re-reference the third term to 100 and the series becomes approximately 66.7, 80 and 100.
The last movement is still 25%. However, the series alone cannot tell us how many books were read: the original reference quantity is missing. This single exercise separates calculation, re-referencing and the limits of what the data reveal.
Conclusion: a scale for change, not a substitute for context
Index numbers are powerful because they simplify comparison. Their usefulness depends on remembering what has been simplified away: original units, absolute size and sometimes the diversity of components hidden inside an average.
The central lesson is to read an index as a relationship to a defined reference. Check denominators, methods and coverage before interpreting the curve. With those habits, rebasing becomes understandable and dramatic-looking numbers become questions that can be tested rather than impressions that must be accepted.
Frequently asked questions
Does an index of 150 mean a 150% increase?
Not when the reference is 100. It means the level is 150% of the reference, which is a 50% increase.
Can an index exceed 100?
Yes. In a conventional reference-based index, 100 is not a maximum. Values above it indicate a level above the reference.
Does changing the reference year change actual growth?
Pure re-referencing does not, apart from rounding effects. Broader rebasing that changes weights or methods may alter results, so documentation matters.
Can I compare two countries’ index levels directly?
Only with care. If each country has its own starting level set to 100, the indices compare relative change, not necessarily absolute output, prices or living standards.
Why does a ten-point rise not always mean 10%?
The percentage depends on the starting value. A rise from 200 to 210 is ten index points but only 5%.
