Index Numbers

Index numbers are statistical measures that show how a particular phenomenon changes over time or in comparison to a reference point (known as the base).

They are used to compare values across different periods or contexts, indicating how a statistic has shifted relative to a starting or reference point.

How do you calculate an index number?

An index number is calculated by dividing a given statistical value by a reference value (base) and multiplying by 100.

$$ \text{index number} = \frac{\text{value}}{\text{base}} \cdot 100 $$

This allows the change to be expressed as a percentage.

Types of Index Numbers

There are two main types of index numbers:

• Fixed-base index number
  In this type, values are always compared to a fixed reference point (such as a base year).

  Example. Suppose the price of a product was 50 euros in 2020 and increased to 60 euros in 2023. Using 2020 as the base year, the index number for 2023 would be: $$ \text{Index number} = \frac{\text{2023 price}}{\text{2020 price}} \times 100 = \frac{60}{50} \times 100 = 120 $$ This indicates that the product's price has increased by 20% compared to 2020. An index number of 120 means the 2023 price is 120% of the 2020 price. All values are compared against 2020 as the reference point.

• Chain-base index number
  In this type, values are compared consecutively, with each value compared to the one immediately before it, rather than to a fixed point. This is useful for measuring period-to-period changes, allowing you to track growth or decline from one period to the next.

  Example: If a product cost 50 euros in 2020 and 55 euros in 2021, the index number relative to 2020 would be: $$ \frac{\text{2021 price}}{\text{2020 price}} \times 100 = \frac{55}{50} \times 100 = 110 $$ In 2022, the price dropped to 53 euros. The index number relative to 2021 (chain base) would be: $$ \frac{\text{2022 price}}{\text{2021 price}} \times 100 = \frac{53}{55} \times 100 = 96.36 $$

The key difference between fixed-base and chain-base index numbers is how values are compared over time:

With a fixed base, comparisons are made consistently against a specific year or reference point, whereas with a chain base, the reference point shifts over time.

A Practical Example

Here’s a table showing how the price evolved from 2020 to 2024, along with the index numbers calculated using both a fixed base (with 2020 as the reference) and a chain base (where each year is compared to the previous one).

Which is better: Fixed or Chain Base?

The right choice depends on the goal of your analysis and the type of trend you're studying.

If you're looking to track the long-term evolution of a phenomenon relative to a stable reference point, a fixed base is more appropriate.

However, if you want to monitor short-term changes or fluctuations between consecutive periods, a chain base gives a more detailed and up-to-date perspective.

Example. A fixed base is ideal for long-term comparisons relative to a consistent reference point (like a base year) and for analyzing overall changes over time (e.g., inflation). A chain base is more useful for tracking short-term changes, highlighting consecutive period fluctuations (e.g., monthly sales or quarterly performance).

And so on.