How to calculate a composite index

A composite index combines the price indices of several items into a single figure that describes an overall price change, for a basket of goods, the ingredients of a product, or a household's spending. Because some items matter more than others, each index $I_{i}$ is multiplied by a weight $w_{i}$, and the weighted mean is taken: $\bar{I} = \frac{\sum I_{i} w_{i}}{\sum w_{i}}$.

The weights may be given as amounts, ratios, or percentages; only their proportions matter. It answers questions such as 'find the composite index for the three ingredients' or 'find the overall price index of the basket'.

A composite index of $132$ means the whole basket costs $132\%$ of its base-year cost, a rise of $32\%$. Each individual index usually comes from the price-index step first.

Reach for a composite index whenever a question lists several items, each with its own price index (or enough data to find one), together with weights, and asks for a single overall figure. Signals are 'composite index', 'weighted index', a table with columns for price index and weight, or a phrase like 'in the ratio $4 : 3 : 3$'.

If only prices are given, first turn each item's prices into a price index, then weight and combine. Watch for weights given as a ratio, you may use the ratio numbers directly as the $w_{i}$, because any common factor cancels between the numerator $\sum I_{i} w_{i}$ and the denominator $\sum w_{i}$.

This is the step that pulls a whole chapter's work into one number, so it is a favourite for the longer, structured questions in the paper.

1. 1

   Find each price index

   If not already given, compute each item's $I_{i} = \frac{P_{1}}{P_{0}} \times 100$ first.

2. 2

   List the weights

   Write each item's weight $w_{i}$; a ratio such as $4 : 3 : 3$ can be used directly.

3. 3

   Multiply index by weight

   Form each product $I_{i} w_{i}$.

4. 4

   Add the products

   Compute $\sum I_{i} w_{i}$, the total of the weighted indices.

5. 5

   Add the weights

   Compute $\sum w_{i}$, the total weight.

6. 6

   Divide

   The composite index is $\bar{I} = \frac{\sum I_{i} w_{i}}{\sum w_{i}}$.

7. 7

   Interpret

   Compare with $100$: the amount above $100$ is the overall percentage rise.

• Forgetting to weight, taking the plain mean $\frac{120 + 130 + 150}{3}$ instead of using the weights.
• Dividing by the number of items instead of by $\sum w_{i}$.
• Multiplying the weights together, or adding $I_{i} + w_{i}$, instead of forming the product $I_{i} w_{i}$.
• Getting a composite index outside the range of the individual indices, it must lie between the smallest and largest.
• Rounding each product early, so the total drifts away from the exact value.

The mistake that costs marks is dropping the weights, students average the three indices and divide by $3$, which ignores that the items matter unequally. In one-to-one lessons our teachers set the work out as a small table: one column for $I_{i}$, one for $w_{i}$, one for the product $I_{i} w_{i}$, so the numerator $\sum I_{i} w_{i}$ and denominator $\sum w_{i}$ can be read straight off.

We finish with a range check, the composite must land between the smallest and largest index, which catches slips instantly. Because SPM Add Math is marked analytically, a correctly formed $\frac{\sum I_{i} w_{i}}{\sum w_{i}}$ already earns method marks even if a product is mis-added.

Lessons are taught in English, while the SPM paper is set bilingually. Every teacher on spmaddmath.com.my is experienced, with lessons online at your own pace.