A composite index joins the indices of several items into one number. The formula is I = Σ(w × I) ÷ Σw: multiply each index by its weight, add the products, then divide by the total weight.
Start with calculating a price index from two values if the single-item formula is not secure. The chapter overview is at Index numbers.
Why use weights?
Items matter differently. A family spends more on food than on stationery, so a rise in food prices should move the overall index more.
The weight records that importance. A weight of 5 against a weight of 2 means the first item counts two and a half times as much.
Worked example: with weights and without
Three groups in a household cost index, with 2020 as the base year:
| Item | Index (I) | Weight (w) | w × I |
|---|---|---|---|
| Food | 120 | 5 | 600 |
| Transport | 110 | 3 | 330 |
| Housing | 105 | 2 | 210 |
| Total | 10 | 1 140 |
Composite index = 1 140 ÷ 10 = 114.
Now average without weights: (120 + 110 + 105) ÷ 3 = 111.67. That is about 2.3 points lower, because the plain average gives housing the same say as food, even though food carries weight 5 and housing only 2.
The weighted answer is higher because the item with the biggest index also has the biggest weight.
The mistake that costs marks
The slip is averaging the indices, or dividing by the number of items instead of the total weight. In the example, dividing 1 140 by 3 gives 380, which is not a sensible index.
A sense check avoids it: a composite index must lie between the smallest and largest item index. Here that is between 105 and 120. If your answer is outside that range, the division is wrong.
Check yourself
Books have index 125 and weight 4. Stationery has index 110 and weight 6. Uniforms have index 90 and weight 10. Find the composite index.
Answer
Products: 125 × 4 = 500, 110 × 6 = 660, 90 × 10 = 900. The total is 2 060.
Total weight = 4 + 6 + 10 = 20.
Composite index = 2 060 ÷ 20 = 103.
Sense check: 103 lies between 90 and 125, and it is close to 90 because uniforms carry the biggest weight.
What to study next
Next, see how to find a missing weight or index in finding missing weights or component indices. Then try the index numbers practice set.
The mistake log and paper-error review helps you record whether your slips come from weights or from division. For a teacher to read your working, see online one-to-one Additional Mathematics tuition.