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Fundamentals of Business Mathematics and Statistics · Index Numbers and Time Series

Cost of Living Index, Base Shifting, Splicing and Deflating

Updated 10 October 2026 · Fact-checked

A cost of living index (consumer price index) measures how the price of a fixed family basket changes over time. Use CLI = ΣPW ÷ ΣW, where P is the price index of each item and W its weight. Base shifting changes the base year, splicing joins two series, and deflating converts nominal values into real values.

Understand Cost of Living Index, Base Shifting, Splicing and Deflating

A cost of living index (also called the consumer price index, CPI) tells you how much more or less a typical family must spend to keep the same standard of living, compared with a base year. A reading of 150 means the same basket now costs 50% more than in the base year.

Different families spend differently, so each item gets a weight. Weights are usually the amounts spent on the item in the base year, or the quantities consumed. Food gets a bigger weight than entertainment because it takes more of the budget.

Base shifting means moving the base of an index series from one year to another. You do this when the old base is too old, or when you want to compare two series that have different bases. Splicing means joining two overlapping index series with different bases into one continuous series, because an old series was discontinued and a new one started.

Deflating means correcting a money (nominal) value for price changes to get the real value. If your salary rises 20% but prices rise 25%, your real income has fallen. Dividing by the price index removes the effect of inflation.

Key formulas to remember

Cost of living index (family budget or aggregate expenditure method)
CLI = ΣPW ÷ ΣW, where P = (p1 ÷ p0) × 100 and W = p0 × q0
P is the price relative of each item. W is the base-year expenditure weight. Also called the weighted average of price relatives.
Aggregate expenditure method
CLI = (Σp1q0 ÷ Σp0q0) × 100
Same result as the family budget method when weights are W = p0q0. This is the Laspeyres index.
Base shifting
New base index = (Old index ÷ Index of the new base year on the old base) × 100
Divide every value of the old series by the old index of the year chosen as the new base.
Splicing (forward, old series to new base)
Spliced index = Old index × (Overlap-year index of new series ÷ 100)
Use this to bring the old series onto the base of the new series. Overlap year is the year common to both series.
Splicing (backward, new series to old base)
Spliced index = New index × (100 ÷ Overlap-year index of the old series)
Use this to extend the new series back onto the old base, with the overlap year taken from the old series.
Deflating (real value)
Real value = (Nominal value ÷ Price index) × 100
Real wage = (Money wage ÷ CPI) × 100. Use the price index of the same year as the nominal value.
Purchasing power of money
Purchasing power = (1 ÷ Price index) × 100
It falls when prices rise.

How to solve Cost of Living Index, Base Shifting, Splicing and Deflating questions

Identify which of the four tasks the question asks for, then apply the matching formula carefully.

  1. 1Read the question and decide the task: cost of living index, base shifting, splicing or deflating.
  2. 2For a cost of living index, list each item with its price index P (or base and current prices) and its weight W.
  3. 3If prices are given, compute P = (p1 ÷ p0) × 100 for each item. If weights are given as quantities, compute W = p0 × q0.
  4. 4Compute ΣW and ΣPW, then divide: CLI = ΣPW ÷ ΣW.
  5. 5For base shifting, find the old index of the year to become the base, then divide every old value by it and multiply by 100.
  6. 6For splicing, identify the overlap year and the base you want. Multiply or divide by the overlap factor as per the formula.
  7. 7For deflating, divide each nominal value by its own year's price index and multiply by 100.
  8. 8Check that the base year gives 100 and the answer is sensible: rising prices should give an index above 100.

Quickest way: Weighted total, then one division

When to use it: Use for CLI numericals where the options are widely spaced and the numbers are simple.

  1. Write P and W in two columns and multiply row by row to get PW.
  2. Add W and PW once each, using round numbers where possible.
  3. Divide ΣPW by ΣW and match the answer to the nearest option.
  4. For base shifting and deflating, a single division by the new base index (then × 100) gives every value. Do one value first and check the pattern.
  5. If the weights sum to 100, ΣPW ÷ 100 is the answer directly.

Common mistakes in Cost of Living Index, Base Shifting, Splicing and Deflating

  • Dividing ΣPW by the number of items instead of ΣW.

    Students confuse the weighted average with a simple average.

    Fix: Always divide by the sum of weights, ΣW, never by n.

  • Using current-year quantities or prices as weights in the family budget method.

    Students mix up base-year and current-year data.

    Fix: Weights are base-year expenditure p0q0 unless the question says the weights are given directly.

  • Multiplying instead of dividing when deflating.

    Students think a higher index means a higher real value.

    Fix: Real value = nominal ÷ index × 100. A higher index shrinks the real value.

  • Using the wrong overlap factor in splicing.

    Students mix up forward and backward splicing.

    Fix: Decide which base you want first. To keep the new series' base, multiply old values by new overlap index ÷ 100. To keep the old base, multiply new values by 100 ÷ old overlap index.

  • Forgetting to multiply by 100 after base shifting.

    The division gives a ratio like 1.25, and students stop there.

    Fix: The new base year must show 100. Check this before finalising.

Worked examples

Example 1

Find the cost of living index for the current year from this data. Item: Food, price index 150, weight 50. Rent, price index 120, weight 20. Clothing, price index 130, weight 10. Fuel, price index 110, weight 10. Others, price index 100, weight 10.

Show the solution
  1. ΣW = 50 + 20 + 10 + 10 + 10 = 100.
  2. PW: Food = 150 × 50 = 7,500. Rent = 120 × 20 = 2,400. Clothing = 130 × 10 = 1,300. Fuel = 110 × 10 = 1,100. Others = 100 × 10 = 1,000.
  3. ΣPW = 7,500 + 2,400 + 1,300 + 1,100 + 1,000 = 13,300.
  4. CLI = 13,300 ÷ 100 = 133.

Answer: Cost of living index = 133

Example 2

A worker's monthly wage was ₹30,000 in the base year (index 100) and is ₹45,000 now, when the consumer price index is 180. Find the real wage now and the real change compared with the base year.

Show the solution
  1. Real wage now = (45,000 ÷ 180) × 100.
  2. 45,000 ÷ 180 = 250, so real wage = 250 × 100 = ₹25,000.
  3. Base-year real wage = ₹30,000, since the index is 100.
  4. Change = 25,000 − 30,000 = −₹5,000, a fall.
  5. Percentage fall = 5,000 ÷ 30,000 × 100 = 16.67% approximately.

Answer: Real wage = ₹25,000, which is ₹5,000 lower than the base-year wage despite the money wage rising.

Exam tips

  • Check whether the question gives price indices P directly or prices p0 and p1. If prices, compute P first.
  • In deflating questions, always divide by the index of the same year as the nominal value.
  • For splicing, read the direction asked: old series to the new base, or new series to the old base.
  • With no negative marking, always attempt every question. Eliminate options that put a base year index at anything other than 100.
  • Many questions have weights that add to 100. Use ΣPW ÷ 100 to save time.

Practice questions from Index Numbers and Time Series

Cost of Living Index, Base Shifting, Splicing and Deflating in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Cost of Living Index, Base Shifting, Splicing and Deflating: frequently asked questions

What is the formula for the cost of living index?

CLI = ΣPW ÷ ΣW, where P is the price index of each item and W is its weight. If you have prices and quantities, it equals (Σp1q0 ÷ Σp0q0) × 100.

What is the difference between base shifting and splicing?

Base shifting moves one index series to a new base year. Splicing joins two series that have different bases and overlap in at least one year into a single continuous series.

What does deflating an index number mean?

Deflating converts a nominal (money) value into a real value by dividing by the price index and multiplying by 100. It removes the effect of price changes so that you can compare values across years.

Why does a rise in the cost of living index reduce purchasing power?

A higher index means the same basket costs more. Each rupee buys less, so purchasing power, which is 100 ÷ index, falls.