CMA Foundation · Fundamentals of Business Mathematics and Statistics

Index Numbers and Time Series for CMA Foundation

Index numbers measure the average change in a group of related variables, such as prices, between a base period and a current period. Time series analysis studies data over time by splitting it into trend, seasonal, cyclical and irregular parts. In MCQs, you apply a formula to simple numbers, so know the formulas and check the base.

What this chapter covers

This chapter has two linked halves. The first half is index numbers. You learn how to compare prices or quantities across two periods using one number, usually with the base period fixed at 100. You build simple and weighted indices, check them with the time reversal and factor reversal tests, and then use them in practice for cost of living, base shifting, splicing and deflating.

The second half is time series. Here you look at data collected over time, such as yearly sales of a Mumbai retailer, and break it into trend (T), seasonal variation (S), cyclical variation (C) and irregular variation (I). You measure trend with moving averages and the least squares method, and then measure seasonal variation.

This chapter connects to the rest of Paper 3. It uses averages and ratios from statistics, and the least squares method reuses the straight-line idea from the algebra and regression work. Index numbers also help you in Paper 2 and Paper 4, where price level changes and inflation appear. Most questions are direct calculations, so this chapter rewards practice.

Paper 3 is fully objective, with 50 MCQs across all of Business Mathematics and Statistics, and there is no negative marking. Index numbers and time series questions are mostly formula-based with small, clean numbers. That makes them quick to score if you have the formulas ready and have practised the routine. Every question you get right here is 2 marks, and the calculations rarely need more than a minute. Since you must score at least 40% in each paper to pass, a chapter where practice converts so directly into marks is worth real effort.

Index Numbers and Time Series: topics in the order to study them

1Index Numbers: Meaning, Uses and TypesStart with the idea of a base year, a current year and what an index measures, because every later formula depends on it.
2Construction of Index Numbers: Simple and Weighted MethodsThis is the core formula set (Laspeyres, Paasche, Fisher and others), so learn it right after the basics.
3Tests of Adequacy: Time and Factor Reversal TestsThese tests check the formulas you just learned, so they make sense only once you can compute the indices.
4Cost of Living Index, Base Shifting, Splicing and DeflatingThese are applications of index numbers, and they use the same base and ratio logic, so they come after construction and tests.
5Time Series: Components and ModelsMove to the second half only after finishing index numbers, and learn the four components and the additive and multiplicative models first.
6Trend Measurement: Moving Averages and Least SquaresTrend is the main component to measure, and you need the models before you can remove or fit it.
7Seasonal Variation MeasurementSeasonal indices build on trend and averages, so study this last.

How to prepare Index Numbers and Time Series

Treat this chapter as a formula chapter with a calculation habit. Aim for accuracy first, then speed.

Write the formulas on one page, grouped as simple index, weighted indices, tests, and time series. Revise this page daily.
For each index formula, practise with a table of two or three items. Name the base period and the weights before you start calculating.
Learn what each test checks. The time reversal test uses P01 × P10 = 1, and the factor reversal test checks that price index × quantity index equals the value ratio. Fisher's index satisfies both.
Practise base shifting, splicing and deflating as short routines. Write the conversion in one line, then substitute.
For time series, practise moving averages with 3-year and 4-year periods. Remember that an even period needs centring.
For least squares, use the shortcut of taking the middle year as origin so that Σx = 0. This keeps numbers small and the arithmetic quick.
Finish with timed MCQ sets. Fisher's index is the geometric mean of Laspeyres and Paasche, so its value lies between the two. Use this only as a check on your last step: it shows whether your √(L × P) arithmetic is sensible, but it does not confirm that your Laspeyres and Paasche values are correct. Check each of those separately, and do not use Fisher to eliminate options before you have calculated both.

Common mistakes in Index Numbers and Time Series

Using current year quantities in Laspeyres or base year quantities in Paasche.
Fix: Remember that Laspeyres uses base quantities (Q0) and Paasche uses current quantities (Q1). Say it aloud once before each question.
Forgetting to multiply by 100, or multiplying twice.
Fix: Check whether the options are around 1 or around 100 and match your answer to that scale.
Taking the wrong base period in base shifting.
Fix: Write the formula first: new index = old index ÷ index of the new base year in the old series × 100. Then substitute.
Using the base shifting formula for splicing.
Fix: Splicing joins two series using an overlap year for which both have an index. To put the old series on the new base, use: spliced index = old index × (new-series index at the overlap year ÷ old-series index at the overlap year). If the overlap year is the new base year, the new-series index is 100, so this becomes old index × 100 ÷ old-series index of that year. To carry the new series back onto the old base, use: spliced index = new index × old-series index of the overlap year ÷ 100.
Not centring a 4-year (even) moving average.
Fix: Take the two-term average of consecutive moving averages so the values line up with actual years.
Making arithmetic errors in least squares because x values are not shifted to the middle year.
Fix: Take the middle year as origin so that Σx = 0. For an even number of years, the origin is midway between the two middle years. Take x in half-year units (…, -3, -1, 1, 3, …), so b is the change per half-year. Multiply b by 2 to get the change per year if the question asks for it.
Mixing up additive and multiplicative models when removing trend or seasonality.
Fix: In the additive model you subtract the trend, and in the multiplicative model you divide by it. Read the model given in the question before you start.

Last-day revision: Index Numbers and Time Series

Index number: a measure of the relative change in a variable between a base period and a current period, with the base usually set at 100.
Simple aggregative price index = (ΣP1 ÷ ΣP0) × 100.
Laspeyres uses base year quantities as weights: (ΣP1Q0 ÷ ΣP0Q0) × 100.
Paasche uses current year quantities as weights: (ΣP1Q1 ÷ ΣP0Q1) × 100.
Fisher's index is the geometric mean of Laspeyres and Paasche: √(L × P).
Time reversal test: P01 × P10 = 1. Factor reversal test: P01 × Q01 = ΣP1Q1 ÷ ΣP0Q0.
Deflated (real) value = current value ÷ price index × 100.
Time series components: trend, seasonal, cyclical and irregular. Additive model: Y = T + S + C + I. Multiplicative model: Y = T × S × C × I.
An even-period moving average must be centred by taking a second two-term average.
Least squares trend line: Y = a + bX. With Σx = 0, a = ΣY ÷ n and b = ΣxY ÷ Σx².
Seasonal variation repeats within a year, while cyclical variation spans longer periods.

Index Numbers and Time Series practice questions

Index Numbers and Time Series in other exams

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

Index Numbers and Time Series: frequently asked questions

Which formulas in Index Numbers and Time Series must I memorise for CMA Foundation?

Learn the simple aggregative index, Laspeyres, Paasche, Fisher, the two reversal tests, the deflating formula, the moving average method and the least squares equations. These cover most direct calculation MCQs. Write them on one page and revise it daily.

Is Fisher's index really the best index?

Fisher's index is called the ideal index because it satisfies both the time reversal and factor reversal tests. It is the geometric mean of Laspeyres and Paasche. For MCQs, remember it as the one that passes both tests.

How do I choose between the additive and multiplicative time series models?

The question usually names the model. In the additive model, the components are added and in the multiplicative model they are multiplied. Use subtraction or division accordingly when you remove a component.

How much time should I spend on this chapter?

Spend enough time to master the formulas and then practise timed MCQs until the calculations feel routine. Since the numbers are usually small, steady practice matters more than long reading. Revise the formula page often, as it is easy to forget which weight goes with which index.