Chain base index numbers Revision Notes for AQA GCSE Statistics

Chain base index numbers

What are chain base index numbers?

A chain base index number is a method of comparing values where each year's data is compared to the previous year's data, rather than to one fixed base year. Think of it like a chain where each link connects to the one immediately before it.

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The key difference from simple index numbers is that the base year changes every year. Instead of always comparing back to year 1 (like simple indices do), chain base indices compare this year to last year, then next year to this year, and so on.

The chain base formula

The formula for calculating a chain base index number is straightforward:

$\text{Chain base index number} = \frac{\text{value this year}}{\text{value last year}} \times 100$

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This formula tells you how this year's value compares to last year's value as a percentage. For example:

If the chain base index = 100, values stayed the same
If the chain base index = 120, values increased by 20%
If the chain base index = 95, values decreased by 5%

Using chain base index numbers to find new prices

You can work backwards from a chain base index number to calculate what a new price should be:

$\text{Price this year} = \frac{\text{chain base index number}}{100} \times \text{price last year}$

For instance, if last year's price was £200 and this year's chain base index number is 115, then:

Price this year = $\frac{115}{100} \times £200 = 1.15 \times £200 = £230$

This means the price has increased by 15% from last year.

Chain base vs simple index numbers

Understanding the difference between these two types is crucial:

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Simple index numbers:

Always compare back to one fixed base year
Formula: $\frac{\text{value}}{\text{value in base year}} \times 100$
The base year never changes

Chain base index numbers:

Compare each year to the immediately previous year
Formula: $\frac{\text{value this year}}{\text{value last year}} \times 100$
The base year changes every year

Worked example 1: CPI analysis

Let's look at Consumer Price Index (CPI) data showing percentage changes over three years.

The table shows CPI percentage changes: 1.5% in 2014, 0% in 2015, and 0.7% in 2016.

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Worked Example: CPI Chain Base Index Analysis

Part (a): Finding the chain base index number from 2015 to 2016

When CPI rises by 0.7% in 2016, the chain base index number is: 100 + 0.7 = 100.7

This means prices in 2016 were 0.7% higher than in 2015.

Part (b): Using chain base index numbers to find gold prices

Starting information:

Gold price at end of 2014: $1199.25 per ounce
Gold price at end of 2016: $1174 per ounce

Using the chain base index numbers:

In 2015: CPI change was 0%, so gold price should remain $1199.25
In 2016: Chain base index = 100.7, so expected gold price = $\frac{100.7}{100} \times \$ 1199.25 = $1207.64$

However, the actual gold price decreased to $1174, showing that gold prices didn't follow the general CPI trend.

Worked example 2: Car price analysis

Now let's examine car prices over three years to calculate chain base index numbers.

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Worked Example: Car Price Chain Base Analysis

Part (a): Calculating chain base index numbers using 2016 as base year

For 2017: Chain base index = $\frac{£18,900}{£18,500} \times 100 = \textbf{102.2}$

For 2018: Chain base index = $\frac{£19,300}{£18,900} \times 100 = \textbf{102.1}$

Part (b): Why the index numbers are different despite equal price rises

Both years saw £400 increases, but the percentages are different:

2016 to 2017: £400 increase on £18,500 = 2.2% rise
2017 to 2018: £400 increase on £18,900 = 2.1% rise

The same absolute increase represents a smaller percentage when the base value is larger. This demonstrates why chain base indices focus on percentage changes rather than absolute changes.

Key exam tips and common mistakes

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Key Points to Remember:

Always remember to:

Identify which year you're comparing to (the base changes each year)
Convert percentage changes to index numbers by adding to 100
Show your working clearly when calculating new prices
Check whether you need to find an index number or use one to find a price

Common exam traps to avoid:

Mixing up simple and chain base index calculations
Forgetting that 0% change gives an index of 100, not 0
Not showing intermediate steps in multi-year calculations
Confusing percentage change with index numbers

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Essential Formula Reminder:

Chain base index numbers compare each year to the previous year, not to one fixed base year
The formula is: $\frac{\text{value this year}}{\text{value last year}} \times 100$
You can find new prices using: $\frac{\text{chain base index}}{100} \times \text{last year's price}$
A chain base index of 120 means a 20% increase from last year
Equal absolute changes give different percentage changes when base values differ

Chain base index numbers (AQA GCSE Statistics): Revision Notes