Introduction to Index Numbers
Welcome to the study notes on Index Numbers! Have you ever heard in the news that "the cost of living has gone up by \(5\%\)" or that "inflation has slowed down"? How do economists and statisticians track the changing prices of thousands of items over time? The answer is Index Numbers.
In this chapter, you will learn how index numbers turn complicated sets of figures into simple, easy-to-compare percentages. Don't worry if this seems a bit mathematical at first — we will break down every single formula and step with real-life examples like pocket money, pizza prices, and gaming consoles!
1. What is an Index Number?
An index number is a statistical measure designed to show the relative change in a variable (such as price, quantity, or cost) over time, compared to a specific starting point called the base year (or base period).
Think of an index number as a scoreboard where the starting point is always reset to \(100\). This makes it super easy to spot percentage changes at a glance!
Key Terms to Learn:
• Base Year (or Base Period): The chosen point in time used as a benchmark for comparison. The index number for the base year is ALWAYS \(100\).
• Current Year (or Given Period): The time period you want to compare against the base year.
• Simple Index Number: An index number calculated for a single item or commodity.
Did you know? The year chosen as a base year is usually a "normal" year without extreme disruptions (like economic crises or natural disasters) so that the comparisons remain fair and meaningful.
2. Calculating Simple Index Numbers
To calculate a simple price index number, you divide the value in the current period by the value in the base period, and then multiply by \(100\).
The Core Formula:
\(\text{Index Number} = \frac{\text{Current Value}}{\text{Base Value}} \times 100\)
We often write this symbolically as:
\(I = \frac{P_n}{P_0} \times 100\)
Where:
• \(P_n\) is the price/value in the current year \(n\)
• \(P_0\) is the price/value in the base year \(0\)
• \(I\) is the index number
Step-by-Step Example:
In the year \(2020\) (the base year), a cinema ticket cost \(£8.00\). In \(2024\), the same ticket cost \(£10.00\). Calculate the price index number for \(2024\) based on \(2020 = 100\).
Step 1: Identify the base value (\(P_0\)) and current value (\(P_n\)).
\(P_0 = £8.00\)
\(P_n = £10.00\)
Step 2: Apply the formula.
\(\text{Index Number} = \frac{10.00}{8.00} \times 100\)
\(\text{Index Number} = 1.25 \times 100 = 125\)
Step 3: State the final answer.
The index number for \(2024\) is \(125\).
Key Takeaway:
An index number has no units (you do not write \(£\) or \(\%\) next to the index number itself, though it represents a percentage relative to \(100\)).
3. Interpreting Index Numbers
The main reason we use \(100\) as the base value is that it makes calculating percentage changes effortless.
How to Read an Index Number:
• If the Index is greater than \(100\): There has been an increase.
Example: An index of \(118\) means an increase of \(118 - 100 = 18\%\) compared to the base year.
• If the Index is less than \(100\): There has been a decrease.
Example: An index of \(93\) means a decrease of \(100 - 93 = 7\%\) compared to the base year.
• If the Index is equal to \(100\): The price or value has remained unchanged.
Working Backwards (Finding the Actual Price):
Sometimes an exam question gives you the index number and the base year price, and asks you to find the current price.
Rearranged Formula:
\(\text{Current Value} = \frac{\text{Index Number} \times \text{Base Value}}{100}\)
Example: The price index for a pair of trainers in \(2023\) is \(140\) based on \(2019 = 100\). If the trainers cost \(£60\) in \(2019\), how much did they cost in \(2023\)?
\(\text{Price in 2023} = \frac{140 \times £60}{100} = £84\)
Common Mistake to Avoid:
Mistake: Saying an index of \(135\) means the price increased by \(135\%\).
Correct: An index of \(135\) means the price is now \(135\%\) of the original, which is an increase of \(35\%\).
4. Chain Base Index Numbers vs. Fixed Base Index Numbers
There are two primary ways to calculate index numbers across several years: Fixed Base and Chain Base.
A. Fixed Base Index Numbers
In a fixed base system, every single year is compared back to the same original starting year.
\(\text{Fixed Base Index} = \frac{\text{Value in Current Year}}{\text{Value in Fixed Base Year}} \times 100\)
Best used for: Looking at long-term trends over many years.
B. Chain Base Index Numbers
In a chain base system, each year is compared directly to the immediately preceding year (the year right before it). The base year moves forward each step!
\(\text{Chain Base Index} = \frac{\text{Value in Current Year}}{\text{Value in Previous Year}} \times 100\)
Best used for: Examining year-on-year changes (short-term fluctuations).
Comparison Example:
Consider the price of a concert ticket over three years:
• Year \(1\): \(£40\)
• Year \(2\): \(£50\)
• Year \(3\): \(£55\)
Fixed Base (Year \(1 = 100\)):
• Year \(1\): \(\frac{40}{40} \times 100 = 100\)
• Year \(2\): \(\frac{50}{40} \times 100 = 125\) (\(25\%\) increase since Year 1)
• Year \(3\): \(\frac{55}{40} \times 100 = 137.5\) (\(37.5\%\) increase since Year 1)
Chain Base:
• Year \(1\): Base / Not calculated (or \(100\))
• Year \(2\): \(\frac{50}{40} \times 100 = 125\) (\(25\%\) increase from Year 1 to Year 2)
• Year \(3\): \(\frac{55}{50} \times 100 = 110\) (\(10\%\) increase from Year 2 to Year 3)
Key Takeaway:
Fixed base compares everything to one anchor point. Chain base links each year to the year before it.
5. Weighted Index Numbers (Composite Index Numbers)
In real life, people spend more money on housing and food than on chewing gum. If chewing gum doubles in price, it doesn't affect your budget much. But if your rent doubles, it has a massive impact!
To reflect this, we use weights to give more important items greater influence when calculating an overall combined index number (also called a composite index).
The Weighted Index Formula:
\(\text{Weighted Index} = \frac{\sum (I \times W)}{\sum W}\)
Where:
• \(I\) is the individual index number for each item
• \(W\) is the weight (importance or proportion spent) for each item
• \(\sum\) means "the sum of" (add them all up)
Memory Aid:
Remember the three steps: Multiply, Add, Divide!
1. Multiply each index by its weight (\(I \times W\)).
2. Add all the products together (\(\sum (I \times W)\)).
3. Divide by the total of the weights (\(\sum W\)).
Step-by-Step Worked Example:
A student tracks three main weekly spending categories: Transport, Food, and Entertainment. The table below shows the index numbers and weights for this year:
• Transport: Index (\(I\)) = \(110\), Weight (\(W\)) = \(3\)
• Food: Index (\(I\)) = \(120\), Weight (\(W\)) = \(5\)
• Entertainment: Index (\(I\)) = \(105\), Weight (\(W\)) = \(2\)
Calculate the overall weighted index number.
Step 1: Calculate \(I \times W\) for each category.
• Transport: \(110 \times 3 = 330\)
• Food: \(120 \times 5 = 600\)
• Entertainment: \(105 \times 2 = 210\)
Step 2: Find the sum of \((I \times W)\).
\(\sum (I \times W) = 330 + 600 + 210 = 1140\)
Step 3: Find the sum of the weights (\(\sum W\)).
\(\sum W = 3 + 5 + 2 = 10\)
Step 4: Divide \(\sum (I \times W)\) by \(\sum W\).
\(\text{Weighted Index} = \frac{1140}{10} = 114\)
Interpretation: Overall, weekly spending across all categories has increased by \(14\%\) (since \(114 - 100 = 14\%\)).
6. Real-World Applications: RPI and CPI
In the UK, government statisticians calculate two very famous weighted index numbers to measure inflation (the rate at which prices are rising):
1. Consumer Price Index (CPI)
• Measures the average change in prices of a representative "basket of goods and services" bought by typical households.
• Used by the Bank of England to set interest rates and measure government inflation targets.
2. Retail Price Index (RPI)
• An older measure of inflation in the UK.
• Includes housing costs such as mortgage interest payments and council tax, which are excluded from the CPI.
• Often used for calculating changes in pension payouts, student loan interest rates, and train fares.
How the Basket of Goods Works: The items in the "basket" are updated every year to reflect modern shopping habits. For example, cassette tapes were removed years ago, while streaming subscriptions and electric vehicles have been added!
7. Quick Summary and Revision Checklist
Before heading into exam questions, make sure you can tick off each of these points:
• Base Year Value: The base year index is always \(100\).
• Simple Index Formula: \(\frac{\text{Current Value}}{\text{Base Value}} \times 100\)
• Percentage Change: An index of \(108.5\) means an increase of \(8.5\%\); an index of \(94.2\) means a decrease of \(5.8\%\).
• Fixed vs. Chain Base: Fixed base compares to a single starting year; chain base compares to the preceding year.
• Weighted Index Formula: \(\frac{\sum (I \times W)}{\sum W}\)
• No Units: Never write currency signs or percentage signs directly attached to an index number value.