Index numbers

What are index numbers?

An index number is a number (or a series of numbers) that shows how a price or value has changed compared with the price or value in a specified earlier time.

A term you’ll see often is the base year.

A base year is the year you choose as the reference point for comparison. It’s usually assigned an index value of $100.$ New base years are introduced from time to time to keep data current. Any year can be used as a base year, but analysts typically choose a recent year.

Here’s the formula for calculating an index number:

$$\frac{\text{Value in any given year}}{\text{Value in base year}}\times 100\%$$

Let’s work through an example to see how this works.

Year	2015	2016	2017	2018	2019	2020
Sales	30000	40000	25000	50000	70000	10000

Now we’ll calculate the index numbers using 2015 as the base year. That means the index number for 2015 is $100$, and every other year is compared to 2015.

Remember: any year can be used as a base year. The “best” choice depends on why you’re calculating the index.

Solution

For 2016:

$$\frac{40000}{30000}\times 100\%$$

$$=133,33$$

So, the index number for 2016 is 133,33.

Now calculate the remaining years the same way:

$$\frac{25000}{30000}\times 100\%$$

$$=83,33$$

$$\frac{50000}{30000}\times 100\%$$

$$=166,67$$

$$\frac{70000}{30000}\times 100\%$$

$$=233,33$$

$$\frac{10000}{30000}\times 100\%$$

$$=33,33$$

Now all together in one table:

Year	2015	2016	2017	2018	2019	2020
Index numbers	100	133,33	83,33	166,67	233,33	33,33

How do you interpret index numbers?

An index number tells you how a value compares to the base year.

• In 2016, the index is 133,33. That means sales were $33.33\%$ above the base year.
• In 2020, the index is 33,33. That means sales were $66.67\%$ below the base year.

Keep in mind that everything is being compared to the base year of 2015, which has an index of $100.$

What happens when a company wishes to change the base year?

In an exam question, you might be asked to calculate index numbers using a new base year. What you use depends on what information you’re given:

• If you’re given the original data and the index numbers, you can use either.
• If you’re given only index numbers, you can still re-base the series using those index numbers.

Original data

Year	2015	2016	2017	2018	2019	2020
Sales	30000	40000	25000	50000	70000	10000

Index numbers

Year	2015	2016	2017	2018	2019	2020
Index numbers	100	133,33	83,33	166,67	233,33	33,33

This means you can use either the original data or the index numbers to calculate a new set of index numbers.

Assume the new base year is 2016. Then 2016 must have an index of $100$, and the other years are recalculated relative to 2016.

Year	2015	2016	2017	2018	2019	2020
Index numbers	75	100	62,5	125	175	25

Keep in mind that you can get the same results whether you use the original data or the index numbers.

Choosing a base year

A base year shouldn’t be too far in the past (for example, $50$ years ago), because conditions may have changed so much that comparisons become less meaningful. For example, it’s hard to make a like-for-like comparison between business results in 1924 and 2024.

A base year should also not be the year with the highest or the lowest results. From the example above, we shouldn’t choose 2019 or 2020 because 2019 has the highest and 2019 has the lowest results. If you choose:

• a year with the highest results, most other years will look worse than they really are
• a year with the lowest results, most other years will look better than they really are

Either choice can mislead users of the index.

Combining a series of index numbers

Entities often change base years over time. That can leave you with index numbers calculated using different base years. If the company wants one continuous series, the index numbers must be combined onto a single base year.

Let’s bring back the index numbers from the beginning.

Original data

Year	2015	2016	2017	2018	2019	2020
Sales	30000	40000	25000	50000	70000	10000

Index numbers (base year 2015)

Year	2015	2016	2017	2018	2019	2020
Index numbers	100	133,33	83,33	166,67	233,33	33,33

The following is the original data from 2021 to 2024, along with the index numbers.

Year	2021	2022	2023	2024
Index numbers	30000	50000	70000	100000

The following are the index numbers (2021 base year)

Year	2021	2022	2023	2024
Index numbers	60	100	140	200

Now let’s combine the previous series with the current series. Keep in mind that the current series already has the current base year of 2022, meaning that we will only need to recalculate 2020 going backwards.

Year	2015	2016	2017	2018	2019	2020	2021	2022	2023	2024
Index numbers	60	80	50	100	140	20	60	100	140	200

Keep in mind that our base year is 2022, the 100 in 2020 it’s because the original data is the same ($\$50000$). Keep in mind that when we combined the series, we used the original data from both sets as follows:

30000	X	100%
50000			=60

In some situations, however, it’s possible to use index numbers rather than original data, especially if the new base year is based on the first data set. In our previous example, if our new base year was 2020 (which is from the first data set), we could use index numbers to recalculate the index numbers, which was impossible to do in our current situation because our base year was based on the new data set. If you are still struggling, please pause and go back at it again because what’s coming next is more challenging!