# Arithmetic or geometric mean to calculate an annual average price index?

I have monthly indices (Consumer Price Indices for 12 months) for a set of countries. I want to calculate the annual average for each country in order to have an average Consumer Price Index for a given year, not month.

Do I calculate the annual average by simply summing all 12 monthly data points (i.e. monthly indices) and divide by 12; or should I be using the geometric mean here?

From Wikipedia I see that

This makes the geometric mean the only correct mean when averaging normalized results; that is, results that are presented as ratios to reference values.

That being said, I'm not sure if the data I'm using can be categorized as "normalized data". Also, perhaps that I am using a reference year (the base year) may further complicate things.

It depends on what you want to use the average for.

A CPI of $x$ means it takes $x$ times as much money now to buy a basket of goods as it did during a reference period. (Sometimes values are multiplied by an arbitrary constant, such as $100$, so that a CPI of $x$ means it takes $x/100$ times as much money as before. The value of that constant does not matter: it merely establishes a unit of measurement of money. It will therefore be set to $1$ in the ensuing calculations.)

• Suppose you want to measure how much people typically needed to spend during the year in toto. You would likely assume they would buy this same basket at regular intervals (rather than varying it or varying the rates at which they purchased it), so without loss of generality let's suppose they bought it once a month, spending $x_i$ during month $i$. Then they spent a total of $$S(x)=\sum_{i\in \{\text{January, February,}\ldots\text{, December}\}} x_i,$$ whence their mean monthly price $S(x)/12$ was the arithmetic mean of the monthly prices.

If you have data about rates of purchase you could instead compute a rate-weighted arithmetic mean.

• Suppose you want to measure how much people with a fixed monthly income were able to purchase on average. This time assume they purchased as much as they could at regular intervals. During month $i$ they could purchase only $1/x_i$ as much as they could during the reference period. Now the total amount of goods purchased is $$H(x) = \sum_{i\in \{\text{January, February,}\ldots\text{, December}\}} \frac{1}{x_i}.$$ In this case the reciprocal of its mean--that is, the harmonic mean $12/H(x)$ of the monthly CPIs--is the right average to use.

Once again it would be possible to weight this harmonic mean by rates of purchase, if such information were available.

In any event, the harmonic mean cannot exceed the arithmetic mean (weighted or not, only provided both use the same weights). The two will be close when the ratio of the largest $x_i$ to the smallest $x_i$ is small--say, well under $2:1$. They will be identical when all the $x_i$ are equal.

The Wikipedia quotation does not apply because it refers to "normalized" values, or ratios, whose denominators could be varying or chosen arbitrarily. That is not the case with an index: the denominator is fixed during the (past) reference period. Its effect is only to establish the units in which money is expressed.

• You state that a "harmonic mean cannot exceed the arithmetic mean". In practice, I suspect this is true, but in theory, couldn't it if there were sufficient deflation? Jan 21 '15 at 18:30
• – whuber
Jan 21 '15 at 19:12
• Thank you for your detailed response. Now I saw a case from the International Labour Organization, where in the notes for the annual average for monthly CPIs for one country they state "Annual average is calculated as geometric mean of monthly indices." Is there indeed any case where the geometric mean would be desired to generate the annual average of monthly CPIs? Jan 21 '15 at 20:27
• I have been thinking hard about that, but have yet to come up with any rationale apart from a hand-waving "if you cannot make up your mind between the arithmetic mean (AM) and harmonic mean (HM), you might take the geometric mean (GM) because it is always the case that $\text{HM}\le\text{GM}\le\text{AM}$." But that's not terribly satisfactory, is it? One might also imagine adopting a model of exponential growth in CPI, for which the GM might be close to a median monthly growth--but then why not just use the median itself if you want that?
– whuber
Jan 21 '15 at 21:14
• Yes, the AM seems more realistic. But the HM could be applicable to, say, anything purchased for fixed amounts at intervals, such as ongoing investments.
– whuber
Jan 23 '15 at 3:40

For annual, it is usually called year-over-year, i.e. you get two price index levels one year apart. So, if your index is monthly, then it is $$inflation_t=\frac{cpi_t}{cpi_{t-12}}-1$$ or $$inflation_t=\ln\frac{cpi_t}{cpi_{t-12}}$$.

• I think you're confused. The year over year change of the annual CPI is indeed the inflation rate between the two yearly periods. I, however, am not interested in that. I have a CPI series for say Jan through Dec for a given year and what to know how to best summarize the CPI for that given year: Jan 21 '15 at 17:34
• If you are interested in CPI level, then get the end of year. Sometimes beginning of the year or mid year are used. You can do average too, depending on what's the purpose. Jan 21 '15 at 17:59