5.2 Calculating the price indices

This subsection concentrates on how the RPI is calculated. Generally the CPI is calculated in a similar way, though some of the details differ. To measure price changes in general, it is sufficient to select a limited number of representative items to indicate the price movements of a broad range of similar items. For each section of the RPI, a number of representative items are selected for pricing. The selection is made at the beginning of the year and remains the same throughout the year. It is designed in such a way that the price movements of the representative items, when combined using a weighted mean, provide a good estimate of price movements in the section as a whole.

For example, in 2012 the representative items in the ‘Bread’ section (which is contained in the ‘Food and catering’ group) were: large white sliced loaf, large white unsliced loaf, large wholemeal loaf, bread rolls, garlic bread. Changes in the prices of these types of bread are assumed to be representative of changes in bread prices as a whole. Note that although the price ratio for bread is based on this sample of five types of bread, the calculation of the appropriate weight for bread is based on all kinds of bread. This weight is calculated using data collected in the Living Costs and Food Survey.

One aim of the RPI is to make it possible to compare prices in any two months, and this involves calculating a value of the price index itself for every month.

So, calculating the RPI involves two kinds of data:

• the price data, collected every month

• the weights, representing expenditure patterns, updated once a year.

Once the price data have been collected each month, various checks, such as looking for unbelievable prices, are applied and corrections made if necessary. Checking data for obvious errors is an important part of any data analysis.

Then an averaging process is used to obtain a price ratio for each item that fairly reflects how the price of the item has changed across the country. The exact details are quite complicated and are not described here. (If you want more details, they are given in the Consumer Price Indices Technical Manual, available from the ONS website. Consumer Price Indices: A brief guide is also available from the same website.)

For each item, a price ratio is calculated that compares its price with the previous January. For instance, for November 2011, the resulting price ratio for an item is an average value of

The next steps in the process combine these price ratios, using weighted means, to obtain 14 subgroup price ratios, and then the group price ratios for the five groups. Finally, the group price ratios are combined to give the all-item price ratio. This is the price ratio, relative to the previous January, for the ‘basket’ of goods and services as a whole that make up the RPI.

The all-item price ratio tells us how, on average, the RPI ‘basket’ compares in price with the previous January. The value of the RPI for a given month is found by the method described in Section 4, that is, by multiplying the value of the RPI for the previous January by the all-item price ratio for that month (relative to the previous January):

Thus, to calculate the RPI for November 2011, the final step is to multiply the value of the RPI in January 2011 by the all-item price ratio for November 2011.

Example 22 is the subject of the following screencast. [Note that references to ‘the unit’ should be interpreted as ‘this course’. The original wording refers to the Open University course from which this material is adapted.]

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Transcript: Screencast 5 Calculating an RPI

INSTRUCTOR

Here’s an example out of the unit of calculating the Retail Prices Index. And what we’re asked to do is to calculate the Retail Prices Index for a particular month, November 2011.

So the first step in doing that is to collect the prices that it’s going to be based on and the weights. Well, actually, I’m not going to ask you to go out and collect hundreds and thousands of prices or anything like that. We’re just going to use the data on prices that have been collected by the Office for National Statistics.

And I’ve got some information off their website. And, similarly, I’ve got the weights from the website as well. And we’re just going to start off at that stage and write down what they are.

So here’s the sort of table that you do these calculations in. Down the side here, we’ve got just the five top-level groups that items are divided into in the calculation of the RPI. And we’ve got a column for the price ratios. We’ll come to that in a minute. And we’ve got a column for the 2011 weights. And I’m just simply going to fill in the numbers and tell you a little bit about what they mean.

So the weight for the food and catering group is 165. And carrying down, the weight for alcohol and tobacco is 88. The weight for housing and household expenditure is 408. And the weight for personal expenditure is 82. And the weight for travel and leisure is 257.

Now there’s a row we need to fill in at the bottom of this table as usual. And that’s the row for the sum. And if you add up these numbers – you can check this yourself, if you like – they come to 1000. Actually, you might not want to check. They always come to 1000 with RPI calculations because they’re designed that way.

And just to remind you of what these things mean, when the weight for food and catering is 165, that means that out of every £1000 the average household spends, 165 of those £1000 go on food and catering. And 88 go on alcohol and tobacco, and so on. And these are derived, again, by the Office for National Statistics statisticians from a survey. They’re the weights that were used in 2011. And they’re based on what people spent their money on, essentially, in the middle of the previous year, 2010.

So what’s the next step? The next step is to calculate the price ratios. You’ve already seen in the table that there’s a column for them. And, again, I’m not going to ask you to go and work these things out for yourselves. You haven’t even got the information to do it from.

So I’m just going to put in the price ratios which I found on the Office for National Statistics website. And they’re price ratios for November 2011, the month we’re interested in, relative to the previous January, that is to January 2011. With the RPI, they’re always used relative to the previous January.

So we’ll just do that again. Let’s write in the first one. The first one for food and catering is 1.030.

And that means that, on average, people spent £1.03 in November to buy the same stuff that they would have spent £1 on in January. It’s the ratio of the price in November to the price in January. And that’s simply what it is.

And you can go in and write in the other ones the same way. So I’ll do that. Alcohol and tobacco is 1.050. So alcohol and tobacco’s gone up a bit more than food and catering had. Housing and household expenditure is 1.037. So that’s gone up somewhere between the previous two.

Personal expenditure had gone up rather a lot. It’s 1.128. And travel and leisure had gone up rather less, 1.026. And we don’t need the sum of those figures. So I’ll leave this space here blank. And those are the price ratios.

So the next step is to calculate the all-item price ratio, the all-item price ratio for November 2011 relative to the previous January, January 2011. And this is usually expressed as a formula. If you call the weights w and the price ratios r, then what you do is you take the products, the price ratio times the weight. And you add them up and you divide that by the sum of the weights. So that’s what we’re going to calculate.

And it’s just a kind of formula for a weighted average. It’s a weighted average of the price ratios weighted by these weights that we had. So let’s go ahead and do that.

This is just a matter of arithmetic. So this is the price ratio times the weight. There’s the price ratio for the food and catering group. There’s the weight. So we multiply that times that. And we write the answer in there. Well, you have to do that in a calculator or something. This comes to 169.950.

We keep all of the accuracy in the calculation. As you probably realise, we’re going to round in the end. But we do that as absolutely the last step so that we don’t lose any accuracy because of rounding that we’d done in the intermediate stage. So it’s just a matter of filling in the rest of the price ratio times weight column in the same kind of way.

So this one turns out to be 92.400. And for housing and household expenditure, it’s 423.096. And the next one’s personal expenditure, that’s 92.496. And then, finally, 263.682.

And then we do need the sum of these. That was in the formula. It’s 1041.624.

And then we just got to work out this all-item price ratio. The formula is here. Sum of rw divided by sum of w, sum of ratios times weights divided by the sum of the weights. So it’s the ratio of those two sums we’ve calculated. 1041.624 divided by 1000. And since the thing we’re dividing by is a nice round number, that’s pretty easy to do. It comes to 1.041624. And, again, that’s an awful lot of decimal places. But we keep the full accuracy because we’re going to round at the end.

So back to the last step in the calculation. And that is to actually do what we wanted, to calculate the RPI. And the RPI, Retail Prices Index, for November 2011 – what we got is the all-item price ratio. We worked that out. And that’s the kind of average amount by which prices have gone up by in November 2011 compared to the previous January.

So if we actually want the Retail Prices Index, we’ve got to multiply the Retail Prices Index for that previous January by this all-item price ratio. And that gives us the Retail Prices Index, put up by the weighted average amount that prices have gone up by. So that means what we do is take the RPI for the previous January, for January 2011, and we multiply it by the all-item price ratio for November 2011. And that’s for November 2011 relative to the previous January.

And so we just need some numbers. So we worked out the all-item price ratio on the table just a bit before. We need the RPI. And, again, you’ve got to look that up on the ONS website or something like that. It was actually 229.0. So that’s the RPI.

We multiply it by the all-item price ratio, which is 1.041624, as we calculated before. And you do that calculation, and it comes to 238.531896.

Again, we’ve kept the full accuracy. But that’s clearly not justified by the accuracy of the data. The RPIs that are published always have just one decimal place. And so we need to round this to one place of decimals.

So how are we going to do that? We’re going to leave this 5 here, because that’s the last one. But the question is, do we round it up to 6? Or do we leave it where it is at 5? You have to look at the next value, which is 3. 3 is less than 5. That is, it’s less than halfway from 0.5 to 0.6. So we round it down.

And what you end up with is 238.5 correct to one decimal place. And that, 238.5, that’s the RPI for November 2011.

End transcript: Screencast 5 Calculating an RPI

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Screencast 5 Calculating an RPI

The same 2011 weights were used to calculate the RPI for every month from February 2011 to January 2012 inclusive. For each of these months, the price ratios were calculated relative to January 2011, and the RPI was finally calculated by multiplying the RPI for January 2011 by the all-item price ratio for the month in question. In February 2012, however, the process began again (as it does every February). A new set of weights, the 2012 weights, came into use. Price ratios were calculated relative to January 2012, and the RPI was found by multiplying the RPI value for January 2012 by the all-item price ratio. This procedure was used until January 2013, and so on.

The process of calculating the RPI can be summarised as follows.

The published value for the RPI in July 2011 was 234.7, slightly different from the value you should have obtained in Activity 21 (that is, 234.6). The discrepancy arises because the government statisticians use more accuracy during their RPI calculations, and round only at the end before publishing the results.

The following activity is intended to help you draw together many of the ideas you have met in this section, both about what the RPI is and how it is calculated.