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# 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.

## Collecting the data

The bulk of the data on price changes required to calculate the RPI is collected by staff of a market research company and forwarded to the Office for National Statistics for processing. Collecting the prices is a major operation: well over 100 000 prices are collected each month for around 560 different items. The prices being charged at a large range of shops and other outlets throughout the UK are mostly recorded on a predetermined Tuesday near the middle of the month. Prices for the remaining items, about 140 of them, are obtained from central sources because, for example, the prices of some items do not vary from one place to another.

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.

## Changing the representative items

The Office for National Statistics (ONS) updates the basket of goods every year, reflecting advancing technology, changing tastes and consumers’ spending habits. The media often have fun writing about the way the list of representative items changes each year.

In the 1950s, the mangle, crisps and dance hall admissions were added to the basket, with soap flakes among the items taken out.

Two decades later, the cassette recorder and dried mashed potato made it in, with prunes being excluded.

Then after the turn of the century, mobile phone handsets and fruit smoothies were included. The old fashioned staples of an evening at home – gin and slippers – were removed from the basket.

So now, in 2012, it is the turn of tablet computers to be added to mark the growing popularity of this type of technology.

That received the most coverage when it was added to the basket of goods, with the ONS highlighting this digital-age addition in its media releases.

But those seafaring captains who once used the then unusual fruit as a symbol to show they were home and hosting might be astonished to find that centuries on, the pineapple has also been added to the inflation basket.

Technically, the pineapple has been added to give more varied coverage in the basket of fruit and vegetables, the prices of which can be volatile.

(Source: BBC News website, 14 March 2012)

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 Calculating the RPI for November 2011

Here are the details of the last two stages of calculation of the RPI for November 2011, after the group price ratios have been calculated, relative to January 2011. The appropriate data are in Table 13.

### Table 13 Calculating the all-item price ratio for November 2011

Group Price ratio: Weight: Ratio weight: Food and catering

1.030

165

169.950

Alcohol and tobacco

1.050

88

92.400

Housing and household expenditure

1.037

408

423.096

Personal expenditure

1.128

82

92.496

Travel and leisure

1.026

257

263.682

Sum

1000

1041.624

You may have noticed that the weights here do not exactly match those in Table 12. That is because the weights here are the 2011 weights, and those in Table 12 are the 2012 weights, and as has been explained, the weights are revised each year.

The all-item price ratio is a weighted average of the group price ratios given in the table. If the price ratios are denoted by the letter r, and the weights by w, then the weighted mean of the price ratios is the sum of the five values of rw divided by the sum of the five values of w. The formula, from Subsection 2.3, is The sums are given in Table 13. (The sum of the weights is 1000, because the RPI weights are chosen to add up to 1000.) Although Table 13 gives the individual values, there is no need for you to write down these individual products when finding a weighted mean (unless you are asked to do so). As mentioned previously, your calculator may enable you to calculate the weighted mean directly, or you may use its memory to store a running total of .

Now the all-item price ratio for November 2011 (relative to January 2011) can be calculated as This tells us that, on average, the RPI basket of goods cost 1.041 624 times as much in November 2011 as in January 2011.

The published value of the RPI for January 2011 was 229.0. So, using the formula, The final result has been rounded to one decimal place, because actual published RPI figures are rounded to one decimal place.

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.] Skip transcript: Screencast 5 Calculating an RPI

#### 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
Screencast 5 Calculating an RPI
Interactive feature not available in single page view (see it in standard view).

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.

## Calculating the RPI

1. The data used are prices, collected monthly, and weights, based on the Living Costs and Food Survey, updated annually.

2. Each month, for each item, a price ratio is calculated, which gives the price of the item that month divided by its price the previous January.

3. Group price ratios are calculated from the price ratios using weighted means.

4. Weighted means are then used to calculate the all-item price ratio. Denoting the group price ratios by and the group weights by , the all-item price ratio is 5. The value of the RPI for that month is found by multiplying the value of the RPI for the previous January by the all-item price ratio:

The weights for a particular year are used in calculating the RPI for every month from February of that year to January of the following year.

## Activity 21 Calculating the RPI for July 2011

Find the value of the RPI in July 2011 by completing the following table and the formulas below. The value of the RPI in January 2011 was 229.0. (The base date was January 1987.)

### Table 14 Calculating the RPI for July 2011

GroupPrice ratio for July 2011 relative to January 2011: 2011 weights: Price ratio weight: Food and catering

1.024

165

Alcohol and tobacco

1.042

88

Housing and household expenditure

1.012

408

Personal expenditure

1.053

82

Travel and leisure

1.030

257

Sum

(Source: Office for National Statistics)

### Discussion

GroupPrice ratio for July 2011 relative to January 2011: 2011 weights: Price ratio weight: Food and catering

1.024

165

168.960

Alcohol and tobacco

1.042

88

91.696

Housing and household expenditure

1.012

408

412.896

Personal expenditure

1.053

82

86.346

Travel and leisure

1.030

257

264.710

Sum

1000

1024.608   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.

## Activity 22 The effects of particular price changes on the RPI

Between February 2011 and February 2012, the price of leisure goods fell on average by 2.3%, while the price of canteen meals rose by 2.8%. Answer the following questions about the likely effects of these changes on the value of the RPI. (No calculations are required.)

(a) Looked at in isolation (that is, supposing that no other prices changed), would the change in the price of leisure goods lead to an increase or a decrease in the value of the RPI?

Would the change in the price of canteen meals (looked at in isolation) lead to an increase or a decrease in the value of the RPI?

### Discussion

The RPI is calculated using the price ratio and weight of each item. Since the weights of items change very little from one year to the next, the price ratio alone will normally tell you whether a change in price is likely to lead to an increase or a decrease in the value of the RPI. If a price rises, then the price ratio is greater than one, so the RPI is likely to increase as a result. If a price falls, then the price ratio is less than one, so the RPI is likely to decrease. Therefore, since the price of leisure goods fell, this is likely to lead to a decrease in the value of the RPI. For a similar reason, the increase in the price of canteen meals is likely to lead to an increase in the value of the RPI.

(b) In each case, is the size of the increase or decrease likely to be large or small?

### Discussion

Both changes are likely to be small for two reasons. First, the price changes are themselves fairly small. Second, leisure goods and canteen meals form only part of a household’s expenditure: no single group, subgroup or section will have a large effect on the RPI on its own, unless there is a very large change in its price.

(c) Using what you know about the structure of the RPI, decide which of ‘Leisure goods’ and ‘Canteen meals’ has the larger weight.

### Discussion

The weight of ‘Leisure goods’ was 33 in 2012 (see Table 12). Since ‘Canteen meals’ is only one section in the subgroup ‘Catering’, which had weight 47 in 2012, the weight of ‘Canteen meals’ will be much smaller than 47. (In fact it was 3.) So the weight of ‘Leisure goods’ is much larger than the weight of ‘Canteen meals’.

(d) Which of the price changes mentioned in the question will have a larger effect on the value of the RPI? Briefly explain your answer.

### Discussion

Since the weight of ‘Leisure goods’ is much larger than the weight of ‘Canteen meals’, and the percentage change in the prices are not too different in size, the change in the price of leisure goods is likely to have a much larger effect on the value of the RPI as a whole.

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