Labour leader Ed Milliband is widely reported today as claiming that tomorrow’s VAT rise will cost the average family £7.50 a week. Unless he is living in a parallel universe where the average household earns far more than is currently the case, that’s not true.

I’m fortunate enough to be earning more than the UK average, by a fairly wide margin. I’ve just had a look through my online bank and credit card statements, and tallied up what I spend on things where VAT is incurred. To make it simple, I’ve assumed that this is everything except housing costs and food bought from the supermarkets (rents and mortgages don’t attract VAT, and neither does most food). In reality, some of it is children’s clothing (since I have two young daughters), which doesn’t attract VAT either. But, on the other hand, some of the things that I’ve excluded from the calculations probably do attract VAT, since the question of which foods do and don’t have VAT can be a bit complicated. So it probably balances out a bit.

Anyway, I’ve worked out that I spend just under £1200 a month on VATable products. So I’ll start with that as a round figure, and do some calculations.

To begin with, how much of what I currently spend goes on tax? The formula to extract VAT from the price paid is price – (price / (1 + (rate / 100)), so let’s do that sum:

1200 – (1200 / (1 + (17.5 / 100)) = 179

(That’s rounded to the nearest whole pound, I’ll stick to that until right at the end when we work out how much a week the rise will cost me).

You can simplify that a bit by doing the internal rate calculation first, since this is simple enough to do in your head. A VAT rate of 17.5% works out as 1.175 when used in the actual calculations, and a rate of 20% works out as 1.2. From now on, I’ll use the decimal rather than percentage rate, as that makes it easier to read. Here’s the first equation again:

1200 – (1200 / 1.175) = 179

So, I’m currently generating £179 a month in VAT for the government. Let’s see what happens when the rate goes up. I’ll assume that I don’t actually cut down on my purchasing at all, so I will pay all the extra. To work out what the new cost will be, we first have to work out how much the ex-VAT figure is for what I’m currently spending. We do that by dividing the amount I spend by the decimal rate:

1200 / 1.175 = 1021

So, again rounded to the nearest pound, I’m actually buying £1021 worth of goods or services each month. To work out what that will cost me after the VAT rise, we multiply that by the new decimal rate:

1021 x 1.2 = 1225

Rounded again, that means my monthly spend will go up to £1225 – an increase of £25 on what I spend now. To get a weekly figure for that, we multiply by the number of months in the year, then divide by the number of days in a year and finally multiply by 7:

((25 * 12) / 365 ) * 7 = 5.75

If you want to break that down step by step, it works out like this:

25 x 12 (number of months) = 300 a year

300 / 365 (number of days) = 0.82 a day

0.82 x 7 (number of days in a week) = 5.74 a week

The reason this is a penny different is that the second time, I’ve rounded each step to the nearest penny, but didn’t the first time. But it’s only an approximation anyway, since I started with a round £1200 rather than trying to calculate my exact spend. What really matters here is that the VAT increase is going to cost me around £5.75 a week. And I’m on more than the average salary, remember.

Ed Milliband, on the other hand, is claiming that it will cost the average household an extra £7.50 a week. That’s assuming, as I have done, that all the extra will actually be paid – people won’t offset it by cutting back a bit.

To end up spending an extra £7.50 a week, you’d need to have an existing VATable spend of around £1565. That’s around £18,780 a year – which is more than the average annual take-home salary after tax.

Obviously, the average household income is more than the average take-home salary, since many households have more than one wage-earner. So what is the average annual household income after tax?

That’s actually a harder question to answer than it seems, since not only are the figures harder to come by but there are three different types of average: the mean, the mode and the median. I’ve had to do a bit of digging using Google to try and get these figures, but as far as I can see this is what they are:

Mode (the most common): around £15,000
Median (the one that’s in the middle): around £20,000
Mean (the one calculated by adding and dividing them all): around £25,000

A household on the modal income can’t possibly be spending £18,780 on VATable products, since they don’t have that amount of money to begin with. A household on the median income could, but only if their expenditure on non-VATable products is not much more than a thousand a year. Given that rent and mortgages, as well as most food, don’t attract VAT, that’s as close to impossible as makes no difference. A household on the mean income could spend £18,780 with just over six grand to spare for non-VATables, but even so that’s extremely unlikely if they’re spending a typical proportion of their income on housing and food.

So the fact is that Ed Milliband’s claim just doesn’t add up. A well-off household will end up spending an extra £7.50 a week due to the VAT rise, yes, but not the average family.

I suspect that Ed, or his advisors, have done their own calculations based on the mean household income and assumed both that the mean itself is towards the upper band of what’s probable and that expenditure on things like rent and mortgages is towards the lower end of normality. If you accept both of those assumptions, then it’s not an unreasonable claim. But I don’t think either of those assumptions are reasonable, and nor do I think that using the mean is the best choice.

There’s another thing here. Most high-earning households are those with two full-time wage-earners. But Ed didn’t say “households”, he said “families”. While a couple with no children is, technically a family, most people would take the word to mean a household which includes children. But households with children are less likely to have two full-time wage-earners than couples without children.

By using the phrase “average family” Ed Milliband is implying that he’s referring to the typical household containing children. But the typical household with children has a disposable income far less than that which would be required to spend an extra £7.50 a week on VAT. Again, based on my research using Google, I reckon that only a quarter of all households will end up spending an extra £7.50 or more a week on VAT as a result of the rate rise. That quarter, of course, includes bankers, footballers, pop stars and senior politicians – all people who earn considerably more than the average salary and distort the mean accordingly.

So, is Ed Milliband deliberately scaremongering, or just unable to do the maths? I’ll let you be the judge of that.