Given the discrete distribution of a nominal variable:
The modal average is the mean.
The median average may be the mean, if the majority agree.
The mean average doesn't exist.
The average average must therefore be the mode, which is the mean.
Know what I mean?
-----Original Message-----
From: email list for Radical Statistics
[mailto:[log in to unmask]]On Behalf Of Stephen Morris
Sent: 25 July 2005 15:22
To: [log in to unmask]
Subject: Re: Mean vs Median (was: Re: New free statistics resource
(fwd))
Dear Mike,
Thank you for your very full response. On re-reading the piece myself I
think it is broadly fair - except that perhaps the word "commentators" is
ambiguous - we didn't mean to point fingers at the IFS but at the media. We
will change that.
We do say "...despite this, even a brief reading of this section of the
report gives the true picture. Maybe next week's headlines will read 'Media
mean median'."
I take your point that the two taken together give a fuller picture than
either of them separately. Hmm - maybe there's another piece in that.
Stephen
-----Original Message-----
From: email list for Radical Statistics [mailto:[log in to unmask]] On
Behalf Of Mike Brewer
Sent: 25 July 2005 14:42
To: [log in to unmask]
Subject: Mean vs Median (was: Re: New free statistics resource (fwd))
Stephen,
It was good of you to send the web link, and brave of you to subject
yourself to
the Radstats critique!
Naturally, I checked out your page about the March 2005 IFS report on
household
incomes, on which I was a lead author, and the mean vs median income
controversy
(I confess to some surprise that was not a Radstats discussion about it at
the
time).
Your web-page find us guilty of incorrectly using the mean to characterise
the
average. I think that is a little harsh on us.
It is true that we implicitly use "average" as a synonym for "arithmetic
mean"
in the first line of our press release
(http://www.ifs.org.uk/pr/hbai05pr.pdf).
This is because we took the view that people might be confused by a headline
that said "Tax rises and new tax credits cut mean income, but reduce
poverty and inequality". But, in our defence, the third paragraph of the PR
discusses the skewness of the income distribution, explains that we use
"average" to mean "mean", and "median" to mean "median", and says that
median
income rose while mean income fell. Furthermore, the relevant paragraph in
our
executive summary reads: "Income growth was particularly sluggish in
2003/04.
Median income increased in real terms by just under £2 per week (an increase
of
0.5 per cent) while mean income fell for the first time since the early
1990s (a
small change of -0.2 per cent)." This makes no mention of "average"; our key
point is that both numbers are low compared to the past few years (the real
statistical crime we committed was that neither of these statistics was
significantly different from zero!).
Of course, what happened in reality is that people in the media had to
decide
which of these 2 numbers was the more interesting and the more applicable,
and,
naturally enough, since we were a week away from a general election
campaign,
the Tory-supporting newspapers decided on a "fall in average income"
headline,
the Treasury objected furiously, and the story became big political as well
as
economic/social affairs news.
But on top of these points, I would argue that it is interesting to look
both at
trends in mean income, as well as median income. Trends in mean income tell
us
about trends in how much "pie" there is to go round, and trends in median
income
tell us about how much of the pie is going to the person in the middle of
society. Our policy point was that direct tax rises had reduced the size of
the
pie enjoyed by households, with richer households bearing most of this fall
in
incomes: this point cannot be made by looking at trends in median income.
Having said all this, your general simplified statistical point - look at
trends
in median when examining skewed distributions, not trends in the mean - is
not
something I am trying to dispute!
best wishes,
Mike
[Incidentally, we, like the Treasury, enjoyed(!) trying to explain the
difference between the mean and the median on live television!]
Stephen Morris wrote:
> And the third discusses when you should use the median or the mean, using
> the example of average household income:
> http://www.conceptstew.co.uk/PAGES/mean_or_median.html
--
Mike Brewer
Programme Director, Direct Tax and Welfare
Institute for Fiscal Studies
7 Ridgmount Street, London, WC1E 7AE
Tel: +44 (0)20 7291 4800 Fax: +44 (0)20 7323 4780
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