You are tasked with finding the average height of 100 houses. With a ruler at arms length and a tape measure you are able to measure the heights +/- 10cm.

When you've found the average height, what is the error band?

Nial.

When I did this at school, I always found it instructive to find the maximum and minimum of the measurements and work with both those figures: e.g.

Max: if all 100 houses are at the max of the error, then your total will be 1000cm larger than actual.
Min: if all 100 houses are at the -max of the error, then your total will be 1000cm smaller than actual.

Thus the Error bands on the total is -1000cm to +1000cm

Now you divide by 100 to get the average, the 100 has no error bars, so it's a simple arithmetic division of the error bars, so the error bars become -10cm to +10cm again.

Which is intuitive, your average can't be any more precise than the worst of your measurements.

Is this wrong?

Also, I didn't read the instructions properly, I think he's measuring the houses by holding a ruler at arm's length, measuring the apparent height on the ruler, and then using trig to work out the actual heights.

Thanks for the replies guys (apart from Martin's pedantic grammar point), the method of measurement isn't important, just that you know the accuracy of each measurement.

Elsewhere someone queried the accuracy quoted in global temperature measurements (0.05 Deg C).

An ardent warmist posted this link....

http://web.archive.org/web/20080402030712/tamino.wordpress.com/2007/07/05/the-power-of-large-numbers/

...which discussed improvements in resolution of a single measurement by performing it multiple times.

This however isn't the same as the global temperature that combines multiple measurements of _different_
things.

The house measurement query was the simplest way I could think of posing the question.

Intuitively I thought there's probably _some_ improvement in accuracy, but after TBYJ's post above I'm not sure.

Nial

Nial,

One of the bugbears of the whole AGW thing for me is that the rise in temps (0.7 degrees) is still within the error bars on the individual readings. Which is fine if you assume the errors in the measurements are symmetrical about the real values. But we have found that they have a bias towards high.

"One of the bugbears of the whole AGW thing for me is that the rise in temps (0.7 degrees) is still within the error bars on the individual readings. "

Aye, but this article is saying that taking a large number of individual readings, over time, reduces the error.

I need to show it doesn't.

Well you might be in trouble Nial.

Whilst the individual readings errors might be large, if the average and the spread moves over time, then this can be measured with an accuracy far more than the individual readings.

E.g. if in year 2000 a measure is 100 plus or minus 50, and in 2010 it's 101 plus or minus 50, then you can say that it's gone up by 1% even if the error is 50%.

Thanks again guys, Martin don't worry I was joking about your pedantry, I should have included a :-) .

"I too think you may be in trouble. If you can take a large number of measurements, where the error in each measurement is statistically independent of the error in the other measurements, then computing the average of all your measurements will generally give a more accurate estimate of the thing you are trying to measure than any one single measurement."

Hmm.

It turns out the builders didn't dig the foundations properly and the houses are _slowly_ subsiding.
If you continually repeated your experiment over the next few years I suppose you would detect the average height reducing, even though each result still had significant error bars.

Obvious when you think about it. I will concede the point elsewhere!

Thanks again,

Nial.

It depends if the error is systematic (your measuring device is such that you get the height too large by ten cm for every house), in which case the overall error on the average is the same as the individual error, or random for each house, some too high, some too low, others just right in which the error on the average decreases as the inverse of the square root of the number of measured values. Here it would be one tenth, just 1cm. For temperature measurements there may be a bit of both, with the added twist that the systematic error can change in time sure to increasing UHI. But the error you'd think of from making a couple of measurements in your house or garden is mostly of the random type.

"others just right in which the error on the average decreases as the inverse of the square root of the number of measured values. Here it would be one tenth, just 1cm."

I'm not sure that's correct Jeremy.

I understand that if you took 100 measurements of the same thing the error would reduce by this amount, but if you are meauring 100 different things I'm not sure you can give the average height +/- 1cm.

What I do accept is that if you continually do the same measurement of the 100 houses you'll be able to spot the long term trends of subsidence.

Possibly.

Martin,

If we are looking at error bars for the result......

> Estimate for average height
> = (1/100) × [SUM (all the actual heights) + SUM (all the measurement errors)]

is that the sum of the actual measurement errors or the potential errors?

> = true average of the 100 houses + final error

If you're summing potential errors you end up with the individual measurement error (or the average of the potential measurement errors).

?

Thanks Martin.

> Could you say exactly what you mean by 'error bars'?

I'm not sure any more!

I hardly slept last night and feel my brain is like mush today!

Perhaps you are thinking of Standard Error, the measure of how well the sample mean approximates the population mean. For a relatively simple reference try here .

Martin A

I sympathise with nial. :-) There is a continuum of statistical understanding.

To a layman whose maths ended at GCSE statistics is a dark art used by politicians to support their dogma; lies, dammed lies and statistics!

To working stiffs like thee and me it was a toolbox of techniques used to judge the quality and reliability of the data we used.

Professional statisticians work at a higher level altogether!