Re: Help with confidence intervals for small sample sets

From: Paige Miller (paige.miller_at_kodak.com)
Date: 10/11/04 

Date: Mon, 11 Oct 2004 08:30:32 -0400

Nils wrote:

> Hello,
>
> I'm trying to construct 95% cofidence intervals for a set of data
> (n=10..25 usually). The data is from a distance measurement device,
> where a constant distance is driven.
>
> After calculating the mean, I calculate the standard deviation.
> Since the sample set is small, I estimate the parent deviation using the
> formula
>
> parentdev = stdev * sqrt(n/n-1)

I think most textbooks call "parentdev" the "population standard
deviation"

> The confidence interval is then mean +- 1.96 * parentdev / sqrt(n)
>
> I realise that I need to use t(a/2) and not z(a/2) here, but that
> doesn't change the results significantly.

I think it makes a bit of difference when n=10, less difference when
n=25. Many textbooks say that, as practical matter (rather than
being extremely rigorous) when n<30, use the t-distribution

> Here is what I obtain:
>
> 100m path offset average= (3069.55/17)= 181 mm
> 100m path offset standard deviation= 53.18 mm
> conf= 180.561713891275 +- 1.96 * 54.8162276614482 / sqrt(17)
> 95% confidence interval = 154.50 .. 206.62 mm
>
> What makes no sense to me: One standard deviation (containing 67% of the
> data) is 53 mm, but the confidence interval is 154mm..206mm
> (206-154=52mm!!)- supposedly implying that 95% of all further trials
> fall within this range.

No!

You have computed a 95% confidence interval for the mean of the
data. You have NOT computed a confidence interval for "all further
trials".

The mean is less variable than an individual trial. This is a
fundamental concept in statistics.

And when you think about it from a layman's point of view, it should
be obvious as well. Think of yearly temperatures ... the mean (at
least here in beautiful Rochester, NY) is around 58 F, some years
the mean is higher, some years the mean is lower, but the mean shows
much less variability than actual temperature. Actual temperature
may range from -10 F to 98 F during the year, but the mean never
moves that much.

> Also, is there a formula for calculating the T tables? I need to be
> able to generate several confidence levels for n=5..100 eventually.

The formula for computing t-distribution values isn't simple, which
is why tables are constructed and widely available in print, as well
as in software. 

-- 
Paige Miller
Eastman Kodak Company
paige dot miller at kodak dot com
http://www.kodak.com
"It's nothing until I call it!" -- Bill Klem, NL Umpire
"When you get the choice to sit it out or dance, I hope you dance" 
-- Lee Ann Womack