Editing Linear trend estimation (section)

=== Noisy time series ===
It is harder to see a trend in a noisy time series. For example, if the true series is 0, 1, 2, 3, all plus some independent normally distributed "noise" ''e'' of [[standard deviation]]{{nbsp}}''E'', and a sample series of length 50 is given, then if ''E''{{nbsp}}={{nbsp}}0.1, the trend will be obvious; if ''E''{{nbsp}}={{nbsp}}100, the trend will probably be visible; but if ''E''{{nbsp}}={{nbsp}}10000, the trend will be buried in the noise.

Consider a concrete example, such as the [[global surface temperature]] record of the past 140 years as presented by the [[Intergovernmental Panel on Climate Change|IPCC]].<ref name=":1">{{cite news|url=http://www.grida.no/publications/other/ipcc_tar/?src=/climate/ipcc_tar/wg1/figspm-1.htm|title=IPCC Third Assessment Report – Climate Change 2001 – Complete online versions|access-date=June 17, 2012|url-status=dead|archive-url=https://web.archive.org/web/20091120181301/http://www.grida.no/publications/other/ipcc_tar/?src=%2Fclimate%2Fipcc_tar%2Fwg1%2Ffigspm-1.htm|archive-date=November 20, 2009}}</ref> The interannual variation is about 0.2{{nbsp}}°C, and the trend is about 0.6{{nbsp}}°C over 140 years, with 95% confidence limits of 0.2{{nbsp}}°C (by coincidence, about the same value as the interannual variation). Hence, the trend is statistically different from 0. However, as noted elsewhere,<ref name=":2" /> this time series doesn't conform to the assumptions necessary for least-squares to be valid.