It is closer to the 0.5 / sqrt (25) = 0.1 cm uncertainty that one would expect if there were
random errors in the measurements.
Not exact matches
The variance
in individual samples is not a result of
measurement errors with known Gaussian
random noise.
However, these
measurements contain non-negligible
random errors and biases owing to the indirect nature of the relationship between the observations and actual precipitation, inadequate sampling, and deficiencies
in the algorithms.
Increasing the number of
measurements in this case, does decrease the
random component of the instrumental
error by 1 / SQRT (n) where n is the the number of observations.
Lots of factors make measuring global temperature a difficult task, such as sparse data
in remote places,
random measurement errors and changes
in instrumentation over time.
The math behind the Law of Large Numbers goes back to Jacob Bernoulli
in 1713, and is based on the statistics of
measurements and
random errors.