The standard normal distribution (mean 0, standard deviation 1) is obtained by transforming any normally distributed variable X via Z = (X − X̄)/σ, enabling probabilities of occurrence to be read from standard tables rather than recomputed for every mean and standard deviation. This underlies the surveying concept of precision, where the spread of repeated observations (standard deviation) indicates reliability of a single observation, while standard error (standard deviation divided by the square root of the number of observations) indicates reliability of the derived mean, since equal standard deviations do not imply equal confidence in the mean when sample sizes differ. When observations carry unequal reliability, they are combined using weights (assigned by judgement, number of observations, or inverse variance) to compute a weighted mean and weighted measures of precision (weighted standard deviation, weighted standard error), extending basic error theory within surveying/measurement science.
The standard normal distribution (mean 0, standard deviation 1) is obtained by transforming any normally distributed variable X via Z = (X − X̄)/σ, enabling probabilities of occurrence to be read from…