Sample Standard Deviation - Estimation - Uncorrected Sample Standard Deviation

Uncorrected Sample Standard Deviation

Firstly, the formula for the population standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by sN, is known as the uncorrected sample standard deviation, or sometimes the standard deviation of the sample (considered as the entire population), and is defined as follows:

s_N = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2},

where are the observed values of the sample items and is the mean value of these observations, while the denominator N stands for the size of the sample.

This is a consistent estimator (it converges in probability to the population value as the number of samples goes to infinity), and is the maximum-likelihood estimate when the population is normally distributed. However, this is a biased estimator, as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/n, and thus is most significant for small or moderate sample sizes; for the bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller mean squared error than the corrected sample standard deviation.

Read more about this topic:  Sample Standard Deviation, Estimation

Famous quotes containing the words uncorrected, sample and/or standard:

    “Such uncorrected visions end in church
    Or registrar:
    A mortgaged semi- with a silver birch;
    Nippers; the widowed mum; having to scheme
    With money; illness; age.”
    —Philip Larkin (1922–1986)

    “As a rule they will refuse even to sample a foreign dish, they regard such things as garlic and olive oil with disgust, life is unliveable to them unless they have tea and puddings.”
    —George Orwell (1903–1950)

    “I don’t have any problem with a reporter or a news person who says the President is uninformed on this issue or that issue. I don’t think any of us would challenge that. I do have a problem with the singular focus on this, as if that’s the only standard by which we ought to judge a president. What we learned in the last administration was how little having an encyclopedic grasp of all the facts has to do with governing.”
    —David R. Gergen (b. 1942)