Question

Compare the Range, Standard Deviation and Root Mean Squared Deviation as statistical measures of data. Explain their meaning, advantages and disadvantages. Give some graphical interpretation to these measures. What is the unbiased estimate of the Standard Deviation? When and why is it recommended for use?

Answer 1

I) Range:

In statistics, the range of a set of data is the difference between the largest and smallest values. Difference here is specific, the range of a set of data is the result of subtracting the smallest value from largest value.

Range = largest obs - smallest obs

Advantages of Range:

1. It is simple to understand and easy to calculate.

2. It is less time consuming.

Disadvantages of Range:

1. It is not based on each and every item of the distribution.

2. It is very much affected by the extreme values.

3. The value of Range is affected more by sampling fluctuations

4. Range cannot be computed in case of open-end distribution.

ii) standard Deviation:

Advantages of Standard Deviation:

Among all measures of dispersion Standard Deviation is considered superior because it possesses almost all the requisite characteristics of a good measure of dispersion. It has the following merits:

1) It is rigidly defined.

2) It is based on all the observations of the series and hence it is representative.

3) It is amenable to further algebraic treatment.

4) It is least affected by fluctuations of sampling.

Disadvantages:

1) It is more affected by extreme items.

2) It cannot be exactly calculated for a distribution with open-ended classes.

3) It is relatively difficult to calculate and understand.

iii) Root Mean Squared Deviation

The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) (or sometimes root-mean-squared error) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents the square root of the second sample moment of the differences between predicted values and observed values or the quadratic mean of these differences. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation and are called errors (or prediction errors) when computed out-of-sample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is a measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent.

The RMSD of an estimator with respect to an estimated parameter is defined as the square root of the mean square error:

iv) Unbiased estimation of standard deviation

In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis

where is the sample (formally, realizations from a random variable X) and is the sample mean.

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