• Probable Error

    The term probable error is seen only in older literature of statistics where it used to donate the product of the product of the standard error and a constant factor 0.6745, thus Where P.E and S.E stand for the probable error and the standard error respectively, by using the above formula the probable error of […]

  • Standard Error of Statistic

    The term standard error has already been introduced very briefly in pervious tutorials while discussing about the sampling distribution of means. In this tutorial we will try to make some more comments on standard error. In the broader sense of the term standard errors of mean, median, standard deviation, coefficient of correlation, regression coefficients etc. […]

  • Confidence Interval Estimate of Variance

    The variance of a population can be estimated using the chi-square variate explained in pervious tutorials. Unlike the and distributions the value of chi-square variate are defined only for positive values. At a level of significance the or are those values of the variate which give an area in the right tail. Also or are […]

  • Confidence Interval Estimate for Difference Between Means

    (1) Large Samples: The difference between two means is of considerable importance in testing the homogeneity of populations. In this tutorial we are concerned about the confidence interval estimate for the difference between two population means. With a non-rigorous logic from central limit theorem we can state that “If we have two populations with means […]

  • Confidence Interval Estimate of Mean with Small Sample

    As long as is known the confidence interval estimate of population mean can be obtained by the method discussed earlier, provided the sample is large. Even if is not known we could replace it by its unbiased estimate , defined by The pervious method of estimation fails to provide good estimations if the sample size […]

  • Confidence Interval Estimate of Mean

    From central limit theorem we know that, is a standard normal variate. From the discussion of introduction to interval estimation we know that P (–1.96 < Z< 1.96) = 0.95 has the least possible range. With this inequality we can construct a 95% confidence interval estimate of the population mean, if we replace Z by […]

  • Introduction to Interval Estimation

    From the discussion on the point estimation we know that is the best possible estimator of the population mean , which is a fixed, usually unknown parameter. It is a matter of common-sense that we have to be extremely lucky to have a sample which has a mean exactly equal to the population mean , […]

  • Consistent Estimator

    An estimator is said to be a consistent estimator of the parameter , if it holds the following conditions: is an unbiased estimator of , i.e. if is biased, it should be unbiased for large values of (in the limit sense) i.e. . Variance of   approaches to zero as becomes very large, i.e., . […]

  • Efficiency of an Estimator

    Among a number of estimators of the same class, the estimator having the least variance is called an efficient estimator. Thus, if we have two estimators and with variances and  respectively, also if then will be an efficient estimator. The ratio of the variances of two estimators denoted by is known as the efficiency of  […]

  • Unbiasedness of an Estimator

    This is probably the most important property that a good estimator should possess. According to this property if the statistic is an estimator of will be an unbiased estimator if the expected value of   equals the true value of the parameter               i.e. Consider the following worked example Example: Show that the sample mean […]

  • Maximum Likelihood Estimation

    The maximum likelihood method is a very rigorous statistical method of estimation. The word likelihood has the same meaning as the word probability. The method of maximum likelihood is not restricted to some specific type of analysis as the least square method; rather its application is universal provided the probability distribution of the population is […]

  • Point Estimation

    In point estimation procedure we make an attempt to compute a numerical value, from sample observations, which could be taken as an approximation to the parameter. The estimators, which are also referred to as statistics (plural of statistic), since based on observations which are random variables are themselves random variables. A number of estimation methods […]