Empirical distribution function  In statistics, an empirical distribution function is the distribution function associated with the empirical measure of a sample. This cumulative distribution function is a step function that jumps up by at each of the data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function. Let be independent, identically distributed real random variables with the common cumulative distribution function . Then the empirical distribution function is defined as where formula_2 is the indicator of event . For a fixed , the indicator formula_3 is a Bernoulli random variable with parameter ; hence formula_4 is a binomial random variable with mean and variance . This implies that formula_5 is an unbiased estimator for . However, in some textbooks, the definition is given as formula_6  The mean of the empirical distribution is an unbiased estimator of the mean of the population distribution.  formula_7  which is more commonly denoted formula_8  The variance of the empirical distribution times formula_9 is an unbiased estimator of the variance of the population distribution.  formula_10  The mean squared error for the empirical distribution is as follows.  formula_11  Where formula_12 is estimator and formula_13 a unknown parameter  For any real number formula_14 the notation formula_15 (read “ceiling of a”) denotes the least integer greater than or equal to formula_14. For any real number a, the notation formula_17 (read “floor of a”) denotes the greatest integer less than or equal to formula_14. If formula_19 is not an integer, then the formula_20-th quantile is unique and is equal to formula_21  If formula_19 is an integer, then the formula_20-th quantile is not unique and is any real number formula_24 such that   formula_25  If formula_26 is odd, then the empirical median is the number  formula_27  If formula_26 is even, then the empirical median is the number   formula_29  Since the ratio approaches 1 as goes to infinity, the asymptotic properties of the two definitions that are given above are the same.