SAMPLING DISTRIBUTIONS DISTRIBUTIONS Population Distribution Sample Distribution Sampling Distribution 2 Distributions Definition The population distribution is the probability distribution of the population data. The sample distribution is the distribution for the sample/data collected. {stem/leaf, dotplot, boxplot, …} 3 Sampling Distribution Definition The probability distribution of a statistic is called its sampling distribution. It gives the various values that the statistic can assume and the probability of each value. Web Applet :: http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html 4 SAMPLING AND NONSAMPLING ERRORS Definition Sampling error is the difference between the value of a sample statistic and the value of the corresponding population parameter. For Example :: In the case of the mean, Sampling error = x assuming that the sample is random and no nonsampling error has been made. 5 SAMPLING AND NONSAMPLING ERRORS Definition The errors that occur in the collection, recording, and tabulation of data are called nonsampling errors. 6 MEAN AND STANDARD DEVIATION OF x Definition The mean and standard deviation of the sampling distribution of x are called the mean and standard deviation of x and are denoted by x and x , respectively. 7 MEAN AND STANDARD DEVIATION OF x Mean of the Sampling Distribution of x The mean of the sampling distribution of x is always equal to the mean of the population. Thus, x 8 MEAN AND STANDARD DEVIATION OF x cont. Standard Deviation of the Sampling Distribution of x The standard deviation of the sampling distribution of x is x n where σ is the standard deviation of the population and n is the sample size. 9 Two Important Observations 1. The spread of the sampling distribution of x is smaller than the spread of the corresponding population distribution, i.e. x x 2. The standard deviation of the sampling distribution of x decreases as the sample size increases. 10 SHAPE OF THE SAMPLING DISTRIBUTION OF x Sampling from a Normally Distributed Population Sampling from a Population That Is Not Normally Distributed 11 Sampling From a Normally Distributed Population If the population from which the samples are drawn is normally distributed with mean μ and standard deviation σ , then the sampling distribution of the sample mean, x ,will also be normally distributed with the following mean and standard deviation, irrespective of the sample size: x and x n 12 Sampling From a Population That Is Not Normally Distributed Central Limit Theorem According to the central limit theorem, for a large sample size, the sampling distribution of x is approximately normal, irrespective of the shape of the population distribution. The mean and standard deviation of the sampling distribution of are x and x n 13 Examples … 14