Normal Curve and Sampling Distribution

Abstract

There is a need to learn about the behavior of normal curve which could only be specified if the population mean (μ) and standard deviation (σ) are known. Any change in the population mean (μ) would bring change in the location of distribution. With a change in the standard deviation (σ), scatter of distribution changes. But the total area under the normal curve always remains the same.

In actual experience we do not know the population. Therefore, the actual mean and standard deviation of the population cannot be determined. Hence, the estimates obtained from a single sample are used to determine the mean and standard error. We assume that the sample mean ($\overline{X}$) and standard deviation (s) are unbiased estimates of the population parameters: mean (μ) and standard deviation (σ). The standard error of the mean for a sample size “n” is determined as $\frac{S}{\sqrt{n}}$.

Sampling distribution can be obtained in two ways: (1) by compiling actual frequencies of observations and (2) by getting frequencies based on mathematical device. Theoretical distribution refers to mathematical models of relative frequencies of a finite number of observations of a variable. We can deduce mathematically the expected frequency distribution on the basis of theoretical considerations. Such distributions are called probability distributions or theoretical distributions. If we toss a coin, we know that the probability of the head or tail is $\frac{1}{2}$ each. Now, if we toss a coin 50 times, the theoretical probability of head would be 25, and that is called expected frequency. Practically if head comes 27 times, that will be called observed frequency.