Parent Distribution and Sample Populations

The parent distribution is obtained from taking the limiting values of the sample as the number of experiments go to $\infty$. This is important because the parent population tells us the exact distribution of the data points. This, in turn, gives us the chance to examine the error associated with making measurements. Experimentally, the best estimation we can get for the mean of the parent population $\mu$ is the mean $\bar{x}$ of the data with the highest number of samples and the best estimation for the standard deviation $\sigma$ is the deviation $s$ obtained from the same data. The following equations² are for a discrete distribution.

For the parent population:

$$\mu = \lim_{N \to \infty} \frac {1}{N}\sum_{i=0}^{N} x_{i}$$ (1)

$$\sigma^{2} \equiv \langle x^{2} \rangle - \langle x \rangle... ..._{N \to \infty} [\frac{1}{N} \sum_{i=0}^{N} (x_{i}- \mu)^{2}]$$ (2)

For the Sample Population:

$$\bar{x} = \frac{1}{N} \sum_{i=0}^{N} x_{i}$$ (3)

$$s^{2} = \frac{1}{N-1} \sum_{i=0}^{N}(x_{i}- \bar{x})^{2}$$ (4)