Due to constraints, it can be difficult to calculate the population mean. Instead, we can sample enough proportion of the population and use the sample mean as an approximate. It is important to perform random sampling to prevent any biased selection that does not effectively represent the entire population. The larger the sample size and the more representative it is of the population, the closer the sample mean tends to be to the population mean.
The sample mean is the average of the sample values, which is the sum of those values divided by the number of samples. Using mathematical notation,
$$ \bar{x} = \frac{1}{N}\sum_{i=1}^{N}x_i $$
The law of large numbers is a mathematical theorem that states that the average of the results obtained from a large number of independent and identical random samples converges to the true value, if it exists. → sampling more data gives us a more accurate approximate of the mean!
A population proportion, generally denoted by $P$, is a parameter that describes a percentage value associated with a population (for e.g. percentage of vehicles which are motorcycles). A population proportion is usually estimated through an unbiased sample statistic obtained from an observational study or experiment. This is known as the sample proportion, denoted by $\hat{p}$.
$$ \hat{p} = \frac{x}{n} $$
where $x$ is the count of successes in the sample and $n$ is the size of the sample obtained from the population.
If all possible observations of the system are present then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance.
Population variance formula:
$$ \sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2 $$