This post is an introduction to point estimation in inferential statistics.
A point estimate is a single calculated value used to estimate a population parameter.
List of point estimates
Point estimates in inferential statistics:
- Mean
- Standard deviation
- Variance
- Proportion
- Standard error
Mean in inferential statistics
The population mean is denoted with the Greek letter \(\mu\).
Standard deviation in inferential statistics
The standard deviation is usually inferred from the variance:
$$s = \sqrt{s^2}$$
All formulas appliance to the variance are applicable to the standard deviation just by applying the squared root.
Variance in inferential statistics
The variance formula in inferential statistics varies slightly compared to the one used in descriptive statistics, in what it is known as the Bessel correction.
Sample variance formula in inferential statistics:
$$s^2 = \frac{\sum_{i=1}^{n}(x_i – \overline{x})^2}{n-1}$$
Take into account that the denominator includes n-1 instead of n in descriptive statistics.
Alternative variance formula:
$$s^2 = \frac{n}{n-1}(\overline{x^2}-(\overline{x})^2)$$
When we talk about the population variance, we use the symbol σ².
In case that the standard deviation is dichotomous and coded as 1 or 0:
$$s^2 = \sqrt{\hat{\pi}(1-\hat{\pi})}$$
This is because:
$$s^2 = \frac{n}{n-1}(\overline{x^2}-(\overline{x})^2) = \frac{1}{n}[(1-\frac{r_1}{n})^2\,r_1+(0-\frac{r_1}{n})^2\,r_0] = \frac{r_1}{n}(1-\frac{r_1}{n}) = \hat{\pi}(1-\hat{\pi})$$
Proportion in inferential statistics
The population proportion is represented with the symbols \(p\) or \(\pi\).
Take note that using the letter \(p\) is an exception to the general rule that parameters are noted with Greek script letters, but historically it has been done this way. Some authors use alternatively \(\pi\).
The uppercase \(\Pi\) is sometimes reserved for population or theoretical proportions, while lowercase \(\pi\) would be reserved for individual proportions.
A sample proportion (noted as \(\hat{p}\)) is the fraction within a sample that possesses a specific characteristic.
Standard error
The standard error is measures the expected variability of a sample statistic (like the mean, proportion, slope, etc.) across repeated samples. Because of this, it can be considered a sampling error.
The standard error will be calculated differently depending on:
- The given statistic.
- The context (e.g. confidence interval vs. hypothesis testing).
For example, the standard error for the sample mean where the samples are independent and identically distributed (i.i.d.) is calculated as:
$$ SE_\bar{x} = \frac{s}{\sqrt{n}} $$
The standard error is dependent of the number of samples n.
The standard error is main to the inferential statistics, though it is used on hypothesis contrast. Because it requires different samples, it cannot be part of descriptive statistics.