Meaning of “point of estimation”
In statistical inference, a point of estimation refers to a point estimate: a single numerical value used to estimate an unknown population parameter. If the parameter is \(\theta\), then a point estimate is written as \(\hat{\theta}\).
Two key objects:
- Point estimator: a rule (a statistic) computed from the sample, such as \(\bar{x}\) or \(\hat{p}\).
- Point estimate: the numerical value produced by that rule for the observed data.
Point estimates differ from interval estimates, which provide a range of plausible values (such as a confidence interval).
Common point estimators
| Parameter (unknown) | Typical point estimator | Point estimate from data |
|---|---|---|
| Population mean \(\mu\) | \(\hat{\mu}=\bar{x}\) | \(\bar{x}\) computed from the sample |
| Population variance \(\sigma^2\) | \(\hat{\sigma}^2=s^2\) | \(s^2\) computed from the sample |
| Population proportion \(p\) | \(\hat{p}=x/n\) | Sample fraction \(x/n\) |
Step-by-step: point estimate of the population mean
Suppose a simple random sample of \(n=10\) observations (e.g., lifetimes in hours) is: 5.1, 5.4, 5.0, 5.6, 5.3, 5.2, 5.5, 5.1, 5.4, 5.3.
The standard point estimator for \(\mu\) is the sample mean: \[ \hat{\mu}=\bar{x}=\frac{1}{n}\sum_{i=1}^{n} x_i. \]
Compute the sum \(\sum x_i = 52.9\) and divide by \(n=10\): \[ \bar{x}=\frac{52.9}{10}=5.29. \]
Therefore, the point estimate (the point of estimation) for \(\mu\) is \(\hat{\mu}=5.29\).
Step-by-step: point estimate of the population variance
For approximately normal data, a standard point estimator for \(\sigma^2\) is the unbiased sample variance.
\[ \hat{\sigma}^2=s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}. \]
Using \(\bar{x}=5.29\), the squared-deviation sum is \(\sum (x_i-\bar{x})^2=0.329\), so \[ s^2=\frac{0.329}{9}=0.03656\quad(\text{rounded}). \]
Thus, the point estimate for \(\sigma^2\) is \(\hat{\sigma}^2 \approx 0.03656\).
Step-by-step: point estimate of a population proportion
If \(x\) successes are observed in \(n\) independent trials, the point estimator for the population proportion \(p\) is \[ \hat{p}=\frac{x}{n}. \]
Example: \(x=12\) successes out of \(n=50\) trials gives \[ \hat{p}=\frac{12}{50}=0.24. \]
How a good point estimator is judged
- Unbiasedness: \(E(\hat{\theta})=\theta\).
- Consistency: \(\hat{\theta}\) tends to \(\theta\) as \(n\) increases.
- Efficiency: among unbiased estimators, smaller variance is preferred.
Visualization: sampling distribution and a point estimate
The curve represents a sampling distribution for an estimator (such as \(\bar{x}\)). The vertical line marks the true parameter \(\theta\), and the dot marks one observed point estimate \(\hat{\theta}\).