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Point Estimation

Estimate μ̂, unbiased σ̂² and sample sd from sample data.

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μ̂=x̄, σ̂²=Σ(xᵢ−x̄)²/(n−1)
enter sample observations

📖 Tutorial | Point Estimation

1. Definition
Point estimation uses a sample statistic to estimate an unknown parameter. Mean estimate μ̂=x̄; unbiased variance s²=Σ(xᵢ−x̄)²/(n−1).
2. Symbols
SymbolMeaning
μ̂estimate of the population mean
s²unbiased sample variance
ssample standard deviation
n−1degrees of freedom for unbiasedness
3. How it works
  • Sample mean x̄=Σxᵢ/n;
  • Unbiased variance divides by n−1;
  • Sample sd s=√s²;
  • A point estimate is a single number; interval estimate gives a range.
4. Steps
  1. Enter sample data (comma separated);
  2. Click Calculate;
  3. Read μ̂, s², s;
  4. Note it is a single-point approximation.
5. Example
Example: 2,4,4,6,6,6,8,8,10. x̄=6; unbiased variance=6; sample sd≈2.449.
6. Pitfalls
A point estimate is a single value, not the true value;
Unbiased variance divides by n−1, not n;
At least two data points are needed.

❓ FAQ | Point Estimation

Point vs interval estimation?
Point: one number; interval: a confidence range with a confidence level.
Why divide variance by n−1?
Only n−1 makes it an unbiased estimate of the population variance.
Is the sample mean always good?
Usually unbiased and consistent, but affected by sampling noise.
Are σ̂² and s² the same?
Here we use unbiased s² (÷n−1); MLE uses ÷n.
What makes a good estimator?
Unbiasedness, efficiency, consistency.