# What is the point estimate?

Computes the point estimate of a proportion (successes ÷ trials) with its Wilson confidence interval, or of a mean from data or a summary with its t confidence interval and margin of error.

- Page: https://www.acalculator.org/statistics/point-estimate-calculator
- JSON spec: https://www.acalculator.org/statistics/point-estimate-calculator.json
- Version: 8808bb626ff2

## Default answer

Example with the default inputs (Estimate Proportion, Successes (x) 26, Trials (n) 200, Confidence level 95%): The point estimate is 0.13, with a 95% confidence interval from 0.0902820228 to 0.1836635187.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | Estimate | A proportion from successes and trials, or a mean from data or from a summary. |
| x | Successes (x) | How many trials had the outcome of interest, such as defects found. |
| n | Trials (n) | The sample size: how many trials or items were checked. |
| data | Data | The sample values, separated by spaces, commas or new lines; at least 2. |
| N | Sample size (N) | How many values are in the sample; at least 2. |
| mean | Sample mean (x̄) | The average of the sample. |
| sd | Sample standard deviation (s) | The standard deviation of the sample (n − 1 in the denominator). |
| cl | Confidence level | The confidence level of the interval, such as 95. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| estimate | Point estimate | The sample proportion p̂ = x ÷ n, or the sample mean x̄. |
| lower | Lower limit | The lower end of the confidence interval (Wilson for a proportion, t for a mean). |
| upper | Upper limit | The upper end of the confidence interval. |
| margin | Margin of error | Half the width of the interval: (upper − lower) ÷ 2. |
| center | Wilson center | The middle of the Wilson interval: (p̂ + z² ÷ 2n) ÷ (1 + z² ÷ n), pulled toward 1/2. |
| se | Standard error | √(p̂(1 − p̂) ÷ n) for a proportion, s ÷ √N for a mean. |
| sd | Standard deviation (s) | The sample standard deviation. |
| critical | Critical value | z (proportion) or t with N − 1 degrees of freedom (mean) for the confidence level. |
| count | Sample size | The number of trials or values in the sample. |

## Method

Proportion: p̂ = x ÷ n; Wilson limits (p̂ + z²/2n ± z√(p̂(1 − p̂)/n + z²/4n²)) ÷ (1 + z²/n). Mean: x̄ ± t(1 − α/2, N − 1) × s ÷ √N.

## Assumptions

- The sample is random and the trials or values are independent.
- The t interval assumes the values come from a roughly normal population, or a large sample.
- z and t are the two-sided critical values for the confidence level: α = 1 − level, upper quantile α/2.

## Worked examples

1. mode = proportion, x = 26, n = 200, cl = 90% gives estimate = 0.13, lower = 0.095773. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.4.1 Confidence intervals (p̂ = N_d ÷ N; the Wilson score limits). https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm (p̂ = 0.13, N = 200: p ≥ 0.09577).
2. mode = summary, N = 195, mean = 9.26146, sd = 0.022789, cl = 95% gives estimate = 9.26146, lower = 9.258241, upper = 9.264679. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N₋₁ × s ÷ √N). https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (N = 195, mean 9.261460, s = 0.022789, t = 1.9723: interval 9.258242 to 9.264679).
3. mode = data, data = 12.1 or 11.8, cl = 95% gives estimate = 12.1, sd = 0.223607, margin = 0.277645. Source: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N₋₁ × s ÷ √N). https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm.

## FAQ

### What is a point estimate?

It is one number from a sample used as the best guess of a population value. The sample proportion p̂ = x ÷ n estimates a population proportion, and the sample mean x̄ estimates a population mean.

### How do I calculate the point estimate of a proportion?

Divide the successes by the trials: p̂ = x ÷ n. If 26 of 200 items are defective, p̂ = 26 ÷ 200 = 0.13.

### How do I find the point estimate from a confidence interval?

For a mean, the point estimate is the middle of the interval: (lower + upper) ÷ 2, and the margin of error is (upper − lower) ÷ 2. A Wilson interval for a proportion is not centred on p̂, so use x ÷ n instead.

### Why does the proportion use the Wilson interval?

The simple interval p̂ ± z√(p̂(1 − p̂)/n) can run below 0 or above 1 and is too narrow for small samples. NIST recommends the Wilson score interval, which always stays between 0 and 1.

### What is the difference between the point estimate and the margin of error?

The point estimate is the single best guess; the margin of error says how far the true value may be from it at the chosen confidence level. The interval is the estimate minus and plus the margin (for a mean).

### Should I use z or t for a mean?

When the population standard deviation is unknown and you use the sample standard deviation s, use t with N − 1 degrees of freedom, as NIST does. For large samples t is close to z.

## Sources

- NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.4.1 Confidence intervals (point estimate p̂ = N_d ÷ N; normal approximation interval; Wilson score limits; example p̂ = 0.13, N = 200, p ≥ 0.09577). https://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm (retrieved 2026-10-02)
- NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.2 Confidence Limits for the Mean (Ȳ ± t₁₋α/₂,N₋₁ s ÷ √N; example N = 195, mean 9.261460, s = 0.022789, interval 9.258242 to 9.264679). https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm (retrieved 2026-10-02)