Confidence Interval Calculator
Print| Margin of Error (e) | ± 0.887 (4.31%) |
| Z-critical Value | 1.9600 |
| Standard Error (SE) | 0.4525 |
In statistics, a confidence interval is an estimated range of values that is likely to contain an unknown population parameter (such as the population mean). Determined through observed sample data, confidence intervals provide researchers with a measure of precision, reflecting the degree of uncertainty associated with sample-based estimations.
Our free Confidence Interval Calculator computes confidence intervals and margins of error assuming the sample mean follows a normal distribution. Simply enter your sample size, sample mean, standard deviation, and target confidence level to get instant, step-by-step calculations.
Understanding Confidence Levels: A Crucial Distinction
A confidence interval is calculated at a chosen confidence level (such as 90%, 95%, or 99%) selected prior to analyzing the data. This level indicates the reliability of the estimation procedure itself, rather than the probability of any single calculated interval containing the true value.
The 95% Confidence Misconception: If you construct 100 independent confidence intervals at a 95% confidence level, you expect 95 of those intervals to contain the true population parameter. However, once you isolate a single specific interval (e.g., [19.713 – 21.487]), you cannot say there is a “95% chance” it contains the parameter. The parameter is a fixed constant, not a random variable; therefore, the specific interval either 100% contains it or 0% contains it. The 95% probability refers strictly to the long-run success rate of the calculation process.
Confidence Interval Notations
Confidence intervals are commonly written in three mathematically equivalent formats:
- Margin of Error Format:
X¯ ± Range(e.g.,20.6 ± 0.887) - Percentage Format:
X¯ ± Percentage(e.g.,20.6 ± 4.3%) - Range Bounds:
[Lower Bound - Upper Bound](e.g.,[19.713 - 21.487])
Confidence Interval Formulas
This calculator computes confidence intervals for normally distributed data with an unknown population mean but a known standard deviation. If the true population standard deviation (σ) is unknown, the sample standard deviation (s) is substituted, provided the sample size is greater than 30 (as s converges to σ under the Law of Large Numbers):
CI = X¯ ± Z × ( σ / √n )
Where:
- X¯: The sample mean (arithmetic average).
- Z: The standard Z-value corresponding to the chosen confidence level.
- σ: The standard deviation (population or sample).
- n: The sample size (total count of elements).
- σ / √n: The standard error of the mean (SEM), reflecting sample distribution variance.
Confidence Interval Calculation Example
Construct a 95% confidence interval for a dataset where the sample mean (X¯) is 22.8, the standard deviation (σ) is 2.7, and the sample size (n) is 100:
- Find the Z-score for a 95% confidence level: Z = 1.960.
- Calculate the standard error of the mean:
SEM = 2.7 / √100 = 2.7 / 10 = 0.27 - Calculate the margin of error:
Margin of Error = 1.960 × 0.27 = 0.5292 - Apply the interval bounds:
CI = 22.8 ± 0.5292CI = [22.2708 - 23.3292]
The true population mean is estimated to lie between 22.2708 and 23.3292.
Z-Values for Confidence Levels
The table below details standard normal Z-scores for various confidence levels:
| Confidence Level | Z-Value |
|---|---|
| 70% | 1.036 |
| 80% | 1.282 |
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
| 99.9% | 3.291 |
Solve standard normal standardizations with our Z-score Calculator or determine required survey sizes using the Sample Size Calculator.
Frequently Asked Questions (FAQ)
What makes a confidence interval narrow?
The width of a confidence interval is determined by three variables: sample size, sample variance, and confidence level. You can narrow the interval (increasing precision) by increasing your sample size (n), selecting a population with lower standard deviation (σ), or choosing a lower confidence level (which decreases the Z-score multiplier).
When should I use a t-distribution instead of a z-distribution?
Use a z-distribution when the population standard deviation is known or the sample size is large (n > 30). Use a t-distribution when the population standard deviation is unknown and the sample size is small (n ≤ 30).
What is standard error of the mean (SEM)?
Standard error of the mean measures the dispersion of sample means around the true population mean. It is calculated by dividing standard deviation by the square root of the sample size: SEM = σ / √n. As sample size increases, standard error decreases.
How does a 99% confidence level affect the margin of error?
A higher confidence level requires a wider interval to guarantee the long-run success of the estimation procedure. A 99% confidence level uses a Z-score of 2.576, which is larger than the 1.960 used for a 95% confidence level. Consequently, the margin of error increases as the confidence level rises.