Confidence Interval Calculator
Use this calculator to compute the confidence interval or margin of error, assuming the sample mean most likely follows a normal distribution.
Use the Standard Deviation Calculator if you have raw data only.
What is the confidence interval?
A confidence interval is a range of values that is likely to contain the true value of a population parameter (such as the mean) with a certain level of confidence. It provides an estimate of the uncertainty associated with a sample statistic.
The confidence level (e.g., 95%) indicates the probability that the confidence interval contains the true population parameter. For example, if you construct 100 different 95% confidence intervals from 100 different samples, approximately 95 of them would contain the true population mean.
Example:
If the confidence interval is 20.6 ± 0.887, this means:
- The sample mean is 20.6
- The margin of error is 0.887
- The confidence interval ranges from 19.713 to 21.487
- We are 95% confident that the true population mean lies within this range
Common Misconceptions:
- The confidence level does NOT mean that 95% of the sample data falls within the interval
- The confidence level does NOT mean that there is a 95% probability that the true value is in the interval (the true value is either in it or not)
- The confidence level refers to the long-run frequency of intervals that contain the true parameter
Different Ways to Express Confidence Intervals:
- As a value with margin of error: 20.6 ± 0.887
- As a percentage: 20.6 ± 4.3%
- As a range: [19.713 - 21.487]
Calculating confidence intervals
This calculator computes confidence intervals for the population mean when the sample mean follows a normal distribution. The calculation assumes:
- The data is normally distributed (or the sample size is large enough for the Central Limit Theorem to apply)
- The population mean is unknown
- The standard deviation is known (or estimated from the sample)
Formula:
Where:
- X = Sample mean
- Z = Z-value for the desired confidence level
- σ = Population standard deviation (or s for sample standard deviation)
- n = Sample size
Example:
Calculate a 95% confidence interval for a sample with:
- Sample mean (X) = 22.8
- Standard deviation (σ) = 2.7
- Sample size (n) = 100
- Z-value for 95% = 1.960
CI = 22.8 ± 1.960 × (2.7/√100)
= 22.8 ± 1.960 × 0.27
= 22.8 ± 0.5292
Confidence Interval: [22.271 - 23.329]
Note on Standard Deviation:
When the sample size is large (typically n ≥ 30), you can use the sample standard deviation (s) instead of the population standard deviation (σ). For smaller samples, you may need to use the t-distribution instead of the normal distribution, which requires a t-value instead of a Z-value.
Z-values for Confidence Intervals
The Z-value (also called Z-score or critical value) is the number of standard deviations from the mean that corresponds to a given confidence level. It is derived from the standard normal distribution.
| Confidence Level | Z Value |
|---|
| 70% | 1.036 |
| 75% | 1.150 |
| 80% | 1.282 |
| 85% | 1.440 |
| 90% | 1.645 |
| 91% | 1.695 |
| 92% | 1.751 |
| 93% | 1.812 |
| 94% | 1.881 |
| 95% | 1.960 |
| 96% | 2.054 |
| 97% | 2.170 |
| 98% | 2.326 |
| 99% | 2.576 |
| 99.5% | 2.807 |
| 99.9% | 3.291 |
| 99.95% | 3.481 |
| 99.99% | 3.891 |
| 99.995% | 4.056 |
| 99.999% | 4.417 |
How to Use the Z-value:
The Z-value is used in the confidence interval formula to determine how many standard errors to add and subtract from the sample mean. Higher confidence levels require larger Z-values, which result in wider confidence intervals.
Important: For sample sizes less than 30, or when the population standard deviation is unknown and must be estimated from the sample, you should use the t-distribution instead of the normal distribution. The t-values are larger than Z-values for the same confidence level, resulting in wider intervals that account for the additional uncertainty.
Applications of Confidence Intervals
Statistics and Research
- Estimating population parameters from sample data
- Comparing means between different groups
- Assessing the precision of survey results
- Quality control and process monitoring
Business and Economics
- Market research and consumer surveys
- Financial forecasting and risk assessment
- Performance metrics and KPIs
- A/B testing and experimentation
Science and Medicine
- Clinical trials and medical research
- Laboratory measurements and error analysis
- Epidemiological studies
- Experimental data interpretation