Confidence Interval Calculator

How This Calculator Works

A confidence interval estimates the range where a true population value is expected to fall.

Step 1: Calculate Standard Error

Standard Error = Standard Deviation ÷ √Sample Size

Step 2: Calculate Margin of Error

Margin of Error = Critical Value × Standard Error

Step 3: Calculate Confidence Interval

Lower Limit = Mean − Margin of Error

Upper Limit = Mean + Margin of Error

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What Is a Confidence Interval?

A confidence interval (CI) is a statistical range used to estimate an unknown population parameter based on sample data. When estimating a population mean, the confidence interval provides a lower and upper bound that are likely to include the true population mean.

Instead of producing a single estimate, a confidence interval expresses the uncertainty associated with sampling. This makes it a more informative way to report results than using only a sample mean.

Understanding Confidence Levels

Every confidence interval is calculated using a chosen confidence level, such as 90%, 95%, or 99%. The confidence level reflects how reliable the estimation method is over repeated sampling.

A common misunderstanding is that a 95% confidence interval means there is a 95% probability that the true population mean lies within the calculated interval. This is not correct.

The correct interpretation is that if the same sampling process were repeated many times and a 95% confidence interval were calculated for each sample, approximately 95% of those intervals would contain the true population mean, while about 5% would not.

For example, if 100 independent samples are collected and a 95% confidence interval is calculated for each one, you would expect roughly 95 of those intervals to include the true population mean.

How Confidence Intervals Are Expressed

Confidence intervals can be written in several equivalent formats.

For example, these all represent the same interval:

• 20.6 ± 0.887
• 20.6 ± 4.3%
• 19.713 to 21.487

Each format communicates the same estimated range using different notation.

How to Calculate a Confidence Interval

This calculator estimates confidence intervals for data that are approximately normally distributed when the population standard deviation is known or can be reasonably approximated.

To calculate a confidence interval, you need:

• The sample mean (x̄)
• The standard deviation (σ)
• The sample size (n)
• The desired confidence level

When the population standard deviation is unknown, the sample standard deviation (s) is often used instead. For larger samples (typically n ≥ 30), the sample standard deviation generally provides a good approximation of the population standard deviation.

For simplicity, this calculator assumes the population standard deviation is known or that the sample size is large enough for the approximation to be appropriate.

The confidence interval is calculated using the following formula:

Confidence Interval = x̄ ± Z × (σ / √n)

Where:

• x̄ = sample mean
• Z = critical value corresponding to the selected confidence level
• σ = standard deviation
• n = sample size

Example Calculation

Suppose you have the following values:

• Sample mean (x̄) = 22.8
• Standard deviation (σ) = 2.7
• Sample size (n) = 100
• Confidence level = 95%

The critical Z-value for a 95% confidence level is 1.96.

Margin of error:

1.96 × (2.7 / √100) = 0.5292

Confidence interval:

22.8 ± 0.5292

This corresponds to a confidence interval of:

22.27 to 23.33

Common Z-Values for Confidence Intervals

Confidence Level	Z-Value
70%	1.036
75%	1.150
80%	1.282
85%	1.440
90%	1.645
95%	1.960
98%	2.326
99%	2.576
99.5%	2.807
99.9%	3.291
99.99%	3.891
99.999%	4.417

These Z-values are commonly used when constructing confidence intervals for normally distributed data and correspond to the selected confidence level.