0.05 Confidence Interval Calculator

A 0.05 confidence interval is a statistical range that estimates the true population mean with 95% confidence. This calculator helps you compute confidence intervals for sample means when the population standard deviation is known.

What is a 0.05 Confidence Interval?

A 0.05 confidence interval provides a range of values that likely contains the true population mean. The "0.05" refers to the significance level (α = 0.05), meaning there's a 5% chance the interval doesn't contain the true mean.

In practical terms, a 95% confidence interval means if you took 100 samples and computed a 95% confidence interval for each, you would expect about 95 of those intervals to contain the true population mean.

Note: This calculator assumes you know the population standard deviation. If you only have sample standard deviation, use a t-distribution instead.

How to Use This Calculator

Enter your sample mean (x̄)
Enter the population standard deviation (σ)
Enter your sample size (n)
Click "Calculate" to get your confidence interval

The calculator will display the lower and upper bounds of your 95% confidence interval.

Formula Explained

The formula for a 95% confidence interval when σ is known is:

Confidence Interval = x̄ ± z*(σ/√n)

Where:

x̄ = sample mean
z = z-score for 95% confidence (1.96)
σ = population standard deviation
n = sample size

The z-score of 1.96 corresponds to the critical value where 2.5% of the area lies in each tail of the standard normal distribution.

Interpreting Results

When you get a confidence interval like "45.2 to 52.8", this means you're 95% confident that the true population mean falls between these two values.

Key points to remember:

The interval doesn't guarantee the true mean is within the range - it's a probabilistic statement
A narrower interval indicates more precise estimation
To increase precision, increase your sample size or reduce the population standard deviation

Worked Example

Suppose you measure the heights of 36 randomly selected adults and find:

Sample mean (x̄) = 170 cm
Population standard deviation (σ) = 10 cm
Sample size (n) = 36

Using the formula:

Margin of Error = 1.96 × (10/√36) = 1.96 × (10/6) ≈ 3.27

Confidence Interval = 170 ± 3.27 = (166.73, 173.27)

You can be 95% confident that the true average height of all adults falls between 166.73 cm and 173.27 cm.

FAQ

What does a 0.05 confidence level mean?

A 0.05 confidence level means there's a 5% chance the confidence interval doesn't contain the true population mean. This is equivalent to 95% confidence that the interval does contain the true mean.

Can I use this calculator for small sample sizes?

Yes, this calculator works for any sample size. However, for very small samples (n < 30), you might want to consider using a t-distribution instead of a normal distribution.

What if I don't know the population standard deviation?

If you only have the sample standard deviation, you should use a t-distribution confidence interval instead. This calculator assumes you know σ.

How does sample size affect the confidence interval?

Larger sample sizes produce narrower confidence intervals because the margin of error decreases as sample size increases. The relationship is inverse square root: doubling the sample size halves the margin of error.