Z Score to Confidence Interval Calculator

This calculator converts a z-score to a confidence interval, helping you understand the range of values that likely contain the true population mean. Learn how to interpret z-scores and confidence intervals in statistics.

What is a Z Score?

A z-score (also called a standard score) measures how many standard deviations an element is from the mean. Z-scores allow you to compare values from different normal distributions.

The formula for calculating a z-score is:

z = (X - μ) / σ

Where:

X = individual raw score
μ = population mean
σ = population standard deviation

Z-scores help determine whether a data point is within a normal range or an outlier.

Z Score to Confidence Interval

Converting a z-score to a confidence interval provides a range of values that likely contains the true population mean. The confidence interval is calculated using the z-score and standard error.

The formula for a confidence interval using a z-score is:

Confidence Interval = X ± (z * σ/√n)

Where:

X = sample mean
z = z-score
σ = population standard deviation
n = sample size

This formula gives you the lower and upper bounds of the confidence interval.

Note: For large samples (n > 30), the z-score can approximate the t-distribution. For smaller samples, consider using a t-score instead.

How to Use This Calculator

Enter your sample mean (X)
Enter the z-score value
Enter the population standard deviation (σ)
Enter the sample size (n)
Click "Calculate" to get the confidence interval

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

Example Calculation

Suppose you have a sample mean (X) of 50, a z-score of 1.96, a population standard deviation (σ) of 10, and a sample size (n) of 100.

Using the formula:

Confidence Interval = 50 ± (1.96 * 10/√100) Confidence Interval = 50 ± (1.96 * 10/10) Confidence Interval = 50 ± 1.96 Lower bound = 50 - 1.96 = 48.04 Upper bound = 50 + 1.96 = 51.96

The 95% confidence interval is (48.04, 51.96).

Frequently Asked Questions

What is the difference between a z-score and a confidence interval?

A z-score measures how many standard deviations a value is from the mean, while a confidence interval provides a range of values that likely contains the true population mean.

How do I choose the right z-score for my confidence interval?

The z-score depends on your desired confidence level. Common z-scores are 1.96 for 95% confidence, 2.58 for 99% confidence, and 1.645 for 90% confidence.

Can I use a z-score for small samples?

For small samples (n < 30), it's better to use a t-score instead of a z-score, as the t-distribution is more appropriate for smaller sample sizes.