Confidence Interval Calculator
Your Confidence Interval is:
Margin of Error:
How Does This Calculator Work?
A confidence interval provides an estimated range of values which is likely to include an unknown population parameter, like the true mean. This calculator computes it using the following steps:
- Find the Standard Error: First, it calculates the standard error of the mean, which measures how much the sample mean is likely to vary from the true population mean. Formula: SE = s / √n (where 's' is the sample standard deviation and 'n' is the sample size).
- Determine the Margin of Error: It then calculates the margin of error by multiplying the standard error by a Z-score that corresponds to your chosen confidence level. Formula: Margin of Error = Z * SE.
- Construct the Interval: Finally, the confidence interval is found by adding and subtracting the margin of error from the sample mean. Formula: CI = x̄ ± Margin of Error.
The Surprising History of the Student's t-distribution
When calculating confidence intervals with small sample sizes (typically n < 30), a Z-score isn't always accurate. Instead, statisticians use a t-score from the "Student's t-distribution." The story behind this name is fascinating.
In the early 1900s, a chemist named William Sealy Gosset was working for the Guinness brewery in Dublin, Ireland. His job was to analyze small samples of barley to find the best-yielding varieties. He realized the standard statistical methods didn't work well for his small batches. He developed a new statistical distribution—the t-distribution—to handle this uncertainty. However, Guinness had a strict policy against its employees publishing research to protect trade secrets. Gosset convinced them to let him publish under a pseudonym, and he chose the humble name "Student." His 1908 paper, "The Probable Error of a Mean," published under this name, became one of the most important papers in the history of statistics.
Explore More Related Tools
While you're here, check out some of our other popular math and statistics calculators:
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Frequently Asked Questions (FAQ)
What does a 95% confidence interval actually mean?
It means that if we were to take many random samples from the same population and construct a confidence interval for each sample, about 95% of those intervals would contain the true population mean. It is a statement about the reliability of the method, not a probability about a single calculated interval.
What if I don't know my standard deviation?
If you have the raw data, you can use our Standard Deviation Calculator to find it first. If you don't have the raw data, you cannot accurately calculate the confidence interval.
Why does this calculator use a Z-score instead of a t-score?
For large sample sizes (typically n > 30), the Z-distribution and the Student's t-distribution are very similar. A Z-score is simpler to use and provides a very close approximation. For smaller samples, a t-score would be more accurate, but this calculator is designed for the more common large-sample case.