95 Confidence Interval with 248 Degrees of Freedom Calculator

This calculator computes the 95% confidence interval for a sample mean when you have 248 degrees of freedom. It's commonly used in statistics to estimate population parameters with a known standard deviation.

What is a 95% Confidence Interval?

A 95% confidence interval is a range of values that is likely to contain the true population parameter with 95% probability. For a sample mean, this interval is calculated using the sample mean, standard deviation, and degrees of freedom.

Key Formula

Confidence Interval = Sample Mean ± (t-value × Standard Error)

Where t-value comes from the t-distribution table with 248 degrees of freedom.

The 95% confidence level means that if you were to take many samples and calculate a 95% confidence interval for each, approximately 95% of these intervals would contain the true population mean.

Degrees of Freedom Explained

Degrees of freedom (df) represent the number of independent pieces of information available in your sample. For a confidence interval calculation, degrees of freedom are typically calculated as:

df = n - 1

Where n is the sample size

With 248 degrees of freedom, your sample size is 249 (n = df + 1). This means you have a relatively large sample size, which typically results in a more precise confidence interval.

How to Calculate

The calculation involves several steps:

Calculate the sample mean (x̄)
Calculate the standard error (SE) = σ/√n
Find the t-value from the t-distribution table with 248 df and 95% confidence level
Calculate the margin of error = t-value × SE
Determine the confidence interval = x̄ ± margin of error

Note: This calculator assumes you know the population standard deviation (σ). If you only have the sample standard deviation (s), you should use the t-distribution instead of the normal distribution.

Interpreting Results

When you get a confidence interval like "45.2 to 48.7", this means you're 95% confident that the true population mean falls within this range. Here's what this implies:

If you took many samples and calculated 95% confidence intervals for each, about 95% of them would contain the true population mean
The interval width depends on your sample size and standard deviation - larger samples give narrower intervals
A 95% confidence interval doesn't mean there's a 95% probability that any particular value is the true mean

In practical terms, this means you can be reasonably confident that the true population mean is between your calculated lower and upper bounds.

FAQ

What does 248 degrees of freedom mean?: It means your sample size is 249 (n = df + 1). With more degrees of freedom, your t-distribution approaches the normal distribution, making your confidence interval more precise.
Can I use this calculator for other confidence levels?: This specific calculator is designed for 95% confidence intervals. For other confidence levels, you would need to adjust the t-value accordingly.
What if I only have the sample standard deviation?: You should use the t-distribution instead of the normal distribution when calculating confidence intervals with sample standard deviations.
How does sample size affect the confidence interval?: Larger sample sizes generally result in narrower confidence intervals because the standard error decreases as sample size increases.
What if my data is not normally distributed?: For small samples (n < 30), your data should be approximately normally distributed. For larger samples, the Central Limit Theorem often ensures the sampling distribution is normal.