One-mean confidence interval
Estimate a population mean with a margin of error.
One-mean confidence interval is one of 2 confidence intervals formulas in the statistics section of this library, and it is used at ap · university level.
Why one-mean confidence interval works
The sample mean misses the truth by a random amount whose typical size is the standard error. Multiplying that by a critical value stretches it into a range that captures the truth in a stated share of repeated samples. The t distribution is used because the spread itself had to be estimated, which needs slightly wider tails than a normal curve.
What each symbol means
$t^*$ comes from confidence level and $n-1$ degrees of freedom.
One-mean confidence interval: when it holds
Use independent random observations and check the sampling distribution is suitable for the sample size and shape.
When it stops applying
The interval assumes independent observations drawn at random from a population that is not strongly skewed, unless the sample is large. Worse, if the sample was not random at all, no width can rescue it: a convenience sample gives an interval centred in the wrong place, and increasing n only makes a confidently wrong answer narrower.
One-mean confidence interval: a worked example
If $\bar x=50,s=10,n=25,t^*=2.064$, the interval is $50\pm4.128$.
The mistake to avoid
What people do: Multiplying the critical value by the sample standard deviation instead of the standard error.
Why it goes wrong: With a mean of 50, a spread of 10, 25 observations and a critical value of 2.064, the correct margin is 4.128. Forgetting the square root of 25 gives 20.64, an interval five times too wide.
Do this instead: Compute the standard error as a separate step first, write it down, and only then multiply it by the critical value.
One-mean confidence interval: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $t^*$ comes from confidence level and $n-1$ degrees of freedom.
- Check the conditions before substituting. Use independent random observations and check the sampling distribution is suitable for the sample size and shape.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Confidence intervals
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where one-mean confidence interval comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about one-mean confidence interval
Does 95 percent mean there is a 95 percent chance the true mean is in my interval?
No. The true mean is a fixed number, and your interval either contains it or does not. The 95 percent describes the method: intervals built this way capture the truth in 95 percent of repeated samples.
How do I find the critical value?
Use n minus 1 degrees of freedom and your confidence level. For 24 degrees of freedom at 95 percent it is 2.0639, and for 9 degrees of freedom the same confidence needs 2.2622.
What makes the interval narrower?
A larger sample, a smaller spread in the data, or accepting a lower confidence level. Only the first of those improves your information rather than lowering your standard of evidence.
When would I use a normal critical value instead of t?
Only when the population standard deviation is genuinely known rather than estimated, which is rare outside textbook problems. For large samples the two critical values are so close that the choice barely matters.