Statistics How To

Confidence Interval For a Sample: Find one in easy steps

Probability and Statistics > T-distribution > Confidence Interval For a Sample

Watch the video or read the steps below:

Confidence Interval For a Sample: Overview

When you don’t know anything about a population’s behavior (i.e. you’re just looking at data for a sample), you need to use the t-distribution to find the confidence interval. That’s the vast majority of cases: you usually don’t know population parameters, otherwise you wouldn’t be looking at statistics!

The confidence interval tells you how confident you are in your results. With any survey or experiment, you’re never 100% sure that your results could be repeated. If you’re 95% sure, or 98% sure, that’s usually considered “good enough” in statistics. That percentage of sureness is the confidence interval.
confidence interval for a sample

Confidence Interval For a Sample: Steps

Sample question:

A group of 10 foot surgery patients had a mean weight of 240 pounds. The sample standard deviation was 25 pounds. Find a confidence interval for a sample for the true mean weight of all foot surgery patients. Find a 95% CI.

Step 1: Subtract 1 from your sample size. 10 – 1 = 9. This gives you degrees of freedom, which you’ll need in step 3.

Step 2: Subtract the confidence level from 1, then divide by two.
(1 – .95) / 2 = .025

Step 3: Look up your answers to step 1 and 2 in the t-distribution table. For 9 degrees of freedom (df) and α = 0.025, my result is 2.262.

df α = 0.1 0.05 0.025 0.01 0.005 0.001 0.0005
tα=1.282 1.645 1.960 2.326 2.576 3.091 3.291
1 3.078 6.314 12.706 31.821 63.656 318.289 636.578
2 1.886 2.920 4.303 6.965 9.925 22.328 31.600
3 1.638 2.353 3.182 4.541 5.841 10.214 12.924
4 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 5.894 6.869
6 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 1.383 1.833 2.262

Step 4: Divide your sample standard deviation by the square root of your sample size.
25 / √(10) = 7.90569415

Step 5: Multiply step 3 by step 4.
2.262 × 7.90569415 = 17.8826802

Step 6: For the lower end of the range, subtract step 5 from the sample mean.
240 – 17.8826802 = 222.117

Step 7: For the upper end of the range, add step 5 to the sample mean.
240 + 17.8826802 = 257.883

That’s how to find the confidence interval for a sample!

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Confidence Interval For a Sample: Find one in easy steps was last modified: September 3rd, 2015 by Stephanie

27 thoughts on “Confidence Interval For a Sample: Find one in easy steps

  1. Mary Johnson

    I found this helpful but I am still not sure about the degrees of freedom. Even in the guided solution on the homework problems there are steps left out. It helps but I have to do alot of searching to understand where some of the numbers come from.

  2. Vanessa DuBarry

    This sample really helped me a lot, because since I am not good at math it really helps that it is explained step by step not like math zone that its so confusing.

  3. Vanessa DuBarry

    proffesor I dont know what is happening but I understand how to do do this but when I do it this way on math zone it says its wrong and they show me how to do it a very different way, and I dont understand it.

  4. Stephanie

    Can you send me an email with the Mathzone question you are working on? And your working out? That way, I’ll be able to compare them and see what’s going on.

  5. Donna Allen

    I found your explanation very helpful. I was a little confused in step 3 where a=0.25, I thought it was 0.025. I think it might have been typed in error. Otherwise, everything else made sense.

  6. April Fulton

    I truly had no idea how to find the degrees of freedom, now from this site and again the step by step process I see I take the sample size and subtract 1. How easy is that!

  7. Lisa Barcomb

    Now this problem made since maybe its because it was a step by step process that I understood and it made a lot of sense regarding the sample size. So once I finished reading the instructions and started to work out the problem it all came to me. Which makes it a whole lot easier on my brain.

  8. Joni Poore

    This was a life saver this week. The book and the CD do NOT give any help with this. Love that table, came in hand a lot this week.

  9. Alison Bryant

    This was really helpful I too was having problems understanding how to find and use the degrees of freedom, but thanks now I understand much better.

  10. omar

    so that finally , what is the conclusion of calculation ? and what is the range of confidence ?


    Civil Eng.

  11. Okeke Paul

    Like seriously this was so helpful… I’m a masscommunication student and I ad to do a course called quatitative techniques .. Had no idea about all this buh I took it on my self to learn it due to a reason dat im a programmer also buh it was challenging and it is still though.. I really got relieved after reading this HOW TO DO IT tips… Thanks a lot

  12. Andale

    Hi, Nteka,

    Could you post your question on our forum? One of our mods will be glad to answer your question.


  13. Amanpreet

    Thanks it was really helpful !
    But why cant we straightaway Use Mean and Standard deviation instead of Standard error we have used in calculation above.

    What is the cut off at which we can consider a sample a good reprensative of mean – where we can suitable use Mean +/- SD for calculations ?

    Thnks in advance

  14. Andale


    Thanks for stopping by. Can you post your question in our forums? Click the Forums tab at the top ^^.


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