How to Find a Confidence Interval For a Sample

Confidence Intervals, T-Distribution 27 Comments

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. This article will show you how in a short series of steps.

Example question:

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

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 it! Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.

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27 Responses to “How to Find a Confidence Interval For a Sample”

  1. Mary Johnson said:

    Oct 20, 09 at 11:17 am

    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 said:

    Oct 23, 09 at 11:00 am

    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. Stephanie said:

    Oct 23, 09 at 11:14 am

    Mary,
    Degrees of freedom is n-1,
    Stephanie

  4. Vanessa DuBarry said:

    Oct 23, 09 at 1:13 pm

    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.

  5. Stephanie said:

    Oct 23, 09 at 3:48 pm

    Vanessa,
    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.
    Stephanie

  6. Donna Allen said:

    Oct 24, 09 at 5:16 pm

    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.

  7. April Fulton said:

    Oct 25, 09 at 8:43 pm

    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!

  8. Jennifer Thomas said:

    Oct 25, 09 at 9:05 pm

    Once again, these step by step instructions were very helpful.

  9. Lisa Barcomb said:

    Nov 01, 09 at 8:16 pm

    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.

  10. angie widdows said:

    Nov 11, 09 at 6:02 pm

    Like it. This is a very good example. Helps to have the table there for reference.

  11. Joni Poore said:

    Mar 07, 10 at 7:20 pm

    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.

  12. Alison Bryant said:

    Mar 22, 10 at 10:30 am

    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.

  13. Mary Brown said:

    Jul 17, 10 at 8:37 am

    This was very helpful to me it explained by steps and I have it Thanks so Much

  14. Mary Brown said:

    Jul 17, 10 at 8:38 am

    This site was very helpful to me and now I understand it very well!

  15. Sherry said:

    Nov 13, 10 at 12:39 pm

    I have a problem the the DF=85, how do I find the value on the table??

  16. Kelly said:

    Mar 01, 11 at 11:32 am

    This was so helpful!

  17. omar said:

    Aug 16, 12 at 1:45 pm

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

    regards,

    Omar
    Civil Eng.

  18. Andale said:

    Aug 17, 12 at 7:44 am

    Omar,

    See steps 6 and 7 for the CI,

    Stephanie

  19. omar said:

    Aug 17, 12 at 2:34 pm

    Dear Stephanie,

    Thank you very much for your reply.

    regards,

    Omar,

  20. Okeke Paul said:

    Oct 16, 12 at 11:53 am

    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

  21. TerrorByteFX said:

    Nov 11, 12 at 2:16 am

    Thanks, you helped simplify the problem I was having.

  22. rushad said:

    Mar 08, 13 at 9:57 am

    what is the t table value if df=59????

  23. Andale said:

    Mar 09, 13 at 4:54 pm

    Rushad,

    For which alpha value? It’ll be very close to the values at df=60.

    If you have a TI-89 you can use that to get an exact answer:
    How to find a t-distribution on the TI-89.

    Stephanie

  24. NTEKA FELICIA said:

    Mar 21, 13 at 5:42 am

    THANK YOU ANYWAY FOR THE INFORMATION BUT I AM REALLY GETTING CONFUSED ON HOW DOES MARGIN OF ERROR RELATE TO THE STANDARD DEVIATION

  25. Andale said:

    Mar 21, 13 at 8:49 am

    Hi, Nteka,

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

    Regards,
    Stephanie

  26. Amanpreet said:

    Apr 01, 13 at 7:31 am

    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

  27. Andale said:

    Apr 01, 13 at 5:57 pm

    Amanpreet,

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

    Thanks,
    Stephanie