Enter N Enter X Enter σ or s Enter Confidence Interval % Rounding Digits

A small sample of 30 units has a mean μ and a standard deviation σ of 4.6. Find a 95% confidence interval of the mean μ

Confidence Interval Formula for μ is as follows:
X - tscoreα * s/√n < μ < X + tscoreα * s/√n where:
X = sample mean, s = sample standard deviation, tscore = statistic with (n - 1) Degrees of Freedom and α = 1 - confidence Percentage

First find degrees of freedom:
Degrees of Freedom = n - 1
Degrees of Freedom = 30 - 1
Degrees of Freedom = 29

Calculate α:
α = 1 - Confidence%
α = 1 - 0.95
α = 0.05

α = ½(α)
α = ½(0.05)
α = 0.025

Find t-score for α0.025 using 29 degrees of freedom:
tscore0.025 = 2.0452 <--- Value can be found on Excel using =TINV(0.05,29)
tscore = 2.0452

Locate Value in the t-chart below:

 DOF α α 0.005 α 0.01 α 0.015 α 0.02 α 0.025 α 0.03 α 0.035 α 0.04 α 0.045 α 0.05 1 63.6559 31.821 21.2051 15.8945 12.7062 10.5789 9.0579 7.9158 7.0264 6.3137 2 9.925 6.9645 5.6428 4.8487 4.3027 3.8964 3.5782 3.3198 3.104 2.92 3 5.8408 4.5407 3.8961 3.4819 3.1824 2.9505 2.7626 2.6054 2.4708 2.3534 4 4.6041 3.7469 3.2976 2.9985 2.7765 2.6008 2.4559 2.3329 2.2261 2.1318 5 4.0321 3.3649 3.0029 2.7565 2.5706 2.4216 2.2974 2.191 2.0978 2.015 6 3.7074 3.1427 2.8289 2.6122 2.4469 2.3133 2.2011 2.1043 2.0192 1.9432 7 3.4995 2.9979 2.7146 2.5168 2.3646 2.2409 2.1365 2.046 1.9662 1.8946 8 3.3554 2.8965 2.6338 2.449 2.306 2.1892 2.0902 2.0042 1.928 1.8595 9 3.2498 2.8214 2.5738 2.3984 2.2622 2.1504 2.0554 1.9727 1.8992 1.8331 10 3.1693 2.7638 2.5275 2.3593 2.2281 2.1202 2.0283 1.9481 1.8768 1.8125 11 3.1058 2.7181 2.4907 2.3281 2.201 2.0961 2.0067 1.9284 1.8588 1.7959 12 3.0545 2.681 2.4607 2.3027 2.1788 2.0764 1.9889 1.9123 1.844 1.7823 13 3.0123 2.6503 2.4358 2.2816 2.1604 2.06 1.9742 1.8989 1.8317 1.7709 14 2.9768 2.6245 2.4149 2.2638 2.1448 2.0462 1.9617 1.8875 1.8213 1.7613 15 2.9467 2.6025 2.397 2.2485 2.1315 2.0343 1.9509 1.8777 1.8123 1.7531 16 2.9208 2.5835 2.3815 2.2354 2.1199 2.024 1.9417 1.8693 1.8046 1.7459 17 2.8982 2.5669 2.3681 2.2238 2.1098 2.015 1.9335 1.8619 1.7978 1.7396 18 2.8784 2.5524 2.3562 2.2137 2.1009 2.0071 1.9264 1.8553 1.7918 1.7341 19 2.8609 2.5395 2.3457 2.2047 2.093 2 1.92 1.8495 1.7864 1.7291 20 2.8453 2.528 2.3362 2.1967 2.086 1.9937 1.9143 1.8443 1.7816 1.7247 21 2.8314 2.5176 2.3278 2.1894 2.0796 1.988 1.9092 1.8397 1.7773 1.7207 22 2.8188 2.5083 2.3202 2.1829 2.0739 1.9829 1.9045 1.8354 1.7734 1.7171 23 2.8073 2.4999 2.3132 2.177 2.0687 1.9783 1.9003 1.8316 1.7699 1.7139 24 2.797 2.4922 2.3069 2.1715 2.0639 1.974 1.8965 1.8281 1.7667 1.7109 25 2.7874 2.4851 2.3011 2.1666 2.0595 1.9701 1.8929 1.8248 1.7637 1.7081 26 2.7787 2.4786 2.2958 2.162 2.0555 1.9665 1.8897 1.8219 1.761 1.7056 27 2.7707 2.4727 2.2909 2.1578 2.0518 1.9632 1.8867 1.8191 1.7585 1.7033 28 2.7633 2.4671 2.2864 2.1539 2.0484 1.9601 1.8839 1.8166 1.7561 1.7011 29 2.7564 2.462 2.2822 2.1503 2.0452 1.9573 1.8813 1.8142 1.754 1.6991

Calculate the Standard Error of the Mean:

 SEM  = σ √n

 SEM  = 4.6 √30

 SEM  = 4.6 5.47723

SEM = 0.8398

Calculate high end confidence interval total:
High End = X + tscoreα * s/√n
High End = 60.5 + 2.0452 x 4.6/√30
High End = 60.5 + 2.0452 x 0.83984125484125
High End = 60.5 + 1.7176433344013
High End = 62.2176

Calculate low end confidence interval total:
Low End = X - tscoreα * s/√n
Low End = 60.5 - 2.0452 x 4.6/√30
Low End = 60.5 - 2.0452 x 0.83984125484125
Low End = 60.5 - 1.7176433344013
Low End = 58.7824

Now we have everything, display our 95% confidence interval:
58.7824 < μ < 62.2176

What this means is if we repeated experiments, the proportion of such intervals that contain μ would be 95%

58.7824 < μ < 62.2176
How does the Confidence Interval for the Mean Calculator work?
Free Confidence Interval for the Mean Calculator - Calculates a (90% - 99%) estimation of confidence interval for the mean given a small sample size using the student-t method with (n - 1) degrees of freedom or a large sample size using the normal distribution Z-score (z value) method including Standard Error of the Mean. confidence interval of the mean
This calculator has 5 inputs.

What 1 formula is used for the Confidence Interval for the Mean Calculator?

SE = σ/√n

For more math formulas, check out our Formula Dossier

What 6 concepts are covered in the Confidence Interval for the Mean Calculator?

confidence interval
a range of values so defined that there is a specified probability that the value of a parameter lies within it.
confidence interval for the mean
a way of estimating the true population mean
degrees of freedom
number of values in the final calculation of a statistic that are free to vary
mean
A statistical measurement also known as the average
sample size
measures the number of individual samples measured or observations used in a survey or experiment.
standard error of the mean
measures how far the sample mean (average) of the data is likely to be from the true population mean
SE = σ/√n