Confidence Interval for the Mean Calculator

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

   

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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 - tscore_α * s/√n < μ < X + tscore_α * 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

Find α spread range:
α = ½(α)
α = ½(0.05)
α = 0.025

Find t-score for α_0.025 using 29 degrees of freedom:
tscore_0.025 = 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
	5.4772255750517

SEM = 0.8398

Calculate high end confidence interval total:
High End = X + tscore_α * 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 - tscore_α * 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:

You have 2 free calculationss remaining

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

Confidence Interval for the Mean Calculator Video

Tags:

• confidence interval for the mean
• confidence interval
• degrees of freedom
• mean
• sample size
• standard error of the mean

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