Confidence Interval Calculator
Calculate confidence intervals for population mean using z-distribution or t-distribution. Automatically selects appropriate distribution based on sample size.
What is a confidence interval and what does it tell you?
A confidence interval is a range of values that likely contains the true population parameter. Example: 95% CI for average height = 165-175cm means "we're 95% confident the true average is between 165-175cm." The interval has a lower bound, upper bound, and confidence level. Wider intervals = more confident but less precise. Narrower = more precise but less confident. Used in polls, medical research, quality control, A/B testing.
What does 95% confidence level actually mean?
95% confidence means if you repeated the study 100 times, about 95 of those intervals would contain the true population value. It does NOT mean 95% probability the true value is in this specific interval. Common levels: 90% (z=1.645), 95% (z=1.96), 99% (z=2.576). Higher confidence = wider interval. Example: Political poll margin of error +/-3% at 95% confidence. If poll shows 52% support, true value likely 49-55%.
When do I use z-score vs t-score for confidence intervals?
Use Z-score when: (1) Sample size >=30, OR (2) Population SD known. Use T-score when: Sample size <30 AND population SD unknown. T-distribution has fatter tails (wider intervals for small samples). As sample size increases, t-distribution approaches z-distribution. Example: n=10, 95% CI → t=2.262 (vs z=1.96). n=100 → t≈1.984 (nearly same as z=1.96). Most real-world cases use t-score.
How does sample size affect confidence interval width?
Larger sample → narrower interval (more precise). Relationship: Width ∝ 1/sqrtn. Doubling sample size reduces width by 30% (sqrt2 ≈ 1.41). Example: n=100, margin of error = +/-10. For +/-5, need n=400 (4x sample for 2x precision). For +/-2.5, need n=1,600. Law of diminishing returns - each improvement costs more. Explains why polls use 1,000-2,000 people (good balance of cost vs precision).
What is the margin of error in a confidence interval?
Margin of Error (MOE) = half the width of confidence interval = z * (s/sqrtn). It's the +/- value reported in polls. Example: "52% +/- 3%" → CI is 49-55%, MOE = 3%. Affected by: (1) Confidence level (higher → larger MOE), (2) Sample size (larger → smaller MOE), (3) Variability (higher SD → larger MOE). Common misconception: MOE only accounts for random sampling error, not bias or non-response.
How do I interpret a confidence interval that includes zero?
When CI includes zero, the result is not statistically significant at that confidence level. Example: Difference between treatments = 5, 95% CI = [-2, 12]. Includes 0 → no significant difference (p>0.05). If CI = [2, 12] (doesn't include 0) → significant difference (p<0.05). For ratios/percentages, check if CI includes 1.0 or 0% respectively. Critical for hypothesis testing, drug trials, A/B tests.
What factors make a confidence interval wider or narrower?
WIDER intervals from: (1) Higher confidence level (99% vs 90%), (2) Smaller sample size, (3) Higher variability (larger SD), (4) Small sample with t-score. NARROWER from: (1) Lower confidence level, (2) Larger sample, (3) Lower variability, (4) Known population SD. Example: same data, 90% CI = [48-52], 95% CI = [47-53], 99% CI = [45-55]. Trade-off between precision and confidence.
How are confidence intervals used in A/B testing?
In A/B tests, CI shows range for difference between variants. Example: Variant A conversion 10%, B conversion 12%, difference = 2%. If 95% CI for difference = [0.5%, 3.5%] → significant (doesn't include 0), implement B. If CI = [-0.5%, 4.5%] → not significant, need more data. Calculate: CI for difference = (pA - pB) +/- zsqrt[(pA(1-pA)/nA) + (pB(1-pB)/nB)]. Minimum detectable effect (MDE) determines required sample size.
What is the relationship between confidence intervals and hypothesis testing?
CI and hypothesis testing are two sides of same coin. If 95% CI doesn't include the null hypothesis value → reject null at alpha=0.05. Example: Testing if mean = 100. If 95% CI = [105, 115] (doesn't include 100) → reject null, p<0.05. If CI = [95, 110] (includes 100) → fail to reject, p>0.05. CI provides more information (effect size + significance), while p-value only shows significance. Best practice: report both CI and p-value.
How do I calculate confidence interval for proportions vs means?
For MEANS: CI = x̄ +/- t(s/sqrtn) or x̄ +/- z(rho�/sqrtn). Uses normal/t-distribution. For PROPORTIONS: CI = p +/- zsqrt[p(1-p)/n]. Uses binomial approximation. Example: Mean salary CI: $50,000 +/- 1.96($5,000/sqrt100) = $49,020-$50,980. Proportion CI: 60% success rate +/- 1.96sqrt[0.6*0.4/100] = 50.4%-69.6%. Proportions require np>=5 and n(1-p)>=5 for normal approximation validity. Use exact methods for small samples.
What are common mistakes when interpreting confidence intervals?
MISTAKE 1: "95% probability true value is in this interval" → Wrong! The interval is fixed, true value either is or isn't in it. Correct: "95% of such intervals contain true value." MISTAKE 2: Comparing overlapping CIs → CIs can overlap while still being significantly different. Need CI of the difference. MISTAKE 3: Ignoring assumptions → Normal distribution, independent samples, random sampling. MISTAKE 4: Confusing CI with prediction interval → CI is for population mean, PI is for individual values (much wider).
How do confidence intervals work with small sample sizes?
Small samples (n<30) require t-distribution with degrees of freedom = n-1. T-distribution has heavier tails → wider intervals. Example: n=5, 95% CI uses t=2.776 (vs z=1.96), ~41% wider interval. As n increases: n=10 → t=2.262, n=20 → t=2.093, n=30 → t=2.042, n=∞ → t=1.96. Also need to verify normality assumption with small samples (use Shapiro-Wilk test, Q-Q plot). For highly skewed data, consider bootstrap methods or transformations.