Calculate confidence intervals with all confidence levels from 50% to 99.9%
ℹ️ Distribution Note
For sample sizes < 30, this calculator uses the t-distribution. For larger samples, it uses the normal distribution (z-scores).
📋 Results
Confidence Interval:
45.04 to 54.96
at 95% confidence level
50.00
Sample Mean
± 4.96
Margin of Error
10.00
Standard Deviation
1.960
Critical Value (z)
📝 Calculation Steps
1. Determine the critical value (z-score or t-score) for the selected confidence level.
2. Calculate the standard error: SE = σ / √n
3. Calculate the margin of error: ME = critical value × SE
4. Calculate the confidence interval: CI = x̄ ± ME
Confidence Interval = x̄ ± (z* × σ/√n)
Where:
x̄ is the sample mean
z* is the critical value from the standard normal distribution
σ is the population standard deviation (or sample standard deviation for large n)
n is the sample size
🎓 Academic Research
Study of 100 students found an average test score of 75 with a standard deviation of 10. The 95% confidence interval for the true population mean score would be:
CI = 75 ± 1.96 × (10/√100) = 73.04 to 76.96
💼 Market Research
Survey of 400 customers showed average satisfaction of 8.2/10 with a standard deviation of 1.5. The 99% confidence interval would be:
CI = 8.2 ± 2.576 × (1.5/√400) = 8.01 to 8.39
🏥 Medical Study
Clinical trial with 30 patients showed average blood pressure reduction of 12 mmHg with a standard deviation of 4. The 90% confidence interval would be:
Calculate confidence intervals with all confidence levels from 50% to 99.9%
ℹ️ Distribution Note
For sample sizes < 30, this calculator uses the t-distribution. For larger samples, it uses the normal distribution (z-scores).
📋 Results
Confidence Interval:
45.04 to 54.96
at 95% confidence level
50.00
Sample Mean
± 4.96
Margin of Error
10.00
Standard Deviation
1.960
Critical Value (z)
📝 Calculation Steps
1. Determine the critical value (z-score or t-score) for the selected confidence level.
2. Calculate the standard error: SE = σ / √n
3. Calculate the margin of error: ME = critical value × SE
4. Calculate the confidence interval: CI = x̄ ± ME
Confidence Interval = x̄ ± (z* × σ/√n)
Where:
x̄ is the sample mean
z* is the critical value from the standard normal distribution
σ is the population standard deviation (or sample standard deviation for large n)
n is the sample size
🎓 Academic Research
Study of 100 students found an average test score of 75 with a standard deviation of 10. The 95% confidence interval for the true population mean score would be:
CI = 75 ± 1.96 × (10/√100) = 73.04 to 76.96
💼 Market Research
Survey of 400 customers showed average satisfaction of 8.2/10 with a standard deviation of 1.5. The 99% confidence interval would be:
CI = 8.2 ± 2.576 × (1.5/√400) = 8.01 to 8.39
🏥 Medical Study
Clinical trial with 30 patients showed average blood pressure reduction of 12 mmHg with a standard deviation of 4. The 90% confidence interval would be:
