Example Data Table
| Example |
Sample Size |
Sample Standard Deviation |
Confidence Level |
Degrees of Freedom |
Lower Standard Deviation |
Upper Standard Deviation |
| Quality control sample |
12 |
4.8000 |
95.00% |
11 |
3.4003 |
8.1498 |
| Exam score variation |
30 |
9.5000 |
99.00% |
29 |
7.0717 |
14.1233 |
| Lab measurement spread |
18 |
2.1500 |
90.00% |
17 |
1.6878 |
3.0103 |
Formula Used
The calculator uses the chi-square distribution to estimate a confidence interval for the unknown population variance and population standard deviation.
When the population mean is unknown:
df = n - 1
SS = (n - 1)s²
When a known population mean is supplied with raw data:
df = n
SS = Σ(xᵢ - μ)²
Variance interval:
Lower variance = SS / χ²(1 - α / 2, df)
Upper variance = SS / χ²(α / 2, df)
Standard deviation interval:
Lower standard deviation = √Lower variance
Upper standard deviation = √Upper variance
Here, α equals 1 minus the confidence level. The method assumes the population is approximately normal.
How to Use This Calculator
- Paste raw sample values into the raw data box when available.
- Leave raw data blank when using summarized sample size and sample standard deviation.
- Enter the confidence level as a percent, such as 95.
- Enter a known population mean only when it is truly known.
- Select decimal places for cleaner reporting.
- Press Calculate to view the interval below the header.
- Use CSV or PDF buttons to download the current calculation.
Understanding the Interval
A population standard deviation confidence interval estimates the range where the true population spread may fall. It uses the sample standard deviation, the sample size, and a selected confidence level. The method assumes the sampled population is approximately normal. This assumption matters because the interval depends on the chi-square distribution. A wider interval means more uncertainty. A narrower interval means the sample gives stronger precision. The calculator helps compare both effects quickly.
Why Spread Matters
Standard deviation describes how far observations usually sit from their mean. In quality control, it can show process consistency. In finance, it can describe volatility. In education, it can show score variation. A confidence interval adds caution to that single estimate. It says the true spread is not known exactly. It is likely within two calculated limits. This makes reporting more honest and more useful.
Statistical Method
The formula begins with sample variance. The variance is multiplied by degrees of freedom, which equals sample size minus one. That value is divided by two chi-square critical values. The larger critical value creates the lower limit. The smaller critical value creates the upper limit. Taking square roots converts variance limits into standard deviation limits. This calculator also reports variance bounds, interval width, and relative precision.
Input Choices
You can paste raw values separated by commas, spaces, or new lines. The tool then calculates sample size, mean, variance, and sample standard deviation. You can also enter a summarized sample size and standard deviation. This helps when only published summary data is available. If raw data is supplied, it has priority over summarized values.
Best Use
Use this calculator when the question concerns an unknown population spread. It is not designed for a known population standard deviation. Choose a confidence level that matches the decision risk. Ninety-five percent is common. Ninety-nine percent gives wider limits. Smaller samples produce wider intervals. Larger samples usually improve precision. Export the result when you need a record for reports, audits, assignments, or reviews.
Interpretation Tips
Do not read the interval as a guarantee for one future value. It estimates the population parameter. Check outliers before trusting the result. Extreme values can inflate spread and widen the final interval substantially today.
FAQs
1. What does this calculator estimate?
It estimates a confidence interval for the unknown population standard deviation. It also reports the matching variance interval and chi-square critical values.
2. When should I use raw data?
Use raw data when you have the original sample observations. The calculator can then compute the sample size, mean, variance, and standard deviation directly.
3. When should I use summarized input?
Use summarized input when you only know the sample size and sample standard deviation. Leave the raw data field empty in that case.
4. Why is the chi-square distribution used?
For normally distributed populations, the scaled sample variance follows a chi-square distribution. That relationship creates the variance and standard deviation interval bounds.
5. What assumption is most important?
The main assumption is approximate population normality. The method can be sensitive when data are strongly skewed or contain extreme outliers.
6. What happens when confidence increases?
Higher confidence creates a wider interval. It gives more assurance, but it also reduces precision because the possible range becomes larger.
7. Can I enter a known population mean?
Yes. Enter it only when the true population mean is known. The calculator then uses a different sum of squares and degrees of freedom.
8. Are CSV and PDF downloads included?
Yes. After entering valid inputs, use the download buttons to save the current results as a CSV file or simple PDF report.