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Confidence Interval Calculator

Use this confidence interval calculator to summarize a data set for confidence interval.

Primary answer
Lower bound
Inputs to verify
Sample mean, Population standard deviation, and Sample size
Use type
Use as an estimate that depends on assumptions.
Keyword intent
confidence interval calculator

Calculator

Confidence Interval Calculator

Calculates lower bound from sample mean, population standard deviation, sample size. Defaults are filled in so you can review a working example before changing inputs.

Enter the sample mean for the measurement being estimated.

Enter the known population standard deviation, not the sample standard deviation.

Enter the number of independent observations.

Choose the two-sided confidence level for the mean interval.

Result

Result reflects the current submitted inputs.

  • Risk B
  • Reviewed 2026-05-26
  • 2 sources
Lower bound48.040036
Upper bound51.959964
Margin of error1.959964
Standard error1
z critical value1.959964
Sample mean50

Breakdown

Confidence level
95%
Known population standard deviation
10
Sample size
100
  • This MVP calculates a two-sided z confidence interval for one population mean.
  • The population standard deviation is known and is not estimated from the sample.
  • The sampling distribution of the sample mean is normal or adequately approximated by normality.
  • Observations are treated as independent.
  • Proportion intervals, t intervals, and two-sample intervals are out of scope.

Accuracy notes

Risk level
B
Reviewed
2026-05-26
Sources
2
Primary result
Lower bound

Formula logic is kept in a pure calculator module with fixtures, source notes, and page-visible assumptions.

Categories

What the result means

Lower bound answers the page's main confidence interval question. Lower endpoint of the two-sided z confidence interval. Read the center or spread metric first, then compare count, minimum, maximum, and sample/population notes. Use upper bound, margin of error, and standard error to explain why lower bound moved when an input changed. Review the raw values and decide whether outliers or missing data should be handled before reporting the result.

Lower boundLower endpoint of the two-sided z confidence interval.
Upper boundUpper endpoint of the two-sided z confidence interval.
Margin of errorHalf-width of the confidence interval.
Standard errorKnown population standard deviation divided by the square root of sample size.

Use the result this way

  1. Start with Lower bound, then use supporting outputs only to explain the primary answer.
  2. Verify sample mean, population standard deviation, and sample size before copying the result.
  3. Choose the mode or method first because it can change which formula is applied and stay within the documented minimum and maximum ranges.
  4. Review the raw values and decide whether outliers or missing data should be handled before reporting the result.

User job

How to use this calculator

Use Confidence Interval Calculator when you need lower bound, then use upper bound and margin of error to check the context for data review, classwork, quality checks, and quick descriptive summaries.

Best for

  • Summarizing a list of values
  • Checking spread, center, or sample assumptions
  • Reviewing a default example before entering your own sample mean and population standard deviation.

Check before relying

  • Confirm whether the data is a sample or population and whether outliers should stay in the list.
  • This MVP calculates a two-sided z confidence interval for one population mean.
  • The population standard deviation is known and is not estimated from the sample.
  • Source context: National Institute of Standards and Technology, reviewed 2026-05-26.

Next useful step

  • P-Value CalculatorUse next when the probability task needs p-value instead of lower bound.
  • Sample Size CalculatorUse next when you need required sample size from confidence level and margin of error after checking lower bound.
  • Z-Score CalculatorUse next when the probability task needs Z-score instead of lower bound.

Formula

For one mean with known population standard deviation, confidence interval = sampleMean +/- zCritical * sigma / sqrt(n). Key assumptions: This MVP calculates a two-sided z confidence interval for one population mean. The population standard deviation is known and is not estimated from the sample. The sampling distribution of the sample mean is normal or adequately approximated by normality.

  • For one mean with known population standard deviation, confidence interval = sampleMean +/- zCritical * sigma / sqrt(n).
  • This MVP calculates a two-sided z confidence interval for one population mean.
  • The population standard deviation is known and is not estimated from the sample.
  • Primary source context: National Institute of Standards and Technology.

Inputs

Enter sample mean, population standard deviation, sample size, and confidence level for data review, summaries, quality checks, and classwork. Before calculating, choose the mode or method first because it can change which formula is applied and stay within the documented minimum and maximum ranges. Sample mean: Enter the sample mean for the measurement being estimated. Population standard deviation: Enter the known population standard deviation, not the sample standard deviation. Sample size: Enter the number of independent observations. Confidence level: Choose the two-sided confidence level for the mean interval.

Sample meanEnter the sample mean for the measurement being estimated.
Population standard deviationEnter the known population standard deviation, not the sample standard deviation.
Sample sizeEnter the number of independent observations.
Confidence levelChoose the two-sided confidence level for the mean interval.

Example

Using the default inputs, Confidence Interval Calculator returns lower bound of 48.040036. Adjust sample mean, population standard deviation, sample size, and confidence level to match your own scenario.

FAQ

How is lower bound calculated here?

For one mean with known population standard deviation, confidence interval = sampleMean +/- zCritical * sigma / sqrt(n). The first assumption to check is: This MVP calculates a two-sided z confidence interval for one population mean.

What does Lower bound mean for confidence interval?

Read the center or spread metric first, then compare count, minimum, maximum, and sample/population notes. Secondary values such as upper bound, margin of error, and standard error are there to explain the primary answer, not to replace it.

What should I enter for Sample mean?

Enter the sample mean for the measurement being estimated. Choose the mode or method first because it can change which formula is applied and stay within the documented minimum and maximum ranges.

How does Population standard deviation change lower bound?

Enter the known population standard deviation, not the sample standard deviation. Changing it can alter lower bound because the formula uses the submitted inputs together. Also compare sample versus population mode, separators, missing values, outliers, and rounding precision.

Why does the confidence interval example show 48.040036 for lower bound?

The default inputs produce 48.040036 for lower bound. Treat that as a format and scale check, then replace every default value with your own inputs.

What should I check before reporting lower bound?

Confirm how the values were parsed, whether the data is a sample or population, and whether outliers or missing values should stay in the set.

Sources

Last reviewed: 2026-05-26

  • Reviewed 2026-05-26 · Source n.d.
    NIST/SEMATECH e-Handbook: What are confidence intervals?National Institute of Standards and Technology. Two-sided mean confidence interval formula using sample mean, known population standard deviation, sample size, and standard normal critical value.
    Scope
    Confidence interval formulas for process/product comparisons, including the mean of a normal population with known standard deviation.
    Supports
    Two-sided mean confidence interval formula using sample mean, known population standard deviation, sample size, and standard normal critical value.
  • Reviewed 2026-05-26 · Source n.d.
    NIST/SEMATECH e-Handbook: Cumulative Distribution Function of the Standard Normal DistributionNational Institute of Standards and Technology. Critical z values used for 90%, 95%, and 99% two-sided confidence levels.
    Scope
    Standard normal cumulative probability table and common critical values.
    Supports
    Critical z values used for 90%, 95%, and 99% two-sided confidence levels.

Disclaimer

This calculator is an educational estimate based on the inputs and assumptions shown on the page.