Confidence Interval from Mean and Standard Deviation Calculator

Confidence Interval from Mean and Standard Deviation Calculator

Results

Confidence Level:

95%

Critical Value (t*):

-

Standard Error:

-

Margin of Error:

-

Lower Bound:

-

Upper Bound:

-

Confidence Interval:

-

Interpretation

Enter values and click calculate to see the interpretation.

Formula Used

CI = x̄ ± t* × (s / √n)

Where:

  • x̄ = sample mean
  • t* = critical value from t-distribution
  • s = sample standard deviation
  • n = sample size
  • SE = s / √n (standard error)

The confidence interval from mean and standard deviation calculator enables researchers, analysts, and students to compute exact confidence intervals using sample statistics. Input your sample mean (X̄), standard deviation (s), and sample size (n) to generate precise 90%, 95%, or 99% confidence ranges — ideal for inferential statistics and statistical inference. Unlike descriptive tools, this calculator uses Z or T critical values to estimate where the true population mean likely lies. Compare this exact confidence range to our Empirical Rule Confidence Interval Calculator for estimated coverage.

Author:

Prof. Aisha Khan, Ph.D. in Statistics, 18+ years in research methodology and data science.

What Is a Confidence Interval?

A confidence interval (CI) is a range of values, derived from a sample, that is likely to contain the true population mean with a specified level of confidence (e.g., 95%). It reflects sampling uncertainty and is expressed as:

Sample Mean ± Margin of Error

For example, a 95% CI of [78.5, 81.5] means: If we repeated this sampling process many times, 95% of the resulting intervals would contain the true population mean.

This is inferential, not descriptive — it estimates a population parameter from sample data.

Graph of sampling distribution showing multiple sample means with 95% confidence interval bars; one highlighted interval contains the true population mean (μ), labeled “95% of intervals capture μ”.

Formula for Confidence Interval Using Mean and Standard Deviation

The calculator uses two formulas based on sample size and distribution assumptions:

Z-Distribution (Large Samples, n ≥ 30)

[ \text{CI} = \bar{X} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} ]

T-Distribution (Small Samples, n < 30)

[ \text{CI} = \bar{X} \pm T_{\alpha/2, df} \times \frac{s}{\sqrt{n}} ] df = n – 1

Term Meaning
(\bar{X}) Sample mean
(s) or (\sigma) Sample or population standard deviation
(n) Sample size
(Zα/2) Critical Z-value (e.g., 1.96 for 95%)
(Tα/2, df) Critical T-value (depends on degrees of freedom)

Rule of Thumb: Use Z when (n \geq 30) or (\sigma) is known. Use T for smaller samples with unknown (\sigma).

How to Use the Confidence Interval Calculator

Follow these steps to calculate confidence interval using mean and standard deviation:

  1. Enter Sample Mean (X̄): e.g., 75

  2. Enter Sample Standard Deviation (s): e.g., 10

  3. Input Sample Size (n): e.g., 25

  4. Select Confidence Level: 90%, 95%, or 99%

  5. Click Calculate: Get lower/upper bounds and margin of error

Note: The tool auto-selects Z or T based on sample size and normality assumptions.

Screenshot of confidence interval calculator with inputs: mean=75, standard deviation=10, sample size=25, 95% confidence level. Output: CI = [70.9, 79.1], Margin of Error = 4.1.

Example: Calculate a 95% Confidence Interval

Given:

  • Sample mean ((\bar{X})) = 75

  • Sample SD (s) = 10

  • Sample size (n) = 25

  • Confidence level = 95%

  • Step-by-Step:

    1. Standard Error = (\frac{s}{\sqrt{n}} = \frac{10}{\sqrt{25}} = 2)

    2. df = 25 – 1 = 24

    3. T-critical value (95%, df=24) ≈ 2.064

    4. Margin of Error = (2.064 \times 2 = 4.128)

    5. CI = (75 \pm 4.128 = [70.872, 79.128])

    Interpretation:

    “We are 95% confident that the true population mean lies between 70.9 and 79.1.”

    For larger samples (n ≥ 30), the Z-value (1.96) would yield a slightly narrower interval.

Empirical Rule vs Confidence Interval (Connection Section)

Aspect Empirical Rule Confidence Interval
Purpose Describes data spread in a population Estimates population mean from a sample
Input Population mean (μ), SD (σ) Sample mean (X̄), SD (s), n
Output % of data in range (68%, 95%, 99.7%) Range likely containing true mean
Type Descriptive Inferential
Assumption Normal distribution Normal sampling distribution

Key Insight:
The Empirical Rule tells you where data falls. A confidence interval tells you where the true mean likely is.

Visual Prompt:
(Insert side-by-side bell curves:

  • Left: Empirical Rule with ±1σ, ±2σ, ±3σ zones (68–95–99.7%)

  • Right: Confidence interval around sample mean with error bars)

Compare this exact confidence range to our Empirical Rule Confidence Interval Calculator for estimated coverage.

Common Confidence Levels

Confidence Level Z-Value T-Value (df=20) Use Case
90% 1.645 ~1.725 Moderate precision
95% 1.96 ~2.086 Standard in research
99% 2.576 ~2.845 High certainty

When to Use This Calculator

  • Use the confidence interval from mean and standard deviation calculator when:

    • Conducting surveys or experiments with sample data

    • Reporting statistical precision in research papers

    • Estimating population parameters in quality control

    • Performing hypothesis testing with margin of error

    It’s essential in statistical modeling, data science, and scientific research.

FAQs

Enter your sample mean, standard deviation, and size into the confidence interval calculator. Select a confidence level (90%, 95%, 99%) to get the range.

 

 

Z is used for large samples (n ≥ 30) or known population SD. T is used for small samples with unknown SD, accounting for greater uncertainty.

 

95% is the standard in most research. Use 90% for less precision, 99% for high-stakes decisions.

 

No. This is for sample-based inference. For full population data, use the Empirical Rule.

 

The Empirical Rule describes data spread (68–95–99.7%). This calculator estimates the population mean from a sample using standard error.

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Conclusion

The confidence interval from mean and standard deviation calculator is a powerful tool for inferential statistics, enabling precise estimation of population means from sample data. Whether you’re in research, quality assurance, or data analysis, this confidence interval calculator delivers statistically reliable results with clear interpretation. For population-level descriptions, compare your results with our Empirical Rule Confidence Interval Calculator. Check our Main Tool.

 

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