Confidence Interval For Mean (T-Distribution)
Use this calculator to compute the confidence interval for population mean when the population standard deviation is unknown. This is the most common scenario in practical statistics.
When to Use This Calculator
- Population standard deviation (σ) is unknown (typical case)
- Sample size is small (n < 30) or population normality is assumed
- Single sample from a population
- Computing a range estimate rather than a single point estimate
- You want to account for sampling variability with a specified confidence level
Common Applications:
- Estimating average test scores from a sample of students
- Estimating average customer satisfaction ratings
- Estimating average product weights or measurements
- Estimating average income in a region
How to Use This Calculator
Step 1: Enter the sample size (n) - the number of observations you collected
Step 2: Enter the sample mean (x̄) - the average of your data
Step 3: Enter the sample standard deviation (s) - measure of variability in your data
Step 4: Select the confidence level (typically 95%)
Step 5: Click the “Calculate” button
Step 6: The calculator will display:
- Standard Error (SE) - the precision of the mean estimate
- t-critical value - from the t-distribution table
- Margin of Error (E) - the range around your estimate
- Lower and Upper Confidence Limits - your final CI interval
| Confidence Interval Calculator for mean | |
|---|---|
| Sample Size ($n$) | |
| Sample Mean ($\overline{x}$) | |
| Sample Standard Deviation ($s$) | |
| Confidence Level ($1-\alpha$) | |
| Results | |
| Standard Error of Mean: ($SE$) | |
| t-critical value: ($t_{\alpha/2,n-1}$) | |
| Margin of Error: ($E$) | |
| Lower Confidence Limits: | |
| Upper Confidence Limits: | |
Theory & Formula
Confidence Interval for Mean (σ Unknown)
When the population standard deviation is unknown, we use the t-distribution to construct the confidence interval.
Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with unknown mean $\mu$ and unknown standard deviation $\sigma$.
The $100(1-\alpha)%$ confidence interval for population mean is:
$$CI = \overline{x} \pm t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}$$
Where:
- $\overline{x}$ = sample mean
- $t_{\alpha/2, n-1}$ = critical t-value (depends on confidence level and degrees of freedom)
- $s$ = sample standard deviation
- $n$ = sample size
- $SE = \frac{s}{\sqrt{n}}$ = standard error of the mean
Margin of Error: $E = t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}$
Assumptions
- Sample is random - Data collected without systematic bias
- Sample is independent - Each observation is independent of others
- Population is normal OR sample size is large (n ≥ 30)
- If n < 30, data should be approximately normally distributed
- If n ≥ 30, the Central Limit Theorem applies
Step-by-Step Procedure
Step 1: Collect Sample Data
Gather your sample data and calculate:
- Sample size: $n$ (number of observations)
- Sample mean: $\overline{x} = \frac{1}{n}\sum_{i=1}^n x_i$
- Sample standard deviation: $s = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \overline{x})^2}$
Step 2: Choose Confidence Level
Select the confidence level:
- 90% (α = 0.10) - Less confident, narrower interval
- 95% (α = 0.05) - Standard choice
- 99% (α = 0.01) - Very confident, wider interval
Step 3: Calculate Standard Error
$$SE = \frac{s}{\sqrt{n}}$$
The standard error measures the precision of your sample mean estimate.
Step 4: Find t-Critical Value
Look up $t_{\alpha/2, n-1}$ from the t-distribution table using:
- Degrees of freedom: df = n - 1
- Significance level: α (based on confidence level)
Critical values for common confidence levels:
| Confidence | α | α/2 |
|---|---|---|
| 90% | 0.10 | 0.05 |
| 95% | 0.05 | 0.025 |
| 99% | 0.01 | 0.005 |
Step 5: Calculate Margin of Error
$$E = t_{\alpha/2, n-1} \times SE = t_{\alpha/2, n-1} \times \frac{s}{\sqrt{n}}$$
Step 6: Construct Confidence Interval
$$CI = [\overline{x} - E, \overline{x} + E]$$
Or equivalently: $\overline{x} \pm E$
Worked Examples
Example 1: Customer Satisfaction Ratings
Scenario: A restaurant wants to estimate the average customer satisfaction rating. They survey 16 customers and record satisfaction scores (1-10 scale).
Data: 7, 8, 8, 9, 7, 6, 8, 9, 8, 7, 7, 8, 9, 8, 7, 6
Solution:
Step 1: Calculate sample statistics
- Sample size: $n = 16$
- Sample mean: $\overline{x} = \frac{7+8+8+…+6}{16} = 7.75$
- Sample std dev: $s = 1.06$
Step 2: Choose confidence level
- Confidence = 95%, so α = 0.05, α/2 = 0.025
Step 3: Calculate standard error $$SE = \frac{1.06}{\sqrt{16}} = \frac{1.06}{4} = 0.265$$
Step 4: Find t-critical value
- df = n - 1 = 15
- $t_{0.025, 15} = 2.131$
Step 5: Calculate margin of error $$E = 2.131 \times 0.265 = 0.565$$
Step 6: Confidence interval $$CI = 7.75 \pm 0.565 = [7.185, 8.315]$$
Interpretation: We are 95% confident that the true average customer satisfaction rating lies between 7.19 and 8.32 on the 1-10 scale.
Example 2: Product Weight Quality Control
Scenario: A manufacturing plant produces cereal boxes. The label claims 500g, but the company wants to ensure the actual average weight is correct. They sample 25 boxes.
Data (weights in grams): 497, 502, 501, 499, 503, 498, 500, 502, 499, 501, 498, 503, 500, 499, 502, 501, 500, 499, 503, 498, 501, 502, 499, 500, 498
Solution:
Step 1: Calculate sample statistics
- Sample size: $n = 25$
- Sample mean: $\overline{x} = 500.0$ g
- Sample std dev: $s = 1.63$ g
Step 2: Choose confidence level
- Confidence = 95%, so α = 0.05
Step 3: Calculate standard error $$SE = \frac{1.63}{\sqrt{25}} = \frac{1.63}{5} = 0.326$$
Step 4: Find t-critical value
- df = 24, $t_{0.025, 24} = 2.064$
Step 5: Calculate margin of error $$E = 2.064 \times 0.326 = 0.673$$
Step 6: Confidence interval $$CI = 500.0 \pm 0.673 = [499.327, 500.673]$$
Interpretation: We are 95% confident that the true average weight of cereal boxes is between 499.3g and 500.7g. Since 500g is within this interval, the production process appears to be on target.
How to Interpret Results
Understanding the Confidence Interval
Example Interval: [73.5, 76.3]
This means:
- We’re 95% confident the true population mean is between 73.5 and 76.3
- NOT: “There’s a 95% probability the mean is in this interval” (incorrect - the mean is fixed, not random)
- CORRECT: “If we repeated this study 100 times, approximately 95 of the resulting intervals would contain the true population mean”
Interpreting Interval Width
Wide Interval (e.g., [60, 90])
- Indicates high uncertainty
- Caused by: small sample size, high variability, or low confidence level
- Solution: Collect more data to narrow the interval
Narrow Interval (e.g., [73.9, 74.1])
- Indicates high precision
- Resulted from: large sample size, low variability, or lots of information
Decision Trees
Q: “Is my estimate precise enough?”
Width of CI = Upper Limit - Lower Limit
If width < desired precision:
→ Your estimate is sufficiently precise
→ Make decisions based on this CI
If width > desired precision:
→ Collect more data (larger sample size)
→ Each 4× sample size → ~50% narrower CI
Q: “Is the population mean likely equal to a specific value?”
Is target value (e.g., 500) within the CI?
If YES:
→ Cannot conclude mean differs from target
→ The data is consistent with target value
If NO:
→ The mean likely differs from target
→ The population mean is probably higher or lower
Key Differences: Z vs T Distribution
| Feature | Z-Distribution | T-Distribution |
|---|---|---|
| When to Use | σ known | σ unknown (typical) |
| Critical Values | Smaller | Larger (wider CI) |
| Intervals | Narrower | Wider |
| Sample Size | Any | Important (affects shape) |
| Conservativeness | May underestimate uncertainty | Accounts for uncertainty in s |
Bottom Line: Use t-distribution (this calculator) in almost all real-world situations.
Related Calculators & Resources
- CI for Mean (Z-Distribution) - When σ is known (rare)
- CI for Two Means - Comparing two groups
- CI for Paired Data - Before-after studies
Tutorial: Complete guide to CI for means Examples: Worked examples with solutions