Calculator
Example Data Table
| Subject | Before | After | Difference |
|---|---|---|---|
| 1 | 82 | 78 | 4 |
| 2 | 76 | 72 | 4 |
| 3 | 91 | 88 | 3 |
| 4 | 69 | 65 | 4 |
| 5 | 85 | 81 | 4 |
| 6 | 88 | 84 | 4 |
Formula Used
First calculate each paired difference: d = X1 − X2 or d = X2 − X1.
Mean difference: d̄ = Σd / n.
Sample standard deviation: sd = √[Σ(d − d̄)² / (n − 1)].
Standard error: SE = sd / √n.
Two-sided confidence interval: d̄ ± tα/2,df × SE.
Degrees of freedom: df = n − 1.
Paired t statistic: t = (d̄ − μd) / SE.
Effect size: Cohen dz = d̄ / sd.
How to Use This Calculator
Select raw paired data when you have both values for each subject.
Enter one pair per line. Separate values by spaces, commas, semicolons, or pipes.
Choose the difference direction carefully. It controls the sign of the result.
Use summary statistics when you already know n, mean difference, and difference standard deviation.
Choose a confidence level such as 90, 95, or 99.
Press Calculate. The result appears above the form.
Use CSV or PDF buttons to export the same calculation.
Dependent Samples Confidence Interval Guide
What This Calculator Measures
A dependent samples confidence interval estimates the average change between two related measurements. The same person, object, patient, machine, or matched unit appears in both measurements. This makes the design different from two independent groups. The calculator focuses on the differences inside each pair. Then it builds an interval around the mean difference. This method is often used for before and after studies. It is also useful for matched case studies.
Why Paired Differences Matter
Paired data removes much of the variation between subjects. Each subject acts as its own control. That can make the estimate more precise. The calculator subtracts one related value from the other. The selected direction controls whether improvement appears positive or negative. Always choose the direction before interpreting the interval.
Understanding the Interval
The interval gives a likely range for the true mean paired difference. A 95 percent interval means the method captures the true mean often. It does not mean one computed interval has a 95 percent chance. The true value is fixed. The interval changes with new samples. Wider intervals show more uncertainty. Narrower intervals show stronger precision.
When Results Are Important
In a two-sided interval, zero is a key reference value. If zero is inside the interval, no clear average change is shown. If zero is outside the interval, the mean change is statistically noticeable. The paired t statistic compares the mean difference with a chosen hypothesized value. Cohen dz adds effect size context. It helps judge practical size, not only statistical evidence.
Best Practices
Use accurate paired observations. Do not mix unmatched rows. Keep units consistent for both measurements. Check outliers before trusting the final result. The t method works best when differences are roughly normal. Larger samples are usually more stable. Report the confidence level, mean difference, interval bounds, and sample size. Also explain the direction of subtraction. Clear reporting prevents wrong conclusions.
FAQs
What is a dependent samples confidence interval?
It is an interval estimate for the mean difference between paired observations. It is used when each data point in one condition is linked to a data point in another condition.
When should I use this calculator?
Use it for before and after scores, matched subjects, repeated measurements, or paired experimental designs. It is not meant for unrelated independent groups.
Why does the difference direction matter?
The direction changes the sign of the mean difference. Before minus after gives a different sign than after minus before. The interval width stays the same.
What does it mean if zero is inside the interval?
For a two-sided interval, zero inside the range means the data does not clearly show a nonzero average paired difference at that confidence level.
What confidence level should I choose?
Common choices are 90, 95, and 99 percent. Higher confidence gives wider intervals. Lower confidence gives narrower intervals but less protection against sampling error.
Can I use summary statistics?
Yes. Select summary mode when you know the sample size, mean difference, and standard deviation of paired differences. You do not need raw pairs then.
What is the standard error?
The standard error measures uncertainty in the mean difference. It equals the standard deviation of differences divided by the square root of sample size.
What is Cohen dz?
Cohen dz is a paired effect size. It divides the mean paired difference by the standard deviation of paired differences. It describes practical change size.