Calculate confidence intervals, margin of error, critical values, and standard error for a population mean. Supports z-intervals and t-intervals with step-by-step statistical explanations.
Confidence Interval Calculator
Choose which standard deviation you have, enter your sample data, and click Calculate.
A t interval requires a sample size of at least 2, because degrees of freedom equal n minus 1.
Zero is a valid standard deviation when every observation in the sample is identical. Standard deviation cannot be negative. If you have raw data but still need its standard deviation, use the Standard Deviation Calculator first, then return here with the result.
Confidence level must be greater than 0% and less than 100%. Confidence levels are expressed as percentages; the Percentage Calculator can help with general percentage math.
Your Result
95% Confidence Interval
72.62 to 77.38
What This Result Means
Your Interval, Visualized
Lower Bound · Sample Mean · Upper Bound
Higher Confidence Creates a Wider Interval
Same Data, Compared at 90% / 95% / 99%
These three intervals use your sample mean, standard deviation, sample size, and method, changing only the confidence level.
A confidence interval is a range of plausible values used to estimate an unknown population parameter from sample data. For this calculator, the parameter being estimated is the population mean.
Suppose a researcher cannot measure every person or item in a population. Instead, the researcher selects a sample and calculates a sample mean. The sample mean provides a single estimate of the unknown population mean, but different random samples will usually produce slightly different means. A confidence interval adds information about that sampling uncertainty by giving a range around the sample estimate.
For example, suppose the sample mean is 75 and a calculated 95% confidence interval is 72.62 to 77.38. The sample mean of 75 is at the center of the interval. The interval extends roughly 2.38 units below and above that sample mean. That 2.38 is the margin of error.
Figure 1
Anatomy of a Confidence Interval
Lower Bound 72.62
Sample Mean 75
Upper Bound 77.38
Margin of Error = 2.38 in each direction from the sample mean of 75, producing the interval 72.62 to 77.38.
What Does a 95% Confidence Interval Mean?
A 95% confidence interval is often misunderstood. It does NOT mean "there is a 95% probability that the true population mean is inside this particular interval."
Under the traditional frequentist interpretation, the population parameter is treated as fixed. Once the interval has been calculated, the true value is either inside that interval or it is not. The 95% describes the long-run performance of the interval-building method.
Imagine repeatedly doing the following: take a random sample, calculate the sample mean, and build a 95% confidence interval using the same method. Repeat that entire process many times. Over many repetitions, approximately 95% of those intervals would be expected to contain the true population mean if the assumptions of the method are satisfied.
Figure 2
Repeated Confidence Intervals
TRUE MEAN
Individual intervals vary. The method is designed for long-run coverage. This is a conceptual illustration, not a literal simulation guarantee.
What Is the Confidence Level?
The confidence level describes the long-run coverage rate of the interval-building procedure. Common confidence levels include 90%, 95%, and 99%. A higher confidence level requires a wider interval when all other inputs remain the same.
Why? Because being more confident requires covering a larger range of plausible values. Suppose the same sample produces a 90% confidence interval of 73.01 to 76.99, a 95% confidence interval of 72.62 to 77.38, and a 99% confidence interval of 71.85 to 78.15. The exact values depend on the inputs and the statistical method, but the relationship is important: higher confidence leads to a larger critical value, which leads to a larger margin of error, which leads to a wider interval.
Figure 3
Higher Confidence Requires a Wider Interval
90%
73.01 to 76.99
95%
72.62 to 77.38
99%
71.85 to 78.15
All three intervals use the same sample mean of 75, sample standard deviation of 12, and sample size of 100. Only the confidence level changes, and the interval widens as confidence increases.
What Is the Margin of Error?
The margin of error is the distance from the sample estimate to either endpoint of the confidence interval. For a mean interval: Confidence Interval = Sample Mean ± Margin of Error.
The margin of error depends on the critical value and the standard error. Formula: Margin of Error = Critical Value × Standard Error.
What Is Standard Error?
Standard error measures the expected sampling variability of a statistic. For a sample mean, the standard error becomes smaller as the sample size becomes larger.
If the population standard deviation σ is known: SE = σ/√n. If population standard deviation is unknown and the sample standard deviation s is used: SE = s/√n.
Suppose s = 12 and n = 100. Then SE = 12/√100 = 12/10 = 1.2.
Figure 4
Standard Error
Sample SD = 12 · Sample Size = 100
↓
√100 = 10
↓
12 ÷ 10
↓
Standard Error = 1.2
Standard error divides the standard deviation by the square root of the sample size, not by the sample size itself.
What Is the Difference Between Standard Deviation and Standard Error?
Standard deviation and standard error are related but they answer different questions. Standard deviation describes how spread out individual observations are. Standard error describes how much a sample statistic, such as the sample mean, would vary across repeated samples.
A sample can have standard deviation = 12 while its sample mean has standard error = 1.2 when n = 100. That does not mean the observations only vary by 1.2. The observations still have a standard deviation of 12. The 1.2 describes uncertainty in the sample mean.
The general process is: (1) identify the sample mean, (2) identify the sample size, (3) determine whether population standard deviation is known, (4) calculate the standard error, (5) select the appropriate critical value, (6) calculate the margin of error, and (7) add and subtract the margin of error from the sample mean.
Consider: Sample Mean = 75. Sample Size = 100. Sample Standard Deviation = 12. Confidence Level = 95%. Population Standard Deviation = Unknown.
Because population standard deviation is unknown and sample standard deviation is being used, use a t interval. Degrees of Freedom: df = n - 1 = 100 - 1 = 99. Standard Error: 12/√100 = 1.2. For a two-sided 95% t interval with df = 99, the critical value is approximately 1.984. Margin of Error: 1.984 × 1.2 ≈ 2.381. Confidence Interval: 75 ± 2.381. Lower Bound: 72.619. Upper Bound: 77.381. Rounded to two decimal places: 72.62 to 77.38.
Figure 5
Confidence Interval Calculation Flow
Sample Data
↓
Choose Z or T
↓
Standard Error
↓
Critical Value
↓
Margin of Error
↓
Lower + Upper Bound
Every confidence interval for a mean follows this same sequence, whether it uses a z or a t critical value.
When Do You Use a Z Confidence Interval?
For this one-sample mean calculator, use a z interval when the population standard deviation σ is known. Formula: x̄ ± z* × σ/√n.
Example: Sample Mean = 50. Population SD = 10. Sample Size = 100. Confidence Level = 95%. Standard Error: 10/√100 = 1. 95% z critical value: 1.96. Margin of Error: 1.96 × 1 = 1.96. Confidence Interval: 50 ± 1.96 = 48.04 to 51.96.
When Do You Use a T Confidence Interval?
Use a t interval for the population mean when population standard deviation is unknown and the sample standard deviation is being used. Formula: x̄ ± t* × s/√n. The critical value depends on confidence level and degrees of freedom. Degrees of Freedom: df = n - 1. The t distribution accounts for additional uncertainty caused by estimating population variability from the sample.
Should You Automatically Switch to Z When the Sample Size Is Above 30?
No. The common classroom shortcut "use z when n is greater than 30" is oversimplified.
If population standard deviation is unknown and sample standard deviation is being used, a t interval remains the standard one-sample mean approach, no matter how large the sample size is. As sample size increases, degrees of freedom increase, and the t distribution becomes increasingly similar to the normal distribution. So the resulting critical values become close, but the underlying choice of method should still be based on whether population standard deviation is known, not on an arbitrary sample-size cutoff.
Figure 6
Z vs T
Standard Normal (Z)T Distribution (small df)
The t distribution has a shorter peak and heavier tails than the standard normal distribution. Small df → more noticeable difference. As df increases, t approaches z.
What Is a Critical Value?
A critical value determines how many standard errors the confidence interval extends from the sample estimate. For a 95% z interval, the critical value is approximately 1.96. For a 99% z interval, the critical value is approximately 2.576. A larger critical value creates a larger margin of error. For a t interval, the critical value depends on degrees of freedom.
What Is the Z-Score for a 95% Confidence Interval?
For a standard two-sided 95% z confidence interval, z* ≈ 1.96. The central 95% of the standard normal distribution lies between approximately -1.96 and +1.96. The remaining 5% is divided between the two tails: 2.5% in each tail. This 1.96 critical value applies to the z-based confidence interval. Do not automatically use 1.96 when a t interval is required.
What Is the Z-Score for a 99% Confidence Interval?
For a standard two-sided 99% z confidence interval, z* ≈ 2.576. Because 2.576 is larger than 1.96, a 99% interval will be wider than a 95% interval when the data are otherwise identical.
What Is the Z-Score for a 90% Confidence Interval?
For a standard two-sided 90% z confidence interval, z* ≈ 1.645.
Common Z Critical Values
Confidence Level
z*
80%
1.282
85%
1.440
90%
1.645
95%
1.960
98%
2.326
99%
2.576
99.5%
2.807
99.9%
3.291
These are standard two-sided z critical values. T critical values depend on degrees of freedom.
How Does Sample Size Affect a Confidence Interval?
Sample size appears inside the square root in the standard error formula: SE = SD/√n.
Suppose the standard deviation stays 20. For n = 25, SE: 20/5 = 4. For n = 100, SE: 20/10 = 2. For n = 400, SE: 20/20 = 1. As sample size increases, standard error decreases. With the same confidence level and variability, this generally produces a narrower confidence interval.
Figure 7
Sample Size and Interval Width
n=25
WIDE
n=100
MEDIUM
n=400
NARROW
Mean, standard deviation, and confidence level held constant at 75, 20, and 95%. Larger n → smaller SE → narrower interval.
Does Four Times the Sample Size Cut the Margin of Error in Half?
Approximately, when the other parts of the formula stay the same. Because standard error is proportional to 1/√n, multiplying sample size by 4 multiplies √n by 2. That cuts the standard-error portion of the margin of error in half. This square-root relationship is important when planning studies.
How Does Standard Deviation Affect the Confidence Interval?
Larger standard deviation means the observations vary more. Because standard deviation appears in the numerator of the standard-error formula: larger SD → larger standard error → larger margin of error → wider confidence interval.
Figure 8
Variability and Interval Width
SD=5
Narrow
SD=10
Wider
SD=20
Widest
Sample size and confidence level held constant at 100 and 95%. Greater variability widens the interval.
What Makes a Confidence Interval Wider or Narrower?
Three of the biggest factors are sample size, standard deviation, and confidence level. Larger sample size generally narrows the interval. Larger variability generally widens the interval. Higher confidence widens the interval.
Figure 9
The Three Big Drivers
🔢
Sample Size ↑
Interval gets narrower
📊
Standard Deviation ↑
Interval gets wider
🎯
Confidence Level ↑
Interval gets wider
Sample size pulls the interval narrower. Standard deviation and confidence level both push it wider.
What Is a 95% Confidence Interval?
A 95% confidence interval is one of the most commonly used statistical intervals. It is built using a procedure designed to capture the true population parameter in approximately 95% of repeated samples under the assumptions of the method.
The exact endpoints depend on sample mean, sample size, variability, and whether the interval uses z or t. There is no universal numerical "95% confidence interval" without those inputs.
How Do You Find a 95% Confidence Interval?
Example using a known population standard deviation: Sample Mean = 100. Population Standard Deviation = 15. Sample Size = 100. 95% z critical value: 1.96. Standard Error: 15/√100 = 1.5. Margin of Error: 1.96 × 1.5 = 2.94. Confidence Interval: 100 ± 2.94 = 97.06 to 102.94.
What Is a 99% Confidence Interval?
A 99% confidence interval uses a higher confidence level than a 95% interval. When everything else is identical, 99% confidence requires a larger critical value and therefore produces a wider interval. This does not mean the 99% interval is always "better." There is a tradeoff: higher confidence comes with less precision in the form of a wider range.
What Does a Narrow Confidence Interval Mean?
A narrower interval indicates more precision in the estimate under the model and assumptions being used. It does not automatically mean the study is unbiased, the data are representative, the measurement process is accurate, or the conclusion is correct. A precise estimate can still be systematically wrong if the underlying sample or method is flawed.
What Does a Wide Confidence Interval Mean?
A wide confidence interval indicates greater uncertainty around the estimate. Possible contributors include small sample size, large variability, and a high confidence level. A wide interval does not mean the calculation failed. It means the data and selected confidence level support a less precise estimate.
Confidence Interval vs Prediction Interval
Do not confuse a confidence interval for the mean with a prediction interval for an individual future observation. A confidence interval estimates a population parameter such as the population mean. A prediction interval attempts to describe where an individual future observation may fall and is typically wider because it must account for individual-level variability as well as uncertainty in the mean. This calculator calculates confidence intervals for a population mean. It is NOT a prediction interval calculator.
Confidence Interval vs Standard Deviation
Standard deviation describes spread among observations. A confidence interval estimates an unknown population parameter. Example: Standard Deviation = 12. Sample Mean = 75. 95% confidence interval = 72.62 to 77.38. These are not interchangeable statistics.
Confidence Interval vs Margin of Error
The margin of error is only half of the confidence interval. Example: Estimate = 75. Margin of Error = 2.38. Confidence Interval = 72.62 to 77.38. The interval contains two endpoints. The margin of error tells you the distance from the estimate to either endpoint.
Original Survey Example
Suppose a researcher randomly samples 100 customers. Average satisfaction score: 82. Sample Standard Deviation: 10. Population Standard Deviation: Unknown. Use a 95% t interval. Degrees of Freedom: 99. Standard Error: 10/√100 = 1. Critical t value: approximately 1.984. Margin of Error: 1.984. Confidence Interval: 82 ± 1.984. Lower: 80.016. Upper: 83.984. Rounded: 80.02 to 83.98.
Interpretation: Using this procedure, the estimated population mean satisfaction score is between approximately 80.02 and 83.98 at the 95% confidence level.
What Assumptions Matter?
A confidence interval is not automatically valid simply because the arithmetic was completed. The statistical method assumes an appropriate sampling process. Important considerations can include independence of observations, representative or properly randomized sampling where required, whether the sampling distribution of the mean is reasonably modeled by the selected procedure, and whether the inputs accurately represent the data. For small samples, strong skew or extreme outliers can be especially important. Do not use the calculator's output as a substitute for good study design.
How Do I Find the Sample Mean?
The sample mean is the arithmetic average of the sample observations. If the sample values are 70, 72, 75, 78, 80, add them: 70+72+75+78+80 = 375. Count: 5. Sample Mean: 375 ÷ 5 = 75. If you still need to calculate the mean of your sample, use the Average Calculator before building the confidence interval.
What If I Only Have Raw Data?
If you have a list of raw observations but do not yet know the mean or standard deviation, first calculate the descriptive statistics. Use the Statistics Calculator or the Standard Deviation Calculator. Then enter the required values into this Confidence Interval Calculator.
Visual Lesson: Confidence Interval Basics
If you prefer a visual explanation after working through the formulas here, Bio-Resource also provides a video covering confidence interval basics and calculation. This is a supplementary external learning resource and not a substitute for the step-by-step math shown above.
What Are Common Confidence Interval Mistakes?
Saying There Is a 95% Probability the True Mean Is Inside This One Interval
That is not the standard frequentist interpretation. Once a sample is collected and an interval is calculated, the true population mean is either inside that specific interval or it is not; the 95% describes the long-run reliability of the method, not a probability for this one result.
Automatically Using Z Because N Is Above 30
Choose z or t based on whether population standard deviation is known, not based on an arbitrary sample-size cutoff.
Confusing Standard Deviation With Standard Error
They measure different things: standard deviation describes spread in the data, and standard error describes sampling variability of the mean.
Forgetting the Square Root of N
Standard error uses SD/√n, not SD/n.
Using the Wrong Confidence Level
A 95% interval and a 99% interval use different critical values.
Using 1.96 for Every Interval
1.96 is the standard two-sided z critical value for 95% confidence. It is not a universal critical value.
Forgetting Degrees of Freedom for T
For a one-sample mean, df = n - 1.
Thinking a Higher Confidence Level Makes the Interval Narrower
Higher confidence makes the interval wider when other inputs are unchanged.
Thinking a Larger Sample Creates a Wider Interval
Larger samples generally reduce standard error and narrow the interval.
Rounding Too Early
Retain precision until the final result.
What Should You Leave This Page Knowing?
Do not leave knowing only "my interval is 72.62 to 77.38." Leave knowing:
I understand what a confidence interval estimates.
I understand what a confidence level means.
I know what a margin of error is.
I know what standard error is.
I know standard deviation and standard error are different.
I know when a z interval is appropriate.
I know when a t interval is appropriate.
I know not to automatically switch to z because the sample size is above 30.
I know what degrees of freedom means for a one-sample t interval.
I know the z critical value for a 95% two-sided interval is approximately 1.96.
I know the z critical value for a 99% two-sided interval is approximately 2.576.
I understand why higher confidence creates a wider interval.
I understand why larger samples generally create narrower intervals.
I understand why larger standard deviations generally create wider intervals.
I know the difference between a confidence interval and a margin of error.
I know the difference between a confidence interval and standard deviation.
I understand that a confidence interval for the mean is not the same as a prediction interval.
I understand that statistical validity depends on more than arithmetic.
I can see every step used to calculate my result.
And most importantly: I can explain what my confidence interval actually means. That is what the CalculateThisWay Confidence Interval Calculator should help someone understand.
Formulas at a Glance
Concept
Formula
Standard Error, Known σ
σ / √n
Standard Error, Unknown σ
s / √n
Z Confidence Interval
x̄ ± z* × σ/√n
T Confidence Interval
x̄ ± t* × s/√n
Degrees of Freedom
n - 1
Margin of Error
Critical Value × Standard Error
Lower Bound
Mean - Margin of Error
Upper Bound
Mean + Margin of Error
Confidence Interval Examples With Answers
Example 1. 95% Z Interval
x̄=50, σ=10, n=100SE: 1z*: 1.96ME: 1.96
CI: 48.04 to 51.96
Example 2. 90% Z Interval
x̄=100, σ=20, n=100SE: 2z*: 1.645ME: 3.29
CI: 96.71 to 103.29
Example 3. 99% Z Interval
x̄=40, σ=5, n=25SE: 1z*: 2.576ME: 2.576
CI: 37.424 to 42.576
Example 4. 95% T Interval
x̄=75, s=12, n=100df: 99SE: 1.2t*: ≈1.984
ME ≈ 2.381. CI: ≈72.62 to 77.38
Example 5. Small Sample T
x̄=20, s=4, n=16df: 15SE: 195% t*: ≈2.131
CI: ≈17.87 to 22.13
Example 6. Sample Size Comparison
Mean=50, SD=10n=25 SE: 2n=100 SE: 1
Standard error changes from 2 to 1 as sample size moves from 25 to 100.
A confidence interval is a range of plausible values, calculated from sample data, used to estimate an unknown population parameter such as the population mean.
What is the confidence interval?
There is no single universal confidence interval. It depends on your sample mean, sample size, variability, and chosen confidence level. Enter your values above to calculate it.
What does a confidence interval mean?
It means the interval was built using a procedure designed so that, over many repeated samples, a certain percentage of such intervals would contain the true population mean.
What does a 95% confidence interval mean?
A 95% confidence level means that if we repeatedly took samples using the same procedure and built an interval each time, about 95% of those intervals would contain the true population mean. It does not mean this one specific interval has a 95% chance of containing the mean.
What is a 99% confidence interval?
A 99% confidence interval uses a higher confidence level than 95%, which requires a larger critical value and produces a wider interval for otherwise identical data.
What is a 90% confidence interval?
A 90% confidence interval uses a lower confidence level than 95%, which requires a smaller critical value and produces a narrower interval for otherwise identical data.
How do you calculate a confidence interval?
Identify the sample mean and sample size, calculate the standard error, select the correct critical value (z or t), multiply to get the margin of error, then add and subtract that margin from the sample mean.
How do you find a confidence interval?
Find the standard error, find the matching critical value for your confidence level, multiply them to get the margin of error, then build the interval as sample mean plus or minus the margin of error.
What is the confidence interval formula?
For a z interval: sample mean plus or minus z* times sigma over the square root of n. For a t interval: sample mean plus or minus t* times s over the square root of n.
What is margin of error?
The margin of error is the distance from the sample estimate to either endpoint of the confidence interval, equal to the critical value multiplied by the standard error.
How do you calculate margin of error?
Multiply the critical value by the standard error: Margin of Error = Critical Value times Standard Error.
What is standard error?
Standard error measures the expected sampling variability of a statistic, such as the sample mean, and it becomes smaller as sample size increases.
How do you calculate standard error?
Divide the standard deviation by the square root of the sample size: Standard Error = SD divided by the square root of n.
What is the difference between standard deviation and standard error?
Standard deviation describes how spread out individual observations are. Standard error describes how much a sample statistic, such as the sample mean, would vary across repeated samples.
What is the difference between confidence interval and margin of error?
The margin of error is only half of the confidence interval. The interval is the sample mean minus the margin of error through the sample mean plus the margin of error.
What is the difference between confidence interval and standard deviation?
Standard deviation describes variability among individual observations in the data. A confidence interval estimates an unknown population parameter. They are not interchangeable statistics.
What is a z confidence interval?
A z confidence interval is used for a population mean when the population standard deviation is known, using the formula sample mean plus or minus z* times sigma over the square root of n.
What is a t confidence interval?
A t confidence interval is used for a population mean when the population standard deviation is unknown and the sample standard deviation is used instead, with degrees of freedom equal to n minus 1.
When should I use z instead of t?
Use a z interval only when the population standard deviation is genuinely known in advance. Otherwise, when you are using the sample standard deviation, use a t interval.
Should I use z when n is greater than 30?
No. That classroom shortcut is oversimplified. If you are using the sample standard deviation, a t interval remains correct regardless of how large the sample size is.
What is degrees of freedom?
For a one-sample t interval for the mean, degrees of freedom equal n minus 1, reflecting that the sample standard deviation is estimated from the same data used for the mean.
What is the z-score for a 95% confidence interval?
For a standard two-sided 95% confidence interval, the z critical value is approximately 1.96.
What is the z-score for a 99% confidence interval?
For a standard two-sided 99% confidence interval, the z critical value is approximately 2.576.
What is the z-score for a 90% confidence interval?
For a standard two-sided 90% confidence interval, the z critical value is approximately 1.645.
Why is a 99% confidence interval wider than a 95% interval?
A 99% interval uses a larger critical value than a 95% interval. A larger critical value produces a larger margin of error, which produces a wider interval.
How does sample size affect a confidence interval?
Sample size appears inside the square root in the standard error formula, so as sample size increases, standard error decreases, which generally narrows the confidence interval.
How does standard deviation affect a confidence interval?
Standard deviation appears in the numerator of the standard error formula, so greater variability produces a larger standard error and generally a wider confidence interval.
Does larger sample size make a confidence interval narrower?
Generally yes, when the standard deviation and confidence level stay the same, because standard error decreases as sample size increases.
What happens if confidence level increases?
Increasing the confidence level requires a larger critical value, which increases the margin of error and widens the interval, assuming other inputs stay the same.
What does a narrow confidence interval mean?
A narrower interval indicates more precision in the estimate under the model and assumptions being used. It does not automatically mean the underlying study is unbiased or correct.
What does a wide confidence interval mean?
A wide confidence interval indicates greater uncertainty, often from a small sample size, large variability, or a high confidence level. It does not mean the calculation failed.
Can a confidence interval be wrong?
Yes. In the long run, a predictable fraction of intervals built by the same method will not contain the true population mean, and violated assumptions can affect validity further.
Does 95% confidence mean 95% probability?
No. Under the standard frequentist interpretation, the population mean is fixed, so this specific interval either contains it or it does not. The 95% describes the long-run reliability of the method.
What assumptions are needed for a confidence interval?
Important assumptions include independence of observations, an appropriate sampling process, and that the sampling distribution of the mean is reasonably modeled by the chosen procedure.
What is the difference between a confidence interval and a prediction interval?
A confidence interval estimates a population parameter such as the mean. A prediction interval estimates where an individual future observation may fall and is typically wider.
Can I calculate a confidence interval from raw data?
Yes. First calculate the sample mean and standard deviation from your raw data, then enter those values, along with your sample size and confidence level, into this calculator.
How do I find the sample mean?
Add every observation together and divide by how many observations there are. For 70, 72, 75, 78, and 80, the sum is 375 and the count is 5, so the sample mean is 75.
How do I find the sample standard deviation?
Calculate how far each value is from the sample mean, square those differences, average them using n minus 1, and take the square root, or use the Standard Deviation Calculator directly.
This calculator estimates a population mean from sample summary statistics.
A z interval is used when population standard deviation is known.
A t interval is used when population standard deviation is unknown and sample standard deviation is supplied.
T degrees of freedom equal n - 1 for this one-sample mean calculation.
Standard error equals SD divided by the square root of sample size.
Margin of error equals critical value times standard error.
Confidence interval endpoints equal sample mean plus or minus margin of error.
Critical values are calculated with adequate numerical precision, using a rational-approximation inverse normal function for z and a numerical inversion of the Student's t distribution (via the regularized incomplete beta function) for t, rather than a fixed lookup table.
Display rounding occurs after the main internal calculation.
Higher confidence produces wider intervals when other inputs stay fixed.
Larger samples generally produce narrower intervals.
Greater variability generally produces wider intervals.
Statistical validity depends on the assumptions and sampling method, not only the formula.
This calculator does not calculate proportion intervals or prediction intervals unless explicitly added as separate supported modes later.
Last reviewed: October 4, 2026
This calculator is provided for general educational and informational purposes only. It is not a substitute for instruction from a qualified statistics teacher or textbook, and it is not research, medical, or professional advice. Always double-check results that will be used for graded coursework, professional work, or any other high-stakes purpose.