Confidence Interval Calculator - Calculate one-sample or two-sample (difference of means) CI (2024)

Use this confidence interval calculator to easily calculate the confidence bounds for a one-sample statistic or for differences between two proportions or means (two independent samples). One-sided and two-sided intervals are supported, as well as confidence intervals for relative difference (percent difference). The calculator will also output P-value and Z-score if "difference between two groups" is selected.

Quick navigation:

  1. Using the confidence interval calculator
  2. What is a confidence interval and "confidence level"
  3. Confidence interval formula
  • Common critical values Z
  • How to interpret a confidence interval
  • Common misinterpretations of confidence intervals
  • One-sided vs. two-sided intervals
  • Confidence intervals for relative difference
  • Using the confidence interval calculator

    This confidence interval calculator allows you to perform a post-hoc statistical evaluation of a set of data when the outcome of interest is the absolute difference of two proportions (binomial data, e.g. conversion rate or event rate) or the absolute difference of two means (continuous data, e.g. height, weight, speed, time, revenue, etc.), or the relative difference between two proportions or two means. You can also calculate a confidence interval for the average of just a single group. It uses the Z-distribution (normal distribution). You can select any level of significance you require.

    If you are interested in a CI from a single group, then to calculate the confidence interval you need to know the sample size, sample standard deviation and the sample arithmetic average.

    If entering data for a CI for difference in proportions, provide the calculator the sample sizes of the two groups as well as the number or rate of events. You can enter that as a proportion (e.g. 0.10), percentage (e.g. 10%) or just the raw number of events (e.g. 50).

    If entering means data, make sure the tool is in "raw data" mode and simply copy/paste or type in the raw data, each observation separated by comma, space, new line or tab. Copy-pasting from a Google or Excel spreadsheet works fine.

    The confidence interval calculator will output: two-sided confidence interval, left-sided and right-sided confidence interval, as well as the mean or difference ± the standard error of the mean (SEM). It works for comparing independent samples, or for assessing if a sample belongs to a known population. For means data the calculator will also output the sample sizes, means, and pooled standard error of the mean. The Z-score (z statistic) and the p-value for the one-sided hypothesis (one-tailed test) will also be printed when calculating a confidence interval for the difference between proportions or means, allowing you to infer the direction of the effect.

    By default a 95% confidence interval is calculated, but the confidence level can be changed to match the required level of uncertainty.

    Warning: You must have fixed the sample size / stopping time of your experiment in advance. Doing otherwise means being guilty of optional stopping (fishing for significance) which will result in intervals that have narrower coverage than the nominal. Also, you should not use this confidence interval calculator for comparisons of more than two means or proportions, or for comparisons of two groups based on more than one metric. If your experiment involves more than one treatment group or has more than one outcome variable you need a more advanced calculator which corrects for multiple comparisons and multiple testing. This statistical calculator might help.

    What is a confidence interval and "confidence level"

    A confidence interval is defined by an upper and lower boundary (limit) for the value of a variable of interest and it aims to aid in assessing the uncertainty associated with a measurement, usually in experimental context, but also in observational studies. The wider an interval is, the more uncertainty there is in the estimate. Every confidence interval is constructed based on a particular required confidence level, e.g. 0.09, 0.95, 0.99 (90%, 95%, 99%) which is also the coverage probability of the interval. A 95% confidence interval (CI), for example, will contain the true value of interest 95% of the time (in 95 out of 5 similar experiments).

    Simple two-sided confidence intervals are symmetrical around the observed mean. This confidence interval calculator is expected to produce only such results. In certain scenarios where more complex models are deployed such as in sequential monitoring, asymmetrical intervals may be produced. In any particular case the true value may lie anywhere within the interval, or it might not be contained within it, no matter how high the confidence level is. Raising the confidence level widens the interval, while decreasing it makes it narrower. Similarly, larger sample sizes result in narrower confidence intervals, since the interval's asymptotic behavior is to be reduced to a single point.

    Confidence interval formula

    The mathematics of calculating a confindence interval are not that difficult. The generic formula used in any CI calculator is the observed statistic (mean, proportion, or otherwise) plus or minus the margin of error, expressed as standard error (SE). It is the basis of any confidence interval calculation:

    CIbounds = X ± SE

    In answering specific questions different variations apply. The formula when calculating a one-sample confidence interval is:

    Confidence Interval Calculator - Calculate one-sample or two-sample (difference of means) CI (1)

    where n is the number of observations in the sample, X (read "X bar") is the arithmetic mean of the sample and σ is the sample standard deviation (&sigma2 is the variance).

    The formula for two-sample confidence interval for the difference of means or proportions is:

    Confidence Interval Calculator - Calculate one-sample or two-sample (difference of means) CI (2)

    where μ1 is the mean of the baseline or control group, μ2 is the mean of the treatment group, n1 is the sample size of the baseline or control group, n2 is the sample size of the treatment group, and σp is the pooled standard deviation of the two samples. The entire expression to the right of ± is the sample estimate of the standard error of the mean (SEM) (unless the entire population has been measured, in which case there is no sampling involved in the calculation).

    In both confidence interval formulas Z is the score statistic, corresponding to the desired confidence level. The Z-score corresponding to a two-sided interval at level α (e.g. 0.90) is calculated for Z1-α/2, revealing that a two-sided interval, similarly to a two-sided p-value, is calculated by conjoining two one-sided intervals with half the error rate. E.g. a Z-score of 1.6448 is used for a 0.95 (95%) one-sided confidence interval and a 90% two-sided interval, while 1.956 is used for a 0.975 (97.5%) one-sided confidence interval and a 0.95 (95%) two-sided interval.

    Therefore it is important to use the right kind of interval: more on one-tailed vs. two-tailed intervals. Our confidence interval calculator will output both one-sided bounds, but it is up to the user to choose the correct one, based on the inference or estimation task at hand. The adequate interval is determined by the question you are looking to answer.

    Common critical values Z

    Below is a table with common critical values used for constructing two-sided confidence intervals for statistics with normally-distributed errors.

    Confidence interval critical values
    Two-sided Confidence levelCritical value (Z)

    For one-sided intervals, use a value for 2x the error. E.g. for a 95% one-sided interval use the critical value for a 90% two-sided interval above: 1.6449.

    How to interpret a confidence interval

    Confidence intervals are useful in visualizing the full range of effect sizes compatible with the data. Basically, any value outside of the interval is rejected: a null with that value would be rejected by a NHST with a significance threshold equal to the interval confidence level (the p-value statistic will be in the rejection region). Conversely, any value inside the interval cannot be rejected, thus when the null hypothesis of interest is covered by the interval it cannot be rejected. The latter, of course, assumes that there is a way to calculate exact interval bounds - many types of confidence intervals achieve their nominal coverage only approximately, that is their coverage is not guaranteed, but approximate. This is especially true in complicated scenarios, not covered in this confidence interval calculator.

    The above essentially means that the values outside the interval are the ones we can make inferences about. For the values within the interval we can only say that they cannot be rejected given the data at hand. When assessing the effect sizes that would be refuted by the data, you can construct as many confidence intervals at different confidence levels from the same set of data as you want - this is not a multiple testing issue. A better approach is to calculate the severity criterion of the null of interest, which will also allow you to make decisions about accepting the null.

    What then, if our null hypothesis of interest is completely outside the observed confidence interval? What inference can we make from seeing a calculation result which was quite improbable if the null was true?

    Logically, we can infer one of three things:

    1. There is a true effect from the tested treatment or intervention.
    2. There is no true effect, but we happened to observe a rare outcome.
    3. The statistical model for computing the confidence interval is invalid (does not reflect reality).

    Obviously, one can't simply jump to conclusion 1.) and claim it with one hundred percent confidence. This would go against the whole idea of the confidence interval. Instead, with can say that with confidence 95% (or other level chosen) we can reject the null hypothesis. In order to use the confidence interval as a part of a decision process you need to consider external factors, which are a part of the experimental design process, which includes deciding on the confidence level, sample size and power (power analysis), and the expected effect size, among other things.

    Common misinterpretations of confidence intervals

    While presenting confidence intervals tend to lead to fewer misinterpretations than p-values, they are still ripe for misuse or bad interpretations. Here are some of the most popular ones, according to Greenland at al. [1].

    Probability statements about specific intervals

    Strictly speaking, an interval computed using any CI calculator either contains or does not contain the true value. Therefore, strictly speaking, it would be incorrect to state about a particular 99% (or any other level) confidence interval that it has 99% probability that it contains the true effect or true value. What you can say is that procedure used to construct the intervals will produce intervals, containing the true value 99% of the time.

    The reverse statement would be that there is just 1% probability that the true value is outside of the interval. This is incorrect, as it is assigning probability to a hypothesis, instead of the testing procedure. What you can say is that, if any null hypothesis not covered by the interval is true, it will fall outside of such an interval only 1% of the time. Results from this confidence interval calculator should under no circ*mstances be interpreted as degrees of belief.

    A 95% confidence interval predicts where 95% of estimates from future studies will fall

    While inexperienced research workers make this mistake, a confidence interval makes no such predictions. Usually the probability with which outcomes from future experiments fall within any specific interval is significantly lower than the interval's confidence level.

    An interval containing the null is less precise than one excluding it

    How precise an interval is does not depend on whether or not it contains the null, or not. The precision of a confidence interval is determined by its width: the less wide the interval, the more accurate the estimate drawn from the data.

    One-sided vs. two-sided intervals

    While presently confidence intervals are customarily given by most researchers in their two-sided form, this can often be misleading. Such is the case where scientists are interested if a particular value below or above the interval can be excluded at a given significance level. A one-sided interval in which one side is plus or minus infinity is appropriate when we have a null / want to make statements about a value lying either above or below the top / bottom limit. By design a two-sided confidence interval is constructed as the overlap between two one-sided intervals at 1/2 the error rate 2.

    For example, if the calculator produced the two-sided 90% interval (2.5, 10), we can actually say that values less than 2.5 are excluded with 95% confidence precisely because a 90% two-sided interval is nothing more than two conjoined 95% one-sided intervals:

    Confidence Interval Calculator - Calculate one-sample or two-sample (difference of means) CI (3)

    Therefore, to make directional statements based on two-sided intervals, one needs to increase the significance level for the statement. In such cases it is better to use the appropriate one-sided interval instead, to avoid confusion.

    Confidence intervals for relative difference

    When comparing two independent groups and the variable of interest is the relative (a.k.a. relative change, relative difference, percent change, percentage difference), as opposed to the absolute difference between the two means or proportions, different confidence intervals need to be constructed. This is due to the fact that in calculating relative difference we are doing an additional division by a random variable: the conversion rate of the control during the experiment, which adds more variance to the estimation.

    In simulations performed [3] using the formulas operating in this confidence interval calculator, the difference a naive extrapolation of a confidence interval with 95% coverage for absolute difference had coverage for the relative difference between 90% and 94.8% depending on the size of the true difference, meaning that it had anywhere from a couple of percentage points to over 2 times worse coverage than the one for absolute difference. At the same time a properly constructed 95% confidence interval for relative difference had coverage of about 95%.

    The formula for a confidence interval around the relative difference (percent effect) is [4]:

    Confidence Interval Calculator - Calculate one-sample or two-sample (difference of means) CI (4)

    where RelDiff is calculated as 2 / μ1 - 1), CV1 is the coefficient of variation for the control and CV2 is the coefficient of variation for the treatment group, while Z is the critical value expressed as standardized score. Selecting "relative difference" in the calculator interface switches it to using the above formula.


    1 Greenland at al. (2016) "Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations", European Journal of Epidemiology 31:337–350

    2 Georgiev G.Z. (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] (accessed Apr 28, 2018)

    3 Georgiev G.Z. (2018) "Confidence Intervals & P-values for Percent Change / Relative Difference", [online] (accessed Jun 15, 2018)

    4 Kohavi et al. (2009) "Controlled experiments on the web: survey and practical guide", Data Mining and Knowledge Discovery 18:151

    Our statistical calculators have been featured in scientific papers and articles published in high-profile science journals by:

    Confidence Interval Calculator - Calculate one-sample or two-sample (difference of means) CI (2024)


    What is the CI for the difference between two means? ›

    Confidence Interval for the Difference of Two Means - Key takeaways. The conditions for constructing a confidence interval for the difference of two means are: The samples are independent. Either the sample size is large enough ( n 1 ≥ 30 and n 2 ≥ 30 ) or the population distribution is approximately normal.

    What is the CI for the difference between group means? ›

    The confidence interval for the difference in means provides an estimate of the absolute difference in means of the outcome variable of interest between the comparison groups. It is often of interest to make a judgment as to whether there is a statistically meaningful difference between comparison groups.

    How to find confidence interval with two samples? ›

    To obtain this confidence interval, compute the difference between the two sample means and then add and subtract the margin of error to obtain the upper and lower limit of this interval. The margin of error is obtained by multiplying the standard error by t*.

    What is the confidence interval for the difference between the means of the two populations? ›

    The confidence interval gives us a range of reasonable values for the difference in population means μ1 − μ2. We call this the two-sample T-interval or the confidence interval to estimate a difference in two population means.

    How do we estimate the difference between two means for two samples? ›

    If the sample means, ˉx1 and ˉx2, each meet the criteria for having nearly normal sampling distributions and the observations in the two samples are independent, then the difference in sample means, ˉx1−ˉx2, will have a sampling distribution that is nearly normal.

    Why is there a difference between 80% CI and 95% CI? ›

    Answer and Explanation:

    The confidence interval with a confidence level of 95% will be wider than that of 80% because the margin of error will be greater and with a wider confidence level, the interval is more imprecise.

    What is the formula for the mean difference between two groups? ›

    This means we take the mean from population 1 (ˉy1) and subtract from it the mean from population 2 (ˉy2). So, our "difference of two means" is (ˉy1 - ˉy2). When studying paired samples means, we are told we are looking at the "mean difference", ˉd.

    How do you tell if there is a difference between two groups? ›

    A t-test is an inferential statistic used to determine if there is a significant difference between the means of two groups and how they are related. T-tests are used when the data sets follow a normal distribution and have unknown variances, like the data set recorded from flipping a coin 100 times.

    How do you compare 95% CI? ›

    If the 95% confidence intervals are known for two sample means, there is a simple test to determine whether those sample means are significantly different. If the 95% CIs for the two sample means do not overlap, the means are significantly different at the P < 0.05 level.

    What is the formula for a confidence interval for a difference of two proportions? ›

    Two proportions confidence interval formula
    CI = p̂1 - p̂2 ± Z1-α/2 √(1(1 - p̂1)2(1 - p̂2)

    What is the formula for mean difference? ›

    The point estimate of mean difference for a paired analysis is usually available, since it is the same as for a parallel group analysis (the mean of the differences is equal to the difference in means): MD = ME – MC.

    How do you know if a confidence interval is one or two tailed? ›

    For a one-tailed test, the critical value is 1.645 . So the critical region is Z<−1.645 for a left-tailed test and Z>1.645 for a right-tailed test. For a two-tailed test, the critical value is 1.96 . So the confidence interval is |Z|<1.96 and the critical regions are where |Z|>1.96 .

    What are confidence intervals for the difference of two means? ›

    The confidence level, 1 – α, has the following interpretation. If thousands of samples of n1 and n2 items are drawn from populations using simple random sampling and a confidence interval is calculated for each sample, the proportion of those intervals that will include the true population mean difference is 1 – α.

    What does the 95% confidence interval of the difference mean? ›

    If a 95% confidence interval includes the null value, then there is no statistically meaningful or statistically significant difference between the groups. If the confidence interval does not include the null value, then we conclude that there is a statistically significant difference between the groups.

    When calculating a 95% confidence interval for the difference between two means, which of the following is true? ›

    Expert-Verified Answer. The statement that is true when calculating a 95% confidence interval for the difference between two means is that when the confidence interval ranges from a negative value to a positive value.

    What is the standard deviation of the difference between two means? ›

    Answer: The expression for calculating the standard deviation of the difference between two means is given by z = [(x1 - x2) - (µ1 - µ2)] / sqrt ( σ12 / n1 + σ22 / n2)

    What are the conditions for the difference of two means? ›

    We use this hypothesis test when the data meets the following conditions. The two random samples are independent. The variable is normally distributed in both populations. If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the t-distribution.

    What is the formula for the difference between two means? ›

    The sampling distribution of the difference between means is all possible differences a set of two means can have. The formula for the mean of the sampling distribution of the difference between means is: μm1m2 = μ1 – μ2.

    How to calculate confidence interval for difference between means in Excel? ›

    Use the following steps to calculate the confidence interval using both formats of the =CONFIDENCE() function in Excel:
    1. Calculate the sample mean. Arrange your data in ascending order in your spreadsheet. ...
    2. Find the standard deviation. ...
    3. Input the alpha value. ...
    4. Type in the confidence function. ...
    5. Calculate the confidence interval.
    Oct 16, 2023


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