Hypothesis Testing Calculator
| $H_o$: | |||
| $H_a$: | μ | ≠ | μ₀ |
| $n$ | = | $\bar{x}$ | = | = |
| $\text{Test Statistic: }$ | = |
| $\text{Degrees of Freedom: } $ | $df$ | = |
| $ \text{Level of Significance: } $ | $\alpha$ | = |
Type II Error
| $H_o$: | $\mu$ | ||
| $H_a$: | $\mu$ | ≠ | $\mu_0$ |
| $n$ | = | σ | = | $\mu$ | = |
| $\text{Level of Significance: }$ | $\alpha$ | = |
The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.
| $\sigma$ Known | $\sigma$ Unknown | |
| Test Statistic | $ z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{{\color{Black} n}}} $ | $ t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $ |
Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.
| Lower Tail Test | Upper Tail Test | Two-Tailed Test |
| $H_0 \colon \mu \geq \mu_0$ | $H_0 \colon \mu \leq \mu_0$ | $H_0 \colon \mu = \mu_0$ |
| $H_a \colon \mu | $H_a \colon \mu \neq \mu_0$ |
In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.
To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.
In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.
To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.
| Lower Tail Test | Upper Tail Test | Two-Tailed Test |
| If $z \leq -z_\alpha$, reject $H_0$. | If $z \geq z_\alpha$, reject $H_0$. | If $z \leq -z_{\alpha/2}$ or $z \geq z_{\alpha/2}$, reject $H_0$. |
| If $t \leq -t_\alpha$, reject $H_0$. | If $t \geq t_\alpha$, reject $H_0$. | If $t \leq -t_{\alpha/2}$ or $t \geq t_{\alpha/2}$, reject $H_0$. |
When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.
| Condition | ||||
| $H_0$ True | $H_a$ True | |||
| Conclusion | Accept $H_0$ | Correct | Type II Error | |
| Reject $H_0$ | Type I Error | Correct | ||
Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.
Hypothesis Testing Calculator Online
Hypothesis testing is a foundational method used in statistics to infer the validity of a hypothesis about a population parameter. The Hypothesis Testing Calculator facilitates this process by automating the computations necessary for the t-test , a method used to compare sample means against a hypothesized mean or against each other. Let’s delve into the formulas this calculator uses to execute one-sample and two-sample t-tests.
One-Sample t-Test
This test is used to determine if the mean (x̄) of your sample is statistically different from a hypothesized population mean (μ₀).
Two-Sample t-Test
Equal variances:, unequal variances (welch’s t-test):.
t = (x̄₁ – x̄₂) / (√((s₁² / n₁) + (s₂² / n₂)))
Table of Critical t-Values
| Confidence Level (%) | df=10 | df=30 | df=50 | df=100 |
|---|---|---|---|---|
| 90 | 1.812 | 1.697 | 1.676 | 1.660 |
| 95 | 2.228 | 2.042 | 2.009 | 1.984 |
| 99 | 3.169 | 2.750 | 2.678 | 2.626 |
These values are crucial in hypothesis testing as they help define the threshold for significance, assisting users of the calculator in interpreting their results accurately.
Using the one-sample t-test formula:
t = (74 – 70) / (8 / √36) = (4 / 1.333) = 3.00
Most Common FAQs
The p-value represents the probability of obtaining test results at least as extreme as the results observed, under the assumption that the null hypothesis is correct. A low p-value (typically below 0.05) indicates strong evidence against the null hypothesis, hence it is usually rejected.
Yes, while the t-test is specifically design for means, the principles of hypothesis testing apply to other parameters such as proportions and variances. Which can also be tested using appropriate versions of hypothesis tests such as the z-test and F-test.
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T-Test calculator
The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t – test.
Enter sample data Reporting results in APA styleOne sample t-test, what is a one sample t-test, how to use the one sample t test calculator, calculators. 
T-Test Calculator for 2 Independent MeansThis simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation. Further Information A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g., males and females). Requirements
Null Hypothesis H0: u1 - u2 = 0, where u1 is the mean of first population and u2 the mean of the second. As above, the null hypothesis tends to be that there is no difference between the means of the two populations; or, more formally, that the difference is zero (so, for example, that there is no difference between the average heights of two populations of males and females).  Z-test CalculatorTable of contents This Z-test calculator is a tool that helps you perform a one-sample Z-test on the population's mean . Two forms of this test - a two-tailed Z-test and a one-tailed Z-tests - exist, and can be used depending on your needs. You can also choose whether the calculator should determine the p-value from Z-test or you'd rather use the critical value approach! Read on to learn more about Z-test in statistics, and, in particular, when to use Z-tests, what is the Z-test formula, and whether to use Z-test vs. t-test. As a bonus, we give some step-by-step examples of how to perform Z-tests! Or you may also check our t-statistic calculator , where you can learn the concept of another essential statistic. If you are also interested in F-test, check our F-statistic calculator . What is a Z-test?A one sample Z-test is one of the most popular location tests. The null hypothesis is that the population mean value is equal to a given number, μ 0 \mu_0 μ 0 : We perform a two-tailed Z-test if we want to test whether the population mean is not μ 0 \mu_0 μ 0 : and a one-tailed Z-test if we want to test whether the population mean is less/greater than μ 0 \mu_0 μ 0 : Let us now discuss the assumptions of a one-sample Z-test. When do I use Z-tests?You may use a Z-test if your sample consists of independent data points and: the data is normally distributed , and you know the population variance ; the sample is large , and data follows a distribution which has a finite mean and variance. You don't need to know the population variance. The reason these two possibilities exist is that we want the test statistics that follow the standard normal distribution N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) . In the former case, it is an exact standard normal distribution, while in the latter, it is approximately so, thanks to the central limit theorem. The question remains, "When is my sample considered large?" Well, there's no universal criterion. In general, the more data points you have, the better the approximation works. Statistics textbooks recommend having no fewer than 50 data points, while 30 is considered the bare minimum. Z-test formulaLet x 1 , . . . , x n x_1, ..., x_n x 1 , ... , x n be an independent sample following the normal distribution N ( μ , σ 2 ) \mathrm N(\mu, \sigma^2) N ( μ , σ 2 ) , i.e., with a mean equal to μ \mu μ , and variance equal to σ 2 \sigma ^2 σ 2 . We pose the null hypothesis, H 0 : μ = μ 0 \mathrm H_0 \!\!:\!\! \mu = \mu_0 H 0 : μ = μ 0 . We define the test statistic, Z , as: x ˉ \bar x x ˉ is the sample mean, i.e., x ˉ = ( x 1 + . . . + x n ) / n \bar x = (x_1 + ... + x_n) / n x ˉ = ( x 1 + ... + x n ) / n ; μ 0 \mu_0 μ 0 is the mean postulated in H 0 \mathrm H_0 H 0 ; n n n is sample size; and σ \sigma σ is the population standard deviation. In what follows, the uppercase Z Z Z stands for the test statistic (treated as a random variable), while the lowercase z z z will denote an actual value of Z Z Z , computed for a given sample drawn from N(μ,σ²). If H 0 \mathrm H_0 H 0 holds, then the sum S n = x 1 + . . . + x n S_n = x_1 + ... + x_n S n = x 1 + ... + x n follows the normal distribution, with mean n μ 0 n \mu_0 n μ 0 and variance n 2 σ n^2 \sigma n 2 σ . As Z Z Z is the standardization (z-score) of S n / n S_n/n S n / n , we can conclude that the test statistic Z Z Z follows the standard normal distribution N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) , provided that H 0 \mathrm H_0 H 0 is true. By the way, we have the z-score calculator if you want to focus on this value alone. If our data does not follow a normal distribution, or if the population standard deviation is unknown (and thus in the formula for Z Z Z we substitute the population standard deviation σ \sigma σ with sample standard deviation), then the test statistics Z Z Z is not necessarily normal. However, if the sample is sufficiently large, then the central limit theorem guarantees that Z Z Z is approximately N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) . In the sections below, we will explain to you how to use the value of the test statistic, z z z , to make a decision , whether or not you should reject the null hypothesis . Two approaches can be used in order to arrive at that decision: the p-value approach, and critical value approach - and we cover both of them! Which one should you use? In the past, the critical value approach was more popular because it was difficult to calculate p-value from Z-test. However, with help of modern computers, we can do it fairly easily, and with decent precision. In general, you are strongly advised to report the p-value of your tests! p-value from Z-testFormally, the p-value is the smallest level of significance at which the null hypothesis could be rejected. More intuitively, p-value answers the questions: provided that I live in a world where the null hypothesis holds, how probable is it that the value of the test statistic will be at least as extreme as the z z z - value I've got for my sample? Hence, a small p-value means that your result is very improbable under the null hypothesis, and so there is strong evidence against the null hypothesis - the smaller the p-value, the stronger the evidence. To find the p-value, you have to calculate the probability that the test statistic, Z Z Z , is at least as extreme as the value we've actually observed, z z z , provided that the null hypothesis is true. (The probability of an event calculated under the assumption that H 0 \mathrm H_0 H 0 is true will be denoted as P r ( event ∣ H 0 ) \small \mathrm{Pr}(\text{event} | \mathrm{H_0}) Pr ( event ∣ H 0 ) .) It is the alternative hypothesis which determines what more extreme means :
The symmetry of the normal distribution gives:
To compute these probabilities, we can use the cumulative distribution function, (cdf) of N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) , which for a real number, x x x , is defined as: Also, p-values can be nicely depicted as the area under the probability density function (pdf) of N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) , due to: Two-tailed Z-test and one-tailed Z-testWith all the knowledge you've got from the previous section, you're ready to learn about Z-tests.
From the fact that Φ ( − z ) = 1 − Φ ( z ) \Phi(-z) = 1 - \Phi(z) Φ ( − z ) = 1 − Φ ( z ) , we deduce that The p-value is the area under the probability distribution function (pdf) both to the left of − ∣ z ∣ -|z| − ∣ z ∣ , and to the right of ∣ z ∣ |z| ∣ z ∣ : 
The p-value is the area under the pdf to the left of our z z z : 
The p-value is the area under the pdf to the right of z z z :  The decision as to whether or not you should reject the null hypothesis can be now made at any significance level, α \alpha α , you desire! if the p-value is less than, or equal to, α \alpha α , the null hypothesis is rejected at this significance level; and if the p-value is greater than α \alpha α , then there is not enough evidence to reject the null hypothesis at this significance level. Z-test critical values & critical regionsThe critical value approach involves comparing the value of the test statistic obtained for our sample, z z z , to the so-called critical values . These values constitute the boundaries of regions where the test statistic is highly improbable to lie . Those regions are often referred to as the critical regions , or rejection regions . The decision of whether or not you should reject the null hypothesis is then based on whether or not our z z z belongs to the critical region. The critical regions depend on a significance level, α \alpha α , of the test, and on the alternative hypothesis. The choice of α \alpha α is arbitrary; in practice, the values of 0.1, 0.05, or 0.01 are most commonly used as α \alpha α . Once we agree on the value of α \alpha α , we can easily determine the critical regions of the Z-test: To decide the fate of H 0 \mathrm H_0 H 0 , check whether or not your z z z falls in the critical region: If yes, then reject H 0 \mathrm H_0 H 0 and accept H 1 \mathrm H_1 H 1 ; and If no, then there is not enough evidence to reject H 0 \mathrm H_0 H 0 . As you see, the formulae for the critical values of Z-tests involve the inverse, Φ − 1 \Phi^{-1} Φ − 1 , of the cumulative distribution function (cdf) of N ( 0 , 1 ) \mathrm N(0, 1) N ( 0 , 1 ) . How to use the one-sample Z-test calculator?Our calculator reduces all the complicated steps: Choose the alternative hypothesis: two-tailed or left/right-tailed. In our Z-test calculator, you can decide whether to use the p-value or critical regions approach. In the latter case, set the significance level, α \alpha α . Enter the value of the test statistic, z z z . If you don't know it, then you can enter some data that will allow us to calculate your z z z for you:
Results appear immediately below the calculator. If you want to find z z z based on p-value , please remember that in the case of two-tailed tests there are two possible values of z z z : one positive and one negative, and they are opposite numbers. This Z-test calculator returns the positive value in such a case. In order to find the other possible value of z z z for a given p-value, just take the number opposite to the value of z z z displayed by the calculator. Z-test examplesTo make sure that you've fully understood the essence of Z-test, let's go through some examples:
Formally, the hypotheses that we set are the following: H 0 : μ = 1000 ml \mathrm H_0 \! : \mu = 1000 \text{ ml} H 0 : μ = 1000 ml H 1 : μ < 1000 ml \mathrm H_1 \! : \mu \lt 1000 \text{ ml} H 1 : μ < 1000 ml We went to a shop and bought a sample of 9 bottles. After carefully measuring the volume of juice in each bottle, we've obtained the following sample (in milliliters): 1020 , 970 , 1000 , 980 , 1010 , 930 , 950 , 980 , 980 \small 1020, 970, 1000, 980, 1010, 930, 950, 980, 980 1020 , 970 , 1000 , 980 , 1010 , 930 , 950 , 980 , 980 . Sample size: n = 9 n = 9 n = 9 ; Sample mean: x ˉ = 980 m l \bar x = 980 \ \mathrm{ml} x ˉ = 980 ml ; Population standard deviation: σ = 30 m l \sigma = 30 \ \mathrm{ml} σ = 30 ml ; And, therefore, p-value = Φ ( − 2 ) ≈ 0.0228 \text{p-value} = \Phi(-2) \approx 0.0228 p-value = Φ ( − 2 ) ≈ 0.0228 . As 0.0228 < 0.05 0.0228 \lt 0.05 0.0228 < 0.05 , we conclude that our suspicions aren't groundless; at the most common significance level, 0.05, we would reject the producer's claim, H 0 \mathrm H_0 H 0 , and accept the alternative hypothesis, H 1 \mathrm H_1 H 1 . We tossed a coin 50 times. We got 20 tails and 30 heads. Is there sufficient evidence to claim that the coin is biased? Clearly, our data follows Bernoulli distribution, with some success probability p p p and variance σ 2 = p ( 1 − p ) \sigma^2 = p (1-p) σ 2 = p ( 1 − p ) . However, the sample is large, so we can safely perform a Z-test. We adopt the convention that getting tails is a success. Let us state the null and alternative hypotheses: H 0 : p = 0.5 \mathrm H_0 \! : p = 0.5 H 0 : p = 0.5 (the coin is fair - the probability of tails is 0.5 0.5 0.5 ) H 1 : p ≠ 0.5 \mathrm H_1 \! : p \ne 0.5 H 1 : p = 0.5 (the coin is biased - the probability of tails differs from 0.5 0.5 0.5 ) In our sample we have 20 successes (denoted by ones) and 30 failures (denoted by zeros), so: Sample size n = 50 n = 50 n = 50 ; Sample mean x ˉ = 20 / 50 = 0.4 \bar x = 20/50 = 0.4 x ˉ = 20/50 = 0.4 ; Population standard deviation is given by σ = 0.5 × 0.5 \sigma = \sqrt{0.5 \times 0.5} σ = 0.5 × 0.5 (because 0.5 0.5 0.5 is the proportion p p p hypothesized in H 0 \mathrm H_0 H 0 ). Hence, σ = 0.5 \sigma = 0.5 σ = 0.5 ;
Since 0.1573 > 0.1 0.1573 \gt 0.1 0.1573 > 0.1 we don't have enough evidence to reject the claim that the coin is fair , even at such a large significance level as 0.1 0.1 0.1 . In that case, you may safely toss it to your Witcher or use the coin flip probability calculator to find your chances of getting, e.g., 10 heads in a row (which are extremely low!). What is the difference between Z-test vs t-test?We use a t-test for testing the population mean of a normally distributed dataset which had an unknown population standard deviation . We get this by replacing the population standard deviation in the Z-test statistic formula by the sample standard deviation, which means that this new test statistic follows (provided that H₀ holds) the t-Student distribution with n-1 degrees of freedom instead of N(0,1) . When should I use t-test over the Z-test?For large samples, the t-Student distribution with n degrees of freedom approaches the N(0,1). Hence, as long as there are a sufficient number of data points (at least 30), it does not really matter whether you use the Z-test or the t-test, since the results will be almost identical. However, for small samples with unknown variance, remember to use the t-test instead of Z-test . How do I calculate the Z test statistic?To calculate the Z test statistic:
Here, we perform a Z-test for population mean μ. Null hypothesis H₀: μ = μ₀. Alternative hypothesis H₁ Significance level α The probability that we reject the true hypothesis H₀ (type I error).
 We want your feedbackPlease add a message. Message received. Thanks for the feedback. Calculator: One-Sample t-TestOne-Sample t-Test CalculatorThis calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability values associated with the test. Please enter the necessary parameter values, and then click 'Calculate'.
An open portfolio of interoperable, industry leading products The Dotmatics digital science platform provides the first true end-to-end solution for scientific R&D, combining an enterprise data platform with the most widely used applications for data analysis, biologics, flow cytometry, chemicals innovation, and more.  Statistical analysis and graphing software for scientists Bioinformatics, cloning, and antibody discovery software Plan, visualize, & document core molecular biology procedures Electronic Lab Notebook to organize, search and share data Proteomics software for analysis of mass spec data Modern cytometry analysis platform Analysis, statistics, graphing and reporting of flow cytometry data Software to optimize designs of clinical trials POPULAR USE CASES T test calculatorA t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator . 1. Choose data entry formatCaution: Changing format will erase your data. 2. Choose a testHelp me choose 3. Enter dataHelp me arrange the data 4. View the resultsWhat is a t test. A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups. This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value. The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:  See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example. How to use the t test calculator
Performing t tests? We can help.Sign up for more information on how to perform t tests and other common statistical analyses. Common t test confusionIn addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight. Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples. ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups. Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests. Assumptions of t testsBecause there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:
Interpreting resultsThe three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test. While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different. If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion. Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page. Graphing t testsThis calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests. Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism. For more informationOur ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests! If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:
Check out our video on how to perform a t test in Prism , for an example from start to finish! Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead. We Recommend:Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.
PLANETCALC Online calculators
Two sample t-TestThe calculator to perform t-Test for the Significance of the Difference between the Means of Two Independent Samples The calculator below implements the most known statistical test, namely, the Independent Samples t-test or Two samples t-test. t-test, also known as Student's t-test, after William Sealy Gosset. "Student" was his pen name. The test deals with the null hypothesis such that the means of two populations are equal. To put it in other words, the difference we find between the means of the two samples should not significantly differ from zero. Again, the test works only if certain assumptions are met. These are:
The calculator displays a level of confidence for both directional and non-directional tests. Let's say you get the result of 96%. Essentially this means that you have 96% confidence that the obtained difference shows something more than simple luck. The chance that you can get the obtained difference and the means of the two samples are the same is only 4%. This is the level of significance you calculate. Now, depending on your chosen level of significance, you can reject or fail to reject your null hypothesis. If you care to find more, you can read excellent explanations here , starting from Chapter 9.  Two samples t-TestSimilar calculators.
Share this pageTeach yourself statistics Bartlett's Test CalculatorBartlett's test is used to test the assumption that variances are equal (homogeneous) across groups. For help in using this calculator, read the Frequently-Asked Questions or review the Sample Problem . To learn more about Bartlett's test, read Stat Trek's tutorial on Bartlett's test .
Frequently-Asked QuestionsInstructions: To find the answer to a frequently-asked question, simply click on the question. What is Bartlett's test?Bartlett's test is used to test the assumption that variances are equal (i.e., homogeneous) across groups. The test is easy to implement and produces valid results, assuming data points within groups are randomly sampled from a normal distribution. Because Bartlett's test is sensitive to departures from normality, a normality test is prudent. Several ways to check for departures from normality are described at: How to Test for Normality: Three Simple Tests . Note: Unlike Hartley's Fmax test , which also tests for homogeneity, Bartlett's test does not assume equal sample sizes across groups. How does Bartlett's test work?Bartlett's test is an actual hypothesis test , where we examine observed data to choose between two statistical hypotheses: H 0 : σ 2 i = σ 2 j for all groups H 0 : σ 2 i ≠ σ 2 j for at least one pair of groups Like many other techniques for testing hypotheses, Bartlett's test for homogeneity involves computing a test-statistic and finding the P-value for the test statistic, given degrees of freedom and significance level . If the P-value is bigger than the significance level, we accept the null hypothesis; if it is smaller, we reject the null hypothesis. What steps (computations) are required to execute Bartlett's test?The steps required to conduct Bartlett's test for homogeneity are detailed below:
s 2 j = [ Σ ( X i, j - X j ) 2 ] / ( n j - 1 ) where X i, j is the score for observation i in Group j , X j is the mean of Group j , n j is the number of observations in Group j , and k is the number of groups. N = Σ n i s 2 p = [ Σ ( n j -1 ) s 2 j ] / ( N - K ) where n j is the sample size in Group j , k is the number of groups, N is the total sample size, and s 2 j is the sample variance in Group j . A = ( N - k ) * ln( s 2 p ) B = Σ [ ( n j - 1 ) * ln( s 2 j ) ] C = 1 / [ 3 * ( k - 1 ) ] D = Σ [ 1 / ( n j - 1 ) - 1 / ( N - k ) ] T = ( A - B ) / [ 1 + ( C * D ) ] where A is the first term in the numerator of the test statistic, B is the second term in the numerator, C is the first term in the denominator , D is the second term in the denominator, and ln is the natural logarithm. It turns out that the test statistic (T) is distributed much like a chi-square statistic with ( k-1 ) degrees of freedom. Knowing the value of T and the degrees of freedom associated with T, we can use Stat Trek's Chi-Square Calculator to find the P-value - the probability of seeing a test statistic more extreme than T.
What should I enter in the field for number of groups?Bartlett's is designed to test the hypothesis of homogeneity among nonoverlapping sets of data. The number of groups is the number of data sets under consideration. What should I enter for significance level?The significance level is the probability of rejecting the null hypothesis when it is true. Researchers often choose 0.05 or 0.01 for a significance level. What should I enter for sample size?Sample size refers to the number of observations in a group. For each group, enter the number of observations in the space provided. Note: Unlike some other tests for homogeneity (e.g., Hartley's Fmax test ), Bartlett's test does not require equal sample sizes across groups. What should I enter for variance?In the fields provided, enter an estimate of sample variance for each group. To compute sample variance ( s 2 j ) for each group, use the following formula: where X i, j is the score for observation i in Group j , X j is the mean of Group j , and n j is the number of observations in Group j . What is degrees of freedom?Bartlett's test computes a test statistic (T) to test for normality. The degrees of freedom ( df ) for a chi-square test of that statistic is: where N is the total sample size across all groups, and k is the number of groups in the sample. What is the test statistic (T)?The test statistic (T) is the statistic used by Bartlett's test to make a decision about whether to accept or reject the null hypothesis of equal variances between groups. When T is very big, we reject the null hypothesis; when T is small, we accept the null hypothesis. The calculator computes a T statistic, based on user inputs. The formulas that the calculator uses to compute a T statistic are given at Bartlett's Test for Homogeneity of Variance . What is the P-value?If you assume that the null hypothesis of equal variance is true, the P-value is the probability of seeing a test statistic (T) that is more extreme (bigger) than the actual test statistic computed from sample data. How does the calculator test hypotheses?Like many other techniques for testing hypotheses, Bartlett's test for homogeneity of variance involves computing a test-statistic and finding the P-value for the test statistic, given degrees of freedom and significance level . If the P-value is bigger than the significance level, the calculator accepts the null hypothesis. Otherwise, it rejects the null hypothesis. Sample ProblemsThe table below shows sample data and variance for five groups. How would you test the assumption that variances are equal across groups?
One option would be to use Stat Trek's Bartlett's Test Calculator . Simply, take the following steps:
Then, click the Calculate button to produce the output shown below: From the calculator, we see that the test statistic (T) is 8.91505. Assuming equal variances in groups and given a significance level of 0.05, the probability of observing a test statistic (T) bigger than 8.91505 is given by the P-value. Since the P-value (0.06326) is bigger than the significance level (0.05), we cannot reject the null hypothesis of equal variances across groups. Note: To see the hand calculations required to solve this problem, go to Bartlett's Test for Homogeneity of Variance: Example 1 . Oops! Something went wrong. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Hypothesis Testing Calculator. The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is ...
Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important. Learn . Courses Career Tracks Projects Upcoming Courses Certificates . Career Track Certificate ...
Alternative to statistical software like SPSS and STATA. DATAtab was designed for ease of use and is a compelling alternative to statistical programs such as SPSS and STATA. On datatab.net, data can be statistically evaluated directly online and very easily (e.g. t-test, regression, correlation etc.). DATAtab's goal is to make the world of statistical data analysis as simple as possible, no ...
This t-test calculator allows you to use either the p-value approach or the critical regions approach to hypothesis testing! Enter your T-score and the number of degrees of freedom . If you don't know them, provide some data about your sample(s): sample size, mean, and standard deviation, and our t-test calculator will compute the T-score and ...
Hypothesis testing is a foundational method used in statistics to infer the validity of a hypothesis about a population parameter. The Hypothesis Testing Calculator facilitates this process by automating the computations necessary for the t-test, a method used to compare sample means against a hypothesized mean or against each other.Let's delve into the formulas this calculator uses to ...
T-Test calculator. The Student's t-test is used to determine if means of two data sets differ significantly. This calculator will generate a step by step explanation on how to apply t - test. Two sample t-test One sample t-test.
Hypothesis Test Calculator. Upload your data set below to get started. Upload File. Or input your data as csv. column_one,column_two,column_three 1,2,3 4,5,6 7,8,9. Submit CSV. Sharing helps us build more free tools. Embed on Your Website.
Clear the table in the Hypothesis test calculator. Copy your data into the table. Select the variables. In the hypothesis test calculator you can calculate e.g. a t-test, a chi-square test, a binomial test or an analysis of variance. If you need a more detailed explanation, you can find more information in the tutorials.
To calculate a t-Test online just select one metric Variable and one nominal Variable with two values. Here you can easily calculate a t-test online, just copy your data into the upper table and select one or two variables. Paired t-test Calculator: If you click on two metric variables, a paired samples t test is calculated.
The p-value is important to reported in a independent samples t-tests, representing the probability of obtaining a difference as extreme as the observed one, assuming the null hypothesis is true. If the p-value is below the chosen significance level (α), typically 0.05, then the null hypothesis is rejected. Calculate Independent t-test online
Step 3: Click the "Perform Test" button. The tool will quickly process the information, performing the hypothesis test and generating a detailed report. Step 4: Review the results. The calculator provides the calculated t-value, critical value, and the test's conclusion. You can also visualize the sample distribution with a dynamic chart ...
t-test calculator is an online statistics tool to estimate the significance of observed differences between the means of two samples when there is a null hypothesis that is no significant difference between the means by using standard deviation. It is necessary to follow the next steps: Enter two samples (observed values) in the box. These values must be real numbers or variables and may be ...
Calculations: Once you input the data and parameters, the calculator performs the necessary statistical tests and calculations. It generates results such as the test statistic, degrees of freedom, and the p-value. Interpretation: Based on the results, the calculator helps you determine whether to reject or fail to reject the null hypothesis.
This test statistic calculator helps to find the static value for hypothesis testing. The calculated test value shows if there's enough evidence to reject a null hypothesis. Also, this calculator performs calculations of either for one population mean, comparing two means, single population proportion, and two population proportions.
Single Sample T-Test Calculator. A single sample t-test (or one sample t-test) is used to compare the mean of a single sample of scores to a known or hypothetical population mean. So, for example, it could be used to determine whether the mean diastolic blood pressure of a particular group differs from 85, a value determined by a previous study.
The one-sample t-test determines if the mean of a single sample is significantly different from a known population mean. The one sample t-test calculator calculates the one sample t-test p-value and the effect size. When you enter the raw data, the one sample t-test calculator provides also the Shapiro-Wilk normality test result and the outliers.
This simple t -test calculator, provides full details of the t-test calculation, including sample mean, sum of squares and standard deviation. A t -test is used when you're looking at a numerical variable - for example, height - and then comparing the averages of two separate populations or groups (e.g., males and females).
Choose the alternative hypothesis: two-tailed or left/right-tailed. In our Z-test calculator, you can decide whether to use the p-value or critical regions approach. In the latter case, set the significance level, α. \alpha α. Enter the value of the test statistic, z. z z.
A P-value calculator is used to determine the statistical significance of an observed result in hypothesis testing. It takes as input the observed test statistic, the null hypothesis, and the relevant parameters of the statistical test (such as degrees of freedom), and computes the p-value. The p-value represents the probability of obtaining ...
This calculator will conduct a complete one-sample t-test, given the sample mean, the sample size, the hypothesized mean, and the sample standard deviation. The results generated by the calculator include the t-statistic, the degrees of freedom, the critical t-values for both one-tailed (directional) and two-tailed (non-directional) hypotheses, and the one-tailed and two-tailed probability ...
A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for ...
The calculator below implements the most known statistical test, namely, the Independent Samples t-test or Two samples t-test. t-test, also known as Student's t-test, after William Sealy Gosset. "Student" was his pen name. The test deals with the null hypothesis such that the means of two populations are equal.
Bartlett's test is an actual hypothesis test, where we examine observed data to choose between two statistical hypotheses: . Null hypothesis: Variance is equal across all groups. H 0: σ 2 i = σ 2 j for all groups. Alternative hypothesis: Variance is not equal across all groups. H 0: σ 2 i ≠ σ 2 j for at least one pair of groups