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This t-statistic calculator takes the busywork out of one-sample t-tests by computing your t-value instantly from four inputs: your sample mean, the population mean, your sample size, and the standard deviation. Just plug in your numbers and get the result you need to move forward with your hypothesis test.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The t-statistic is one of the most widely used values in statistical analysis, and understanding what it tells you matters just as much as calculating it. 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Think of it as a signal-to-noise ratio: the \"signal\" is the difference between what you observed and what you expected, and the \"noise\" is how spread out your data is.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The larger the absolute value of your t-statistic, the stronger the evidence that your sample mean is genuinely different from the population mean — not just different by random chance.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"For example, a t-statistic of 0.5 suggests your sample mean is close to the population mean relative to the spread of your data. A t-statistic of 4.2 suggests a much more meaningful difference. Whether that difference is statistically significant depends on your degrees of freedom and chosen significance level, but the t-statistic is the starting point for making that call.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"William Sealy Gosset developed the t-distribution in 1908 while working as a chemist at Guinness Brewery. 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You need to compare it against a critical value for your degrees of freedom and significance level (usually α = 0.05). If your |t| exceeds the critical value, you reject the null hypothesis.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"For a quick rule of thumb with large samples (n > 30): a |t| greater than about 2.0 is usually significant at the 0.05 level. 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Her class averaged 76.4 with a standard deviation of 8.2.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"x̄ = 76.4, μ = 72, n = 35, s = 8.2","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Standard error = 8.2 / √35 = 8.2 / 5.916 = 1.386","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"t = (76.4 - 72) / 1.386 = 4.4 / 1.386 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"3.17","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"With 34 degrees of freedom, a t-statistic of 3.17 exceeds the critical value of 2.032 (two-tailed, α = 0.05). 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A trial with 45 participants shows an average of 7.3 hours with a standard deviation of 1.1 hours.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"x̄ = 7.3, μ = 7.0, n = 45, s = 1.1","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Standard error = 1.1 / √45 = 1.1 / 6.708 = 0.164","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"t = (7.3 - 7.0) / 0.164 = 0.3 / 0.164 = ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"1.83","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"With 44 degrees of freedom, the critical value at α = 0.05 (two-tailed) is approximately 2.015. Since 1.83 < 2.015, the result is not statistically significant — the observed increase could be due to chance. The researcher might consider increasing the sample size for a follow-up study.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"When to Use a One-Sample T-Test","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The one-sample t-test (and this calculator) is the right choice when you:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Have ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"one sample","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" and want to compare its mean to a known or hypothesized population value","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Are working with ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"continuous data","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" (measurements, scores, times — not categories)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Don't know the population standard deviation","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" and need to estimate it from your sample","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Have a ","type":"text","version":1},{"detail":0,"format":1,"mode":"normal","style":"","text":"reasonably normal distribution","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" or a sample size above 30 (the Central Limit Theorem helps with larger samples)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"When to use something else:","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Comparing two groups?","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" You need a two-sample t-test (independent or paired)","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Know the population standard deviation?","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Use a z-test instead","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Non-normal data with small samples?","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Consider a non-parametric test like the Wilcoxon signed-rank test","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Comparing three or more groups?","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Use ANOVA","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Common Mistakes to Avoid","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Using population standard deviation instead of sample standard deviation.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" If you're using Excel, make sure you use STDEV.S (sample) and not STDEV.P (population). This is one of the most common errors in introductory statistics and will give you a z-statistic rather than a t-statistic.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Forgetting to check assumptions.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" The t-test assumes your data comes from a roughly normal distribution. With small samples (under 30), severe skewness or outliers can distort your results. Always plot your data first — a histogram or normal probability plot takes seconds and can save you from drawing wrong conclusions.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Confusing statistical significance with practical significance.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" A large sample size can make tiny differences statistically significant. If your t-test shows that a new study method increases test scores by 0.3 points out of 100 with p < 0.05, the result is statistically significant but practically meaningless. Always consider effect size alongside your t-statistic.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Using the wrong type of t-test.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" This calculator performs a one-sample t-test. If you're comparing the means of two different groups, you need a two-sample (independent) t-test. If you're comparing before-and-after measurements on the same subjects, you need a paired t-test.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Tips for Stronger Statistical Analysis","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h2"},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Increase your sample size when possible.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Larger samples reduce the standard error, making your t-test more sensitive to real differences. Going from n = 10 to n = 40 cuts your standard error roughly in half.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Report confidence intervals alongside your t-statistic.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" A t-statistic tells you whether a difference exists, but a confidence interval tells you the likely range of that difference — which is often more useful for decision-making.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":2},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Check for outliers before running your test.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" A single extreme value in a small sample can dramatically shift your sample mean and standard deviation, producing misleading results.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":3},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Document your hypotheses before collecting data.","type":"text","version":1},{"detail":0,"format":0,"mode":"normal","style":"","text":" Deciding what you're testing after seeing the results (called \"HARKing\" — Hypothesizing After Results are Known) inflates your false positive rate and undermines the validity of your analysis.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"listitem","version":1,"value":4}],"direction":null,"format":"","indent":0,"type":"list","version":1,"listType":"bullet","start":1,"tag":"ul"}],"direction":null,"format":"","indent":0,"type":"root","version":1}}},"id":"69b487d47999bd000420104e"}]}]}],"$L34"]}],"$L35"]}]]}]}]}]
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34:["$","div",null,{"className":"bg-white rounded-lg shadow-sm p-8","children":[["$","h2",null,{"className":"typo-large mb-6 text-mist-950","children":"Frequently Asked Questions"}],["$","div",null,{"className":"space-y-6","children":[["$","div","0",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What is the t-statistic formula?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The one-sample t-statistic equals the difference between your sample mean and the population mean, divided by the standard error: t = (x̄ - μ) / (s / √n). The standard error (s / √n) accounts for both the variability in your data and your sample size.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","1",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What does a t-statistic tell you?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"It tells you how many standard errors your sample mean is from the hypothesized population mean. A t-statistic of 2.5 means your sample mean is 2.5 standard errors away from the population mean — suggesting the difference is unlikely due to random chance alone.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","2",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How do I find degrees of freedom for a one-sample t-test?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Degrees of freedom equal your sample size minus one (df = n - 1). If you have 25 observations, your degrees of freedom are 24. You need this value to look up critical values in a t-table or to calculate a p-value.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","3",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What's the difference between a t-statistic and a z-statistic?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Both measure how far a sample mean is from a population mean in standard error units. The key difference: use a z-statistic when you know the population standard deviation, and a t-statistic when you estimate it from your sample. With large samples (n > 30), the two values converge and the practical difference becomes negligible.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],["$","div","4",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Can a t-statistic be negative?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Yes. A negative t-statistic simply means your sample mean is below the population mean. The sign tells you the direction of the difference, while the absolute value tells you the magnitude. A t-statistic of -2.8 carries the same strength of evidence as +2.8.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}],"$L36","$L37","$L38","$L39","$L3a"]}]]}]
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36:["$","div","5",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"How large should my sample be for a t-test?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"There's no strict minimum, but the t-test becomes more reliable as sample size increases. With fewer than 15 observations, your data should be approximately normally distributed. Above 30, the Central Limit Theorem provides reasonable protection against non-normality. For detecting small effects, you may need 100 or more observations.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
37:["$","div","6",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What is a critical t-value?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A critical t-value is the threshold your t-statistic must exceed to reject the null hypothesis at a given significance level. It depends on your degrees of freedom and whether you're running a one-tailed or two-tailed test. For example, with 20 degrees of freedom and α = 0.05 (two-tailed), the critical value is approximately 2.086.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
38:["$","div","7",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What's the difference between a one-tailed and two-tailed t-test?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"A two-tailed test checks whether your sample mean differs from the population mean in either direction (higher or lower). A one-tailed test checks for a difference in only one direction. Use two-tailed unless you have a strong theoretical reason to test only one direction — one-tailed tests are easier to achieve significance with but are less conservative.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
39:["$","div","8",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"Can I use this calculator for a paired t-test?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Yes, with a small adjustment. For paired data (before-and-after measurements on the same subjects), calculate the difference for each pair first, then enter the mean of those differences as your sample mean, zero as the population mean, the number of pairs as your sample size, and the standard deviation of the differences as your standard deviation.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
3a:["$","div","9",{"className":"pb-6 last:pb-0","children":[["$","h3",null,{"className":"text-lg font-medium text-mist-950 mb-3","children":"What assumptions does the t-test make?"}],["$","div",null,{"className":"prose max-w-none text-mist-600","children":["$","$L31",null,{"content":{"root":{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"The one-sample t-test assumes: (1) your data are continuous, (2) observations are independent of each other, (3) the data are approximately normally distributed (especially important for small samples), and (4) the sample is randomly drawn from the population. Violations of normality matter less with larger samples, but independence is always critical.","type":"text","version":1}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"root","version":1}}}]}]]}]
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