One-Sample T-Test - Chapter 205 - Procedure overview and assumptions

Sep 30, 2026

NCSS One-Sample T-Test procedure focuses on inference about a population mean or median using a single sample. It provides confidence intervals (mean or median), the t-test, z-test, and nonparametric alternatives such as the randomization test, quantile (sign) test, and Wilcoxon Signed-Rank test. The documentation explains the research questions, outlines key assumption checks for both parametric and nonparametric approaches, and includes an example workflow using a Weight dataset to test whether the mean differs from 130.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages29

Reading Time1 h 13 min

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Introduction

Research Questions

Technical Details

Null and Alternative Hypotheses

Assumptions

One-Sample T-Test Assumptions

Wilcoxon Signed-Rank Test Assumptions

Quantile (Sign) Test Assumptions

Data Structure

Example 1 – Running a One-Sample Analysis

NCSS Statistical Software [NCSS.com](NCSS.com) Chapter 205 One-Sample T-Test | Introduction
This procedure provides several reports for making inference about a population mean based on a single sample. These reports include confidence intervals of the mean or median, the t-test, the z-test, and nonparametric tests including the randomization test, the quantile (sign) test, and the Wilcoxon Signed-Rank test. Tests of assumptions and distribution plots are also available in this procedure.
Another procedure that produces a large amount of summary information about a single sample is the Descriptive Statistics procedure. While it is not as focused on hypothesis testing, it contains many additional descriptive statistics, including minimum, maximum, range, counts, trimmed means, sums, mode, variance, Skewness, Kurtosis, coefficient of variation, coefficient of dispersion, percentiles, additional normality tests, and a stem-and-leaf plot. | | --- | | Research Questions
For the one-sample situation, the typical concern in research is examining a measure of central tendency (location) for the population of interest. The best-known measures of location are the mean and median. For a one-sample situation, we might want to know if the average waiting time in a doctor’s office is greater than one hour, if the average refund on a 1040 tax return is different from $500, if the average assessment for similar residential properties is less than $120,000, or if the average growth of roses is 4 inches or more after two weeks of treatment with a certain fertilizer.
One early concern should be whether the data are normally distributed. If normality can safely be assumed, then the one-sample t-test is the best choice for assessing whether the measure of central tendency, the mean, is different from a hypothesized value. On the other hand, if normality is not valid, one of thenonparametric tests, such as the Wilcoxon Signed Rank test or the quantile test, can be applied. | Technical Details The technical details and formulas for the methods of this procedure are presented in line with the Example 1 output. The output and technical details are presented in the following order: • Descriptive Statistics • Confidence Interval for 􀟤 with 􀟪 Unknown • Confidence Interval for 􀟤 with 􀟪 Known • Confidence Interval for the Median • Bootstrap Confidence Intervals • Confidence Interval of 􀟪 • T-Test and associated power report 205-1 © NCSS, LLC. All Rights Reserved. NCSS Statistical Software [NCSS.com](NCSS.com) One-Sample T-Test • Z-Test • Randomization Test • Quantile (Sign) Test • Wilcoxon Signed-Rank Test • Tests of Assumptions • Graphs Null and Alternative Hypotheses The basic null hypothesis is that the population mean is equal to a hypothesized value, 􀜪0 : 􀟤 = 􀜪􀝕􀝌􀝋􀝐ℎ􀝁􀝏􀝅􀝖􀝁􀝀 􀜸􀜽􀝈􀝑􀝁 with three common alternative hypotheses, 􀜪􀯔 : 􀟤 ≠ 􀜪􀝕􀝌􀝋􀝐ℎ􀝁􀝏􀝅􀝖􀝁􀝀 􀜸􀜽􀝈􀝑􀝁 , 􀜪􀯔 : 􀟤 < 􀜪􀝕􀝌􀝋􀝐ℎ􀝁􀝏􀝅􀝖􀝁􀝀 􀜸􀜽􀝈􀝑􀝁 , or 􀜪􀯔 : 􀟤 > 􀜪􀝕􀝌􀝋􀝐ℎ􀝁􀝏􀝅􀝖􀝁􀝀 􀜸􀜽􀝈􀝑􀝁 , one of which is chosen according to the nature of the experiment or study. Assumptions This section describes the assumptions that are made when you use one of these tests. The key assumption relates to normality or nonnormality of the data. One of the reasons for the popularity of the t-test is its robustness in the face of assumption violation. However, if an assumption is not met even approximately, the significance levels and the power of the t-test are invalidated. Unfortunately, in practice it often happens that more than one assumption is not met. Hence, take the steps to check the assumptions before you make important decisions based on these tests. There are reports in this procedure that permit you to examine the assumptions, both visually and through assumptions tests. One-Sample T-Test Assumptions The assumptions of the one-sample t-test are: 1. The data are continuous (not discrete) . 2. The data follow the normal probability distribution. 3. The sam

FAQs

What does the One-Sample T-Test procedure aim to infer?

It makes inference about a population mean (or median-related quantities) using a single sample. The procedure delivers confidence intervals and hypothesis tests suited to the normality conditions of the data.

When should the one-sample t-test be used instead of nonparametric tests?

Use the one-sample t-test when normality can be safely assumed to assess whether the mean differs from a hypothesized value. If normality is not valid, nonparametric options such as the Wilcoxon Signed-Rank test or the quantile (sign) test can be applied.

What key assumptions are required for the one-sample t-test?

The one-sample t-test assumes continuous (not discrete) data, that the data follow a normal probability distribution, and that the sample is a simple random sample where each population member has an equal chance of selection.

Sep 30, 2026

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