The Probable Error of a Mean
When dealing with small sample sizes, using the sample standard deviation as an estimate for the population standard deviation in a normal distribution calculation results under-estimates the uncertainty, leading to false statistical confidence.
The author derived the exact probability distribution of the ratio of the sample mean deviation to the sample standard deviation, introducing what is now known as Student's t-distribution.
Proved that for small samples, the distribution of the mean has heavier tails than the normal distribution, and provided tables of critical values for statistical significance.
The derivation assumed that the parent population from which samples are drawn is normally distributed.
Used universally in experimental sciences, A/B testing, and quality control to assess statistical significance when the sample size is small ($n < 30$) and the population variance is unknown.
📇 Summary flashcard — 13 analytical fields for this paper
خلاصه
Introduced Student's t-distribution to calculate sample mean confidence when the sample size is small and the population variance is unknown.
نمای سریع
The mathematical formulation of the t-distribution for small sample sizes.
یافتههای کلیدی
Proved that for small samples, the distribution of the mean has heavier tails than the normal distribution, and provided tables of critical values for statistical significance.
هدف
To develop a reliable method to test hypotheses on small data samples without over-estimating confidence.
روش
The author derived the exact probability distribution of the ratio of the sample mean deviation to the sample standard deviation, introducing what is now known as Student's t-distribution.
نتایج
Determined the exact curve of t-distributions for varying degrees of freedom, showing they converge to the Normal distribution as sample size increases.
نتیجهگیری
Analyses on small experimental groups must adjust for the added variance of calculating the standard deviation from the sample itself.
مفاهیم کلیدی
t-distribution، statistics، hypothesis-testing، sample-size
مطالعهی بیشتر
https://doi.org/10.1093/biomet/6.1.1
تحلیل
This paper bridged the gap between theoretical large-sample statistics and practical engineering and biology experiments, where large datasets are often hard to acquire.
محدودیتها
The derivation assumed that the parent population from which samples are drawn is normally distributed.
کارهای آینده
Ronald Fisher later generalized the t-distribution to regression analyses and Analysis of Variance (ANOVA).
کاربرد عملی
Used universally in experimental sciences, A/B testing, and quality control to assess statistical significance when the sample size is small ($n < 30$) and the population variance is unknown.
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