What is Bessel's correction (÷n-1)?

Bessel's correction divides by (n−1) instead of n when calculating sample variance. This makes the sample variance an unbiased estimator of the population variance. Without the correction, using ÷n systematically underestimates the true population variance. Named after Friedrich Bessel. Used in: Excel STDEV(), Python numpy.std(ddof=1), R sd(), MATLAB std(). Default formula in AP Statistics (US) and A-Level (UK). Japan and China often teach ÷n (population) as the primary formula.

When should I divide by n vs n-1?

Divide by n (population variance): when you have data on the ENTIRE population — e.g., all employees in a company, all products produced in one day. Divide by n−1 (sample variance): when your data is a SAMPLE from a larger population — e.g., a survey of 100 people from a city of 1 million, test scores from one class used to estimate all students. In practice, sample variance (÷n−1) is nearly always the correct choice because you almost never have data on the complete population.

Variance Calculator — Population & Sample Variance and Standard Deviation

Variance Calculator

Calculate variance and standard deviation for both population (÷n) and sample (÷n−1) from any data set. Shows the deviation table step-by-step.

Quick Answer

Sample variance s² = Σ(x−x̄)² / (n−1) — use for data samples (nearly always). Population σ² = Σ(x−μ)² / n — only if you have ALL the data. Excel STDEV() = sample. STDEVP() = population.

Enter numbers (comma, space, or new line separated)

Country / Tool	Default Formula	Calculator Key	Function
🇺🇸 US (AP Stats, Excel)	Sample (÷n−1)	Sx on TI-84	STDEV() in Excel
🇬🇧 UK (A-Level, Casio)	Sample (÷n−1)	σn−1 on Casio	STDEV() in Excel
🇩🇪 Germany (Gymnasium)	Sample (÷n−1)	s on calc	numpy std(ddof=1)
🇯🇵 Japan (高校数学B)	Population (÷n)	σ on Casio FX	numpy std(ddof=0)
🇨🇳 China (高中数学)	Population (÷n)	σn on calc	STDEVP() in Excel
🇮🇳 India (CBSE/ISC)	Population (÷n)	σ on CASIO fx-82	STDEVP() in Excel

Frequently Asked Questions

Why do Japan and China teach population variance as the default?

In Japan and China's high school curricula (数学B / 高中数学), descriptive statistics focuses on characterising a given data set rather than making inferences about a broader population. Since the data set in hand IS the "population" being described, dividing by n is appropriate. When students reach university-level statistics, they learn sample variance (÷n−1). US and UK curricula prioritise inferential statistics from early on, so ÷n−1 is introduced first.

How does Python calculate standard deviation?

Python's numpy uses population formula by default: numpy.std(data) divides by n. For sample standard deviation, use numpy.std(data, ddof=1). Python's statistics module uses sample formula by default: statistics.stdev(data) divides by n−1. pandas DataFrame.std() also uses n−1 by default. R's sd() function uses n−1. Always check the ddof (delta degrees of freedom) parameter.

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