{"id":44710,"date":"2025-06-03T11:42:57","date_gmt":"2025-06-03T03:42:57","guid":{"rendered":"https:\/\/www.wukongsch.com\/blog\/?p=44710"},"modified":"2026-05-21T14:18:57","modified_gmt":"2026-05-21T06:18:57","slug":"variance-formula-in-ma","status":"publish","type":"post","link":"https:\/\/www.wukongsch.com\/blog\/variance-formula-in-ma-post-44710\/","title":{"rendered":"Variance Formula: Population, Sample, Calculator and Examples"},"content":{"rendered":"
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<\/span>Quick Answer: What Is the Variance Formula?<\/span><\/h2>\n\n\n\n

Variance measures how far a set of data values spreads out from the mean. A small variance means the values are close to the mean, while a large variance means the values are more spread out.<\/p>\n\n\n\n

There are two main variance formulas:<\/p>\n\n\n\n

Type of variance<\/th>Formula<\/th>Use when<\/th><\/tr><\/thead>
Population variance<\/td>\u03c3\u00b2 = \u03a3(x\u1d62 – \u03bc)\u00b2 \/ N<\/td>You have data for every member of the group<\/td><\/tr>
Sample variance<\/td>s\u00b2 = \u03a3(x\u1d62 – x\u0304)\u00b2 \/ (n – 1)<\/td>You have only a sample and want to estimate the population variance<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

In these formulas:<\/p>\n

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  • x\u1d62 means each data value<\/li>\n\n\n\n
  • \u03bc means the population mean<\/li>\n\n\n\n
  • x\u0304 means the sample mean<\/li>\n\n\n\n
  • N means the total number of values in a population<\/li>\n\n\n\n
  • n means the total number of values in a sample<\/li>\n\n\n\n
  • \u03a3 means “add them all together”<\/li>\n<\/ul>\n\n\n\n

    To calculate variance, find the mean, subtract the mean from each value, square each difference, add the squared differences, and divide by N for population variance or n – 1 for sample variance.<\/p>\n\n\n\n

    <\/span>Variance Formula: Population vs Sample<\/span><\/h2>\n\n\n\n

    The formula you use depends on whether your data represents an entire population or only a sample.<\/p>\n\n\n\n

    If you collect the test scores of every student in one class, you can use the population variance formula because you have the full group. If you collect scores from only 20 students to estimate the performance of a much larger grade level, you should use the sample variance formula.<\/p>\n\n\n\n

    Feature<\/th>Population variance<\/th>Sample variance<\/th><\/tr><\/thead>
    Symbol<\/td>\u03c3\u00b2<\/td>s\u00b2<\/td><\/tr>
    Mean used<\/td>\u03bc<\/td>x\u0304<\/td><\/tr>
    Denominator<\/td>N<\/td>n – 1<\/td><\/tr>
    Best for<\/td>Complete data set<\/td>Partial data set<\/td><\/tr>
    Common example<\/td>Heights of all students in one class<\/td>Heights of 30 students chosen from a school<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    The most important difference is the denominator. Population variance divides by N, while sample variance divides by n – 1. This adjustment helps a sample give a better estimate of the population’s true variance.<\/p>\n\n\n

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    \"Variance<\/figure><\/div>\n\n\n
    Population Variance Formula<\/strong><\/th>Sample Variance Formula<\/strong><\/th><\/tr>
    \u03c3\u00b2 = \u03a3(x_i – \u03bc)\u00b2 \/ N<\/td>s\u00b2 = \u03a3(x_i – x\u0304)\u00b2 \/ (n – 1)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

    <\/span>Online<\/strong> Variance Calculator<\/strong><\/span><\/h2>\n\n\n\n
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    Step-by-Step Variance Lab<\/h3>\n

    Learn how to calculate Variance & Standard Deviation<\/p>\n <\/div>\n\n

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