File:Scaled chi squared.svg
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Summary
A plot of the scaled chi-squared probability distribution, the probability distribution of the <a href="https://en.wikipedia.org/wiki/sample_variance" class="extiw" title="en:sample variance">sample variance</a> s2,
- <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56ef5cdeadd1c7680021d383d32abb9143f08eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:5.676ex;" alt="{\displaystyle s^{2}={\frac {\;\sigma ^{2}}{\nu }}\;\chi _{\nu }^{2}}">
where s2 is the mean of the squares of ν independent normal distributions of mean 0 and standard deviation σ; plotted here for the case σ = 1.0.
Equivalently this is also a plot of the distribution of the variance ratio s2/σ2 for any value of σ. As ν increases s2/σ2 becomes more sharply peaked close to 1
Licensing
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File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 11:53, 6 January 2017 | | 270 × 270 (54 KB) | 127.0.0.1 (talk) | A plot of the scaled chi-squared probability distribution, the probability distribution of the <a href="https://en.wikipedia.org/wiki/sample_variance" class="extiw" title="en:sample variance">sample variance</a> <i>s</i><sup>2</sup>, <dl><dd><span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msup><mi>s</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mspace width="thickmathspace"></mspace><msup><mi>σ<!-- σ --></mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></mrow><mi>ν<!-- ν --></mi></mfrac></mrow><mspace width="thickmathspace"></mspace><msubsup><mi>χ<!-- χ --></mi><mrow class="MJX-TeXAtom-ORD"><mi>ν<!-- ν --></mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle s^{2}={\frac {\;\sigma ^{2}}{\nu }}\;\chi _{\nu }^{2}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56ef5cdeadd1c7680021d383d32abb9143f08eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:12.377ex; height:5.676ex;" alt="{\displaystyle s^{2}={\frac {\;\sigma ^{2}}{\nu }}\;\chi _{\nu }^{2}}"></span></dd></dl> <p>where <i>s</i><sup>2</sup> is the mean of the squares of ν independent normal distributions of mean 0 and standard deviation σ; plotted here for the case σ = 1.0. </p> Equivalently this is also a plot of the distribution of the variance ratio <i>s</i><sup>2</sup>/σ<sup>2</sup> for any value of σ. As ν increases <i>s</i><sup>2</sup>/σ<sup>2</sup> becomes more sharply peaked close to 1 |
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