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Equation 2 · From Origins to Frontier: A History of Evolutionary Biology and Ecology

What does this equation mean?

Var⁡(Δp)=p(1−p)2Ne,\operatorname{Var}(\Delta p) = \frac{p(1-p)}{2N_e},

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Start withp(1-p)
Divide by2N_e
This relates toVar(Δ p)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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Δp\Delta p

Symbol Δ p

Δ p is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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pp

Symbol p

the frequency.

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NeN_e

Symbol N_e

NeN_e occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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fraction

fraction

Divide the expression above the line by the one below it.

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change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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p(1−p)p(1-p)

Numerator: p(1-p)

The complete quantity above the fraction bar.

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2Ne2N_e

Denominator: 2N_e

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Sewall Wright’s 1931 paper, “Evolution in Mendelian Populations,” added the piece Fisher’s and Haldane’s more deterministic treatments lacked: a formal account of genetic drift, the random change in allele frequency from generation to generation caused by finite population size [ 3 ] . Wright showed that in a randomly mating population of effective size NeN_e , the variance in allele frequency change per generation from sampling alone is Var⁡(Δp)=p(1−p)2Ne\operatorname{Var}(\Delta p) = \frac{p(1-p)}{2N_e}. for a diploid population starting at allele frequency p . This single expression carries a real consequence: small populations can fix or lose alleles by chance alone, independent of whether those alleles are beneficial, neutral, or mildly…
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Sewall Wright’s 1931 paper, “Evolution in Mendelian Populations,” added the piece Fisher’s and Haldane’s more deterministic treatments lacked: a formal account of genetic drift, the random change in allele frequency from generation to generation caused by finite population size [ 3 ] . Wright showed that in a randomly mating population of effective size NeN_e , the variance in allele frequency change per generation from sampling alone is Var⁡(Δp)=p(1−p)2Ne\operatorname{Var}(\Delta p) = \frac{p(1-p)}{2N_e}. for a diploid population starting at allele frequency p . This single expression carries a real consequence: small populations can fix or lose alleles by chance alone, independent of whether those alleles are beneficial, neutral, or mildly deleterious, and the smaller the effective population, the faster that random fixation proceeds. Wright’s “shifting balance” framework, in which small subpopulations explore a rugged fitness landscape partly by drift and partly by selection, remains debated in its strong form, but the underlying mathematics of drift is uncontroversial and became a permanent part of population genetics.

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