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Given an elliptic curve E over a function field K = ℚ(T₁,...,Tₙ), we study the behavior of the canonical height $h{̂}_{E_{\omega }}$ of the specialized elliptic curve $E_{\omega }$ with respect to the height of ω ∈ ℚⁿ. We prove that there exists a uniform nonzero lower bound for the average of the quotient $\left(h{̂}_{E_{\omega }}\left(P_{\omega }\right)\right)/h\left(\omega \right)$ over all nontorsion P ∈ E(K).