1. **State the problem:** Write the equation in standard form for the circle with diameter endpoints at $\left(\frac{19}{2}, \frac{15}{2}\right)$ and $\left(\frac{11}{2}, \frac{21}{2}\right)$.\n\n2. **Find the center of the circle:** The center is the midpoint of the diameter. Use the midpoint formula: $$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$\nSubstitute the points: $$\left(\frac{\frac{19}{2} + \frac{11}{2}}{2}, \frac{\frac{15}{2} + \frac{21}{2}}{2}\right) = \left(\frac{\frac{30}{2}}{2}, \frac{\frac{36}{2}}{2}\right) = \left(\frac{15}{2}, \frac{18}{2}\right) = \left(7.5, 9\right)$$\n\n3. **Calculate the radius:** The radius is half the length of the diameter. First, find the distance between the endpoints using the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$\nSubstitute the points: $$d = \sqrt{\left(\frac{11}{2} - \frac{19}{2}\right)^2 + \left(\frac{21}{2} - \frac{15}{2}\right)^2} = \sqrt{\left(-4\right)^2 + \left(3\right)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$\nRadius $r = \frac{d}{2} = \frac{5}{2} = 2.5$.\n\n4. **Write the equation in standard form:** The standard form of a circle's equation is $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the center and $r$ is the radius.\nSubstitute $h=7.5$, $k=9$, and $r=2.5$: $$ (x - 7.5)^2 + (y - 9)^2 = (2.5)^2 = 6.25 $$\n\n**Final answer:** $$ (x - 7.5)^2 + (y - 9)^2 = 6.25 $$