In this problem, you are asked to find the percentage error in the estimation of the focal length of a convex lens using a 2-meter long scale with a least count of 0.2 cm.

First, let's determine the object distance (u), image distance (v), and focal length (f) of the lens.

1. Object distance (u): It's the distance between the object pin and the convex lens. The object pin is at the 80 cm mark, and the convex lens is at the 1 m (100 cm) mark, so the object distance is $$u = 100 - 80 = 20~cm$$.

2. Image distance (v): It's the distance between the image pin and the convex lens. The image pin is at the 180 cm mark, and the convex lens is at the 1 m (100 cm) mark, so the image distance is $$v = 180 - 100 = 80~cm$$.

3. Focal length (f): Using the lens formula, we can calculate the focal length:
   
   $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{80} + \frac{1}{20} = \frac{5}{80}$$ So, $$f = \frac{80}{5} = 16~cm$$.

Now, we will calculate the error in the focal length (df) using the given least count (0.2 cm). The error in the object distance and image distance will both be 0.2 cm.

1. Error in the focal length (df): We can use the formula for the error in the focal length:
   
   $$\frac{df}{f^2} = \frac{0.2 \times 2}{6400} + \frac{0.2 \times 2}{400}$$

Solving for df:

$$df = \frac{16 \times 16 \times 0.2 \times 6800 \times 2}{6400 \times 400} = 0.136 \times 2$$

1. Percentage error in the focal length: Finally, we will calculate the percentage error using the formula:
   
   $$\frac{df}{f} = \frac{0.0085 \times 2}{1} = 1.70$$

So, the percentage error in the estimation of the focal length is 1.70%.