Scale and Scale Factor

1:100,000 is a ratio, on a map it is referred to as map scale. The one represents a unit of measurement that represents 100,000 units of measurements in the real world. For example if I measure 1 inch on the map, I should be able to measure 100,000 inches on the earth for the same area.

The map scale is found by dividing the distance on the map by the distance in the real world, see this link. If I measure -100W by 40N to -100W by 35N on a map I get 2.77 cm. I then measure the real world distance using the NOAA’s ellipsoidal measurement tool to come up with 55,493,612.86 cm. By using these numbers in the equation I get the scale, 2.77cm ÷ 55,493,612.86cm = 1:20,000,000 (to get 1:20,000,000 exactly I would have had to measure the map as 2.774680643 cm). For this particular map 1:20,000,000 is the principle scale, ie the displayed scale.

However, map scale is not static, it is not the same everywhere on the map. It depends on the projection used and how the projection distorts the world. In an equidistant projection the scale is correct between certain points. The equidistant conic projection‘s scale is correct for the meridians and the standard parallels. So in the example above, the map was projected with the World Equidistant Projection and the principle scale was 1:20,000,00 as shown. If I move to a new location and measure -105W,30N to -100W,30N on a map I come up with 2.6 cm. I then measure the real distance on the earth using NOAA’s tool to and come up with 48,239,311.01 cm. Putting these numbers into the scale equation yields; 2.6 cm ÷ 48,239,311.01 cm = 1:18,555,581. This scale was taken from a location that was NOT along the standard parallels or along the meridians, it is referred to as as local scale.
To find the scale factor we divide local scale by principle scale. Using our examples above 1:18,555,581 ÷ 1:20,000,000 = 1.077959. Or in other words the local scale has been exaggerated by 107% This is all rather confusing and actually makes my head hurt, a really good explanation of all this is found in Portraits of the Earth: A Mathematician Looks at Maps by Timothy G Freeman ISBN 0821832557, chapter 3.