When temperatures are officially recorded for hourly weather observations and climate reports, they are measured at a height of between 1.25 and 2 metres (4.1 and 6.6 ft) above the ground in special shelters, called Stevenson screens. (The shelter is named after the father of writer and poet Robert Louis Stevenson.) Meteorologists call this temperature the "surface temperature," and it is what is reported on the radio and TV (and internet and newspapers, reports, etc.). The distinction is important for the following reason.
During clear and calm nights, the temperature at the ground or some surface near the ground can become much cooler than the "surface temperature". The radiation of heat away from the ground is the cause of this drop. The coldest air, therefore, forms near the ground, and being heavier than the air above it remains there.
If we were to make measurements of temperature from the surface to the height of the official "surface temperature" measurement every few centimetres or inches, we would find the air temperature increases as we move upward from the ground. Meteorologists call this a surface temperature inversion.
Since cold air is heavy air, in the absence of wind, the coldest air will remain nearest the ground, thus allowing surface temperatures to continue to fall. Thus, under such conditions -- clear and calm nights -- the ground temperature may fall below the freezing point while the temperature measured officially at was still above freezing. This is particularly common in the autumn and winter when nights are long allowing more time for cooling to occur.
Now frost is a covering of ice crystals on the surface produced by the depositing of water vapor to a surface cooler than 0° C (32° F). The deposition occurs when the temperature of the surface falls below the frost point. Similarly, dew forms when the air or surface temperature falls below the dew point temperature. (Note that the water vapor goes directly from gas to ice. Therefore, frost is not frozen dew.)
Thus, if the temperature on the ground or an object such as a bush or a car windshield near the ground falls below the frost point, frost crystals may form. But the measured "surface temperature" may still be above freezing.
This is the most common way in which frost may form when the official surface temperature is still above the freezing point.