Elementary Statstics

5.2.5.3. Smoothed Frequency Curve

A smoothed frequency curve can be drawn through the various points of the polygon. The curve is drawn freehand in such a manner that the area included under the curve is approximately the same as that of the polygon. The object of drawing a smoothed frequency curve is to eliminate as far as possible accidental variations that might be present in the data. While smoothing a frequency polygon the fact that it is really derived from the histogram should always be kept in mind. This would imply that the top of the curve would overtop the highest point of the polygon particularly when the magnitude of class-interval is large. The curve should look as regular as possible and sudden turns should be avoided. The extent of smoothing would, however, depend upon the nature of the data. If it is a natural phenomenon like tossing of coin, smoothing may be freely resorted to as such phenomenon normally has symmetrical curves, but if the phenomenon is social or economic the curve is generally skewed and as such smoothing cannot be carried too far.

For drawing a smoothed frequency curve it is necessary to first draw the polygon and then smooth it out. As discussed earlier, the polygon can be constructed even without first constructing a histogram by plotting the frequencies at the mid-points of class-intervals. This may save some time but the smoothing of the polygon cannot be done properly without a histogram. Hence, it is desirable to proceed in a sequence, i.e. first  draw a histogram than a polygon and lastly  smooth it to obtain the smoothed frequency curve. This curve should begin and end at the base line and as a general rule it may be extended to the mid-points of the class-intervals just outside the histogram. The area under the curve should represent the total number of frequencies in the entire distribution.

The following points should be kept in mind while smoothing a frequency curve :

1. Only frequency distribution based on samples should be smoothed.
2. Only continuous series should be smoothed.
3. The total area under the curve should be equal to the area under the original histogram or polygon.