Analogue scales use mechanical components such as springs or levers to determine weight and display their measurements with a moving dial or with graduated beams and sliding weights (poises).
Digital scales use an electronic load sensor such as a load cell to determine weight and convert this to a digital signal to be displayed on an LCD or LED.
Notice that this is not the same as digital vs analog when it comes to electrical circuits. Scales which use electrical power to function, but are not digital (no microprocessors) are simply called Electronic Scales, not Analog Scales. The term Analog Scale in the industry typically refers to mechanical scales.
Digital and electronic scales are more common these days as their price to performance ratio has greatly improved. They are also much easier to use than older analog scales. Accuracy is not a matter of analog or digital and both can be accurate or inaccurate depending on several factors just like a digital and an analog wrist watch can both have the wrong time. Generally speaking though, digital scales offer much finer display resolution.
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