Why digital multimeter more accurate than analog multimeter?

An analogue instrument is analogous to the unit being measured. For instance, a thermometer uses a fluid, such as Mercury, which reacts to heat, expanding and contracting, and thus providing a direct reading upon a calibrated scale. Similarly, a stadiometer measures a person's standing height by sliding an arm down a scale until it touches the persons head. In both cases, a reading can be taken directly off the scale, thus the scale is analogous to the quantity being measured.

Digital instruments, on the other hand, must encode these analogue measurements into digital information. However, analogue information is infinite. That is, between any two fixed points upon a scale there are an infinite number of variables, just as the number of real values between 0 and 1 is infinite. Digital information cannot encompass the full range of infinite values, therefore digital information must employ a degree of precision; values must be rounded. However, humans do this anyway. If we're measuring distance in millimetres then we will round to the nearest millimetre.

The amplitude of a sonic waveform at any given moment in time also has infinite range, from -1 to +1. When we digitise analogue audio signals we use approximation by scaling the analogue value to an integer in the range -32,768 to +32,767 (assuming 16-bit precision). This inevitably means some data will be lost. However, one hopes, the loss is so minimal that it will not be missed. If we need to capture the signal more accurately, we simply increase the number of bits to give a wider range of scaled values. So 24-bit would allow a range of -8,388,607 to +8,388,607.