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Manufacturer application note. The oversampling and Averaging section discusses uncorrelated noise and dither; Table 1 gives fourfold sample-count increases per extra output bit. No individual author is established by the consulted HTML page.

dither is an intentionally introduced fluctuation that makes a quantizer's output vary between codes, reducing signal-correlated quantization artifacts. A suitable distribution can make the averaged code carry information about a constant voltage smaller than one conversion step. The improvement requires filtering or averaging; extra noise does not improve an individual conversion or repair arbitrary calibration errors.

A small constant input can repeatedly produce the same analogue-to-digital converter code, so averaging those codes cannot reveal its position within a conversion interval. dither makes adjacent codes toggle; their relative frequencies then encode changes smaller than one conversion step. With adequate oversampling, low-pass filtering and averaging reduce the added, sufficiently uncorrelated noise while retaining the slow signal.

For independent sample errors, the averaged standard deviation scales as $N^{-1/2}$; ideally four times as many samples gain about one effective bit. For example a constant input $0.2\Delta$ rounded to levels separated by $\Delta$ would always return zero. Adding uniform dither on $[-\Delta /2,\Delta /2]$ gives output $\Delta$ with probability $0.2$ and zero otherwise: the expected average is $0.2\Delta$, with error $0.4\Delta /\sqrt{N}$. The benefit concerns the filtered estimate, not a more accurate individual conversion. Excessive dither wastes range, and averaging does not correct calibration bias, correlated drift or arbitrary converter nonlinearity.^[ref]

Let a constant input be $x=(k+f)\Delta$ with $0\le f\le 1$, between consecutive quantization levels $k\Delta$ and $(k+1)\Delta$. Add independent centered uniform dither $U\sim \mathrm{Uniform}[-\Delta /2,\Delta /2]$ before nearest-level rounding, and assume that neither code clips.

The upper code is selected exactly when $U\ge (1/2-f)\Delta$. This subinterval occupies fraction $f$ of the uniform distribution's support, so $Q=k\Delta +\Delta B$ with $B$ a Bernoulli random variable of parameter $f$. Thus $\mathbb{E}[Q]=x$ and $\mathrm{Var}(Q)=f(1-f){\Delta }^2$. Averaging $N$ independent conversions givesFor example, an input one quarter of a step above the lower code produces the upper code one quarter of the time. Its averaged code can estimate a sub-step voltage, with a finite remaining variance. Centering, independence and absence of clipping are assumptions of this calculation; a shifted dither distribution can bias the result.