One changing sample moves both mean and range in a finite ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The first row is a balanced four-sample ledger

With top sample eight, the average is five and the range is six.

Vˉ=5,R=6\bar V=5,\quad R=6
Eight-volt top sampleAverage and range are both sourced from the sample list.rawSamples=2,4,6,8 Vbaseline=0 Vsamples=2,4,6,8 Vcount=4sum=20 Vaverage=5 Vmin=2 Vmax=8 V

Raising one sample moves both average and range

Changing only the top sample to twelve raises the average to six.

Vˉ=6,R=10\bar V=6,\quad R=10
Twelve-volt top sampleThe count stays fixed while the spike changes the totals.rawSamples=2,4,6,12 Vbaseline=0 Vsamples=2,4,6,12 Vcount=4sum=24 Vaverage=6 Vmin=2 Vmax=12 V

A larger spike keeps the count but widens the ledger

The third row shows why one outlier changes more than a caption.

spikeVˉrange8561261016714\begin{array}{c|c|c}\text{spike}&\bar V&\text{range}\\8&5&6\\12&6&10\\16&7&14\\\end{array}
Sixteen-volt top sampleThe range and average labels are recomputed together.rawSamples=2,4,6,16 Vbaseline=0 Vsamples=2,4,6,16 Vcount=4sum=28 Vaverage=7 Vmin=2 Vmax=16 V