When a data point joins or leaves, don't try to "adjust" the old answer in your head — just build the new list and recompute each measure from scratch. It's safer, and it makes the differences between the measures jump out.
Worked Example — Adding a Value
Say the list is 3, 5, 6, 7, 9. The sum is 30 over 5 values, so the mean is 30 ÷ 5 = 6; the median is the middle one, 6; the range is 9 − 3 = 6. Now we add a 19. New list: 3, 5, 6, 7, 9, 19. The new sum is 49 over 6 values, so the mean jumps to about 8.2. The median? With six values it's the average of the middle two (6 and 7), so 6.5 — barely moved. And the range leaps to 19 − 3 = 16, because 19 is a brand-new biggest value. One number added; three very different reactions.
Worked Example — Removing a Value
Take that same 3, 5, 6, 7, 9 and remove the 9. New list 3, 5, 6, 7: sum 21 over 4 values, mean 5.25; median is the average of 5 and 6, so 5.5; range is 7 − 3 = 4. The range dropped because we deleted the biggest number. If you'd instead removed a middling value like the 6, the range wouldn't change at all.
The One Trap
Don't forget that adding or removing a value changes how many there are — so the median can flip between "single middle" and "average of two", and your divisor for the mean changes too. People lose marks by dividing the new sum by the old count. Re-count every time.