Question
The mean of 10 numbers is 25. If one number is replaced by , what is the new mean? If instead the same number is replaced by twice its value, find the new mean in terms of .
Solution — Step by Step
If mean of 10 numbers is 25, total sum is .
The sum increases by . New sum . New mean:
The sum increases by (since ). New sum . New mean:
If , mean stays at (replacing 0 with 0 is a non-change, but , so no change — correct). If , mean increases by 5 — replacing one number with one larger raises sum by , divided by values gives — correct.
Final answers: Case 1 mean ; Case 2 mean .
Why This Works
Mean is sum/count. When you replace one element, only the sum changes (count is constant). The change in sum equals (new value) − (old value). Divide that change by the count to get the change in mean.
This is faster than recomputing the sum from scratch. Useful for “what if” problems.
Alternative Method
Direct: original sum . Sum without is . Add new value back. Divide by . Same approach, written differently.
Students often subtract the old value once but forget to add the new one. Mean change formula: , not written in two parts. Combine into one step.