A reliable method
Recover totals before combining means. Sort values before finding a median. Apply the stated quartile convention explicitly when a small data set is used.
The common trap
Equal group sizes guarantee that the unweighted average of group means is the combined mean; otherwise use group-size weights. Adding a constant shifts the center without changing standard deviation.
Make the connection
Distinguish the population described by the data from a broader population you hope to discuss. Arithmetic accuracy does not repair an unrepresentative sample.
Contrast the cases
Two observations with mean 10 and eight with mean 20 have combined mean 18, not 15: group totals are 20 and 160. Equal group sizes would justify averaging the two means. Adding 5 to all observations raises mean and median by 5 while preserving deviations from the mean, so standard deviation stays fixed. Multiplying every value by 5 instead multiplies standard deviation by 5.
Transfer and explain
Transfer: groups of sizes 3 and 1 have means 8 and 20. The combined mean is 11. Then explain why shifting all four observations by −2 changes that mean to 9 but leaves their spread unchanged.
One group has 3 values with mean 10; another has 5 values with mean 20. What is the combined mean?
- Group totals are 30 and 100.
- Add totals and divide by the combined size 8.
- The combined mean is 65/4.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference