A reliable method
For combined groups, add group totals and divide by the combined count. For an even-sized list, average the two central observations.
The common trap
Do not average two group means unless their sizes match. Distinguish a population variance from its square root, the standard deviation.
Make the connection
If a quartile convention matters, use the one stated in the question. Small data sets can have different quartiles under different conventions.
Worked example: choose the requested quantity
A class of 3 students has mean score 60; another class of 6 students has mean 90. What is the combined mean? Convert each mean into a total by multiplying by its group size. Add the totals and divide by the total number of students. The requested value is 80.
Contrast and diagnose an error
The dot plot shows all six observations. What is their median? Dot plot with one observation at each of 3, 4, 7, 9, 12, 14. The six observations are already in increasing order. Average the third and fourth observations. The requested value is 8. A tempting answer is 17/2: Use the midrange from extreme values.
Transfer practice and check
A population consists of the two values 3 and 7. What is its population variance? Try it before reading on. Check: 4. The mean is halfway between the two values. Each deviation is 2 in magnitude; average their squared deviations.
A class of 3 students has mean score 60; another class of 6 students has mean 90. What is the combined mean?
- Convert each mean into a total by multiplying by its group size.
- Add the totals and divide by the total number of students.
- The requested value is 80.
Compare the other answer choices
Divide the sum of means by the number of size parts.
Assign the larger weight to the wrong group.
Average group means without size weights.
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.
ACT Mathematics content