SHSAT: Mean, median and range: think in totals
Find missing or added values from a mean, combine groups correctly, find medians, and predict how changes to the data affect each measure.
How do you answer mean questions when one value is missing or a new value is added?
Short answer
Multiply the mean by the count to get the total. Compare totals to find a missing or added value, and sort the data before finding a median.
Know first: Adding and dividing whole numbers, Ordering numbers
Step 1 of 5
A mean is a total in disguise
The mean is the total divided by the count. Turn it around and you get the most useful fact in this topic: total = mean × count. Missing values, added values and combined groups all come down to totals.
The median is the middle value after sorting; with an even count, it's the average of the two middle values. The range is greatest minus least.
| Measure | How to find it | If you add 5 to every value | If you double every value |
|---|---|---|---|
| Mean | total ÷ count | goes up 5 | doubles |
| Median | middle of the sorted list | goes up 5 | doubles |
| Range | greatest − least | stays the same | doubles |
Adding the same number to every value shifts the data but doesn't spread it out.
Step 2 of 5
Work with totals
Worked example: Find a missing value
The mean of six numbers is 14. Five of them are 6, 10, 11, 19 and 20. What is the sixth number?
- A14
- B18
- C66
- D84
- 1
Find the total the mean requires
14 × 6 = 84.
- 2
Add the known values
6 + 10 + 11 + 19 + 20 = 66.
- 3
Subtract
84 − 66 = 18. Check: (66 + 18) ÷ 6 = 14.
- 4
See why the other choices are there
14 assumes the missing number equals the mean. 66 is the sum of the known values. 84 is the total of all six.
Answer
18
Wrong: 10 students average 80 and 30 students average 88. The combined mean is (80 + 88) ÷ 2 = 84.
Right: Combined total = 10 × 80 + 30 × 88 = 800 + 2640 = 3440. Mean = 3440 ÷ 40 = 86.
The larger group pulls the mean toward its own average. Only equal-sized groups can be averaged directly.
Step 3 of 5
Sort, total, then decide
Check yourself · Question 1
What is the median of 4, 9, 2, 7, 12, 9, 1, 6?
Answer: A
Sort first: 1, 2, 4, 6, 7, 9, 9, 12. With eight values, the median is the average of the 4th and 5th: (6 + 7) ÷ 2 = 6.5.
Check yourself · Question 2
Five values have a mean of 8. A sixth value is added, and the mean of all six is 9. What value was added?
Answer: C
The first total is 8 × 5 = 40. The new total is 9 × 6 = 54. The added value is 54 − 40 = 14.
Trap answer: 9
Treating the mean as a data value
- Why it looks right
- The new mean is 9, so it feels like the new value should be 9 too.
- Why it's wrong
- Adding a 9 would pull a mean of 8 up only slightly, to 49 ÷ 6. To raise the mean by a full 1, the new value has to be well above 9.
- The right answer
- 14, the difference between the totals 54 and 40.
Check yourself · Question 3
A data set has a range of 11. Every value is doubled, then decreased by 5. What is the new range?
Answer: D
Doubling every value doubles the gap between the greatest and least values: 22. Subtracting 5 from both ends doesn't change their difference.
Common mistake
Averaging the two group means to get the overall mean.
- Why it's tempting
- It works in class when the groups happen to be the same size.
- Do this instead
- Turn each mean into a total by multiplying by its count, add the totals, then divide by the combined count.
Step 4 of 5
What to remember
Remember
- Total = mean × count. Missing, added and corrected values all come from comparing totals.
- Sort before finding a median; with an even count, average the two middle values.
- Adding a constant shifts the mean and median but not the range; multiplying scales all three.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what total did the mean require, and how did you use it to find the missing number?