The decision and its conditions
A summary can constrain missing data without revealing every observation. Multiply a mean by the number of observations to reconstruct the required total. A frequency table stores counts, so the modal value comes from the largest count rather than the largest category label. When a new observation is inserted, locate its sorted position before identifying the median. A combined mean may also reveal a group size: write one total-weight equation and solve it, then verify that the count is positive and integral.
Worked example
Four inspections average 9 minutes. Three took 5, 8, and 11 minutes. How long did the fourth take? The required total is 4×9=36 minutes. The known times total 24 minutes. The missing time is 36−24=12 minutes.
A contrasting decision
Given figure data: Days late and frequencies. Columns Days late, Deliveries. Rows: (2, 3), (4, 7), (7, 4), (9, 2). The frequency table describes late deliveries in month 1. What is the modal number of days? The mode is a data value associated with the greatest frequency. The largest frequency is 7, attached to 4 days. The modal value is 4 days, not seven deliveries.
Diagnose a competing method
The first example’s tempting answer 9 follows this mistake: Use the desired mean as though every observation must equal it. A mean and a count determine the total, so recover that total first.
Transfer before revealing
A sorted dataset is 4, 6, 9, 11. A new observation 1 is added. What is the median of the five observations? Pause and identify the requested quantity. Solution: The new observation belongs before 4. The reordered list is 1, 4, 6, 9, 11. With five observations, the third value is the median: 6.
Check the boundary of the method
When a new observation is inserted, locate its sorted position before identifying the median. Read the value beside the largest frequency; do not report the frequency itself. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
Four inspections average 9 minutes. Three took 5, 8, and 11 minutes. How long did the fourth take?
- The required total is 4×9=36 minutes.
- The known times total 24 minutes.
- The missing time is 36−24=12 minutes.
Compare the other answer choices
Use three observations when reconstructing the four-observation total.
Use the desired mean as though every observation must equal it.
Report the existing total rather than the missing observation.
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.
GED Mathematical Reasoning Assessment Guide for Educators