Question 2·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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A data set contains 13 measurements. The mean of the measurements is $52$ , and the median is $50$ . Exactly one measurement in the data set is equal to $50$ .

An adjustment is made to the original 13 measurements: each measurement less than $50$ is increased by $4$ , and each measurement greater than $50$ is decreased by $3$ . The measurement equal to $50$ is not changed. Then one additional measurement of $50$ is added to the data set.

What is the mean of the 14 measurements after these changes?

Your answer

A median is the middle value after the data are sorted. If an odd-sized set has only one measurement equal to its median, the other measurements split evenly below and above it. For a changed mean, track the total and the number of measurements separately. Count how many receive each adjustment, then use Desmos for the arithmetic. Neither adjustment applies to every measurement.

Hints

Hint 1
With 13 measurements, the median is the 7th value in order. Because exactly one value equals 50, how many measurements must be below 50, and how many must be above it?
Hint 2
Each measurement below 50 adds 4 to the total; each one above 50 subtracts 3. Use the counts on each side of the median to find the net change.
Hint 3
The mean is the total divided by the count. Start with the original mean times 13, apply the adjustments, then account for the additional 50 in both the total and the count.

Step-by-step

Track the total through each change
Step 1
Count the measurements on each side
In sorted order, the median of 13 measurements is the 7th value. That value is 50, and exactly one measurement equals 50, so the first 6 are below 50 and the last 6 are above it.
Step 2
Recover the original total
The mean is the total divided by the count, so multiply the given mean by 13. Type $13\times52$ in Desmos. It shows $676$ , the original total: $13(52)=676$ .
Step 3
Find the net effect of the adjustments
The 6 lower measurements each gain 4, while the 6 higher ones each lose 3. Type $6(4)-6(3)$ in Desmos. It shows $6$ , so the total rises by 6: $6(4)-6(3)=6$ . Don't add 4 to all 13 measurements.
Step 4
Add the new measurement
Type $676+6+50$ in Desmos. It shows $732$ , the new total: $676+6+50=732$ . Add only the additional 50; the original 50 was already in 676. Adding a measurement also brings the count to 14.
Step 5
Divide by the new count
Divide the new total by the new count. Type $732/14$ in Desmos; it shows $52.285714\ldots$ . Tap the fraction button to get the exact mean, $\frac{366}{7}$ . The mean of the 14 changed measurements is $\frac{366}{7}$ . Grid in 366/7.

Question 2·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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