Super-Hard SAT Data Analysis Question 2 | Free SAT Prep

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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

For a mean question involving changes to a data set, work with the total rather than trying to list all the values. First use the median and the number of data values to determine how many values belong to each group. Then track each group’s total change, include any added or removed values, and divide by the new number of data values.

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Use the median's position

In an ordered data set of 13 values, the median is the seventh value. Use the fact that only one value equals $50$ to count the values below and above $50$ .

Track the total, not each individual value

Convert the original mean into a total. Then calculate how much the two groups of adjustments change that total.

Account for the new number of measurements

After finding the adjusted total, add the new measurement of $50$ and divide by 14.

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Use arithmetic to verify

Algebra is faster because the key reasoning is determining that 6 measurements are below $50$ and 6 are above $50$ . To verify the arithmetic, enter the following expression:

((13)(52)+6(4)-6(3)+50)/14

Check the exact fraction

Enter 366/7 on a new expression line. It has the same value as the first expression, confirming the mean.

Step-by-step Explanation

Determine how many measurements are adjusted each way

With 13 measurements, the median is the seventh measurement when the data are ordered. Since exactly one measurement equals $50$ , there are 6 measurements less than $50$ and 6 measurements greater than $50$ .

Find the change in the total

The original total is $13(52)=676$ .

Increasing each of 6 measurements by $4$ increases the total by $6(4)=24$ . Decreasing each of 6 measurements by $3$ decreases the total by $6(3)=18$ .

Thus, the adjusted total is

676+24-18=682.

Include the added measurement

Adding one measurement of $50$ gives a total of $682+50=732$ across 14 measurements. Therefore, the new mean is

\frac{732}{14}=\frac{366}{7}.

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