Pick a simple set of 3 or 4 numbers (like 10, 12, 14), find the mean, median, and minimum. Then subtract 5 from each number (to get 5, 7, 9, for example) and recompute those statistics. Which changed?

Some statistics tell you where the data sit on the number line (like an average or a smallest value). Others tell you about how spread out the data are from the center. Which type of statistic should stay the same if every value is shifted by the same amount?

Once you know which type of statistic is unchanged by adding or subtracting the same number to all data values, look at the four answer options and pick the one that belongs to that type.

In Desmos, type something like L1 = [10, 13, 17, 20, 25] to represent a small set of finish times (any reasonable list will work).

Type L2 = L1 - 5 to create a new list where 5 has been subtracted from every time, just like the officials do.

Enter these expressions:

• mean(L1), median(L1), min(L1), stdev(L1)
• mean(L2), median(L2), min(L2), stdev(L2)

Look at the numerical outputs and identify which statistic has the same value for both L1 and L2; that is the statistic that does not change when you subtract 5 from every data value, and it corresponds to the correct answer choice.

The clock starting 5 seconds late means every recorded time is 5 seconds greater than the cyclist’s actual time.

If a cyclist’s actual time is $T$, then the recorded time is $T+5$. When officials correct the times, they subtract 5 from each recorded time, so all corrected times are $T+5-5=T$.

Mathematically, subtracting 5 from every data value is called a shift: every number in the data set moves left by 5 on the number line.

Think about what happens if you subtract 5 from every number in a list.

• Mean (average): If each value in a set goes down by 5, the sum of all values goes down by $5$ times the number of values, so when you divide by the same number of values, the mean also goes down by 5.
• Median (middle value): All values move left by 5, including the middle one. The order stays the same, but the actual middle number is 5 less.
• Minimum (smallest value): The smallest value in the list also has 5 subtracted from it, so it becomes 5 less.

So for these three statistics, the numerical value changes when you subtract 5 from every data point.

Now consider a different kind of statistic that depends on how far each data point is from the center of the data (for example, from the mean), not on the absolute location of the data on the number line.

If you subtract 5 from every value in the data set, the mean also decreases by 5, so:

• Each data point moves left by 5.
• The mean moves left by 5.

That means the distance between each data point and the mean stays exactly the same as before. A statistic that is built from these distances will not change when you add or subtract the same number to every data value.

Among the four choices—mean, median, minimum, and standard deviation—only standard deviation is defined using how far data values are from the mean (it measures the spread of the data around the mean).

Because subtracting 5 from every time does not change any of those distances, the standard deviation stays the same numerical value for both the recorded times and the actual times.

So the correct answer is D) Standard deviation.