Some measures depend mainly on the position of values in the ordered list (like the middle), while others depend on the actual distances between values. Which type is more likely to change when one side is pushed up and the other side is pushed down?

For each option (median, mean, sum, interquartile range), ask: if I move all low values down by 15 and all high values up by 15, does that measure have to stay the same, or can it change?

One of the answer choices is based on the spread of the middle half of the data, not just a single central value. How does stretching the lower and upper halves in opposite directions affect that spread?

In Desmos (calculator or table), pick a small, symmetric example list to mimic the situation. For instance, in a table column or as a list, enter something like $L_1 = [90, 92, 94, 96, 104, 106, 108, 110]$, which has a clear center.

In the expression list, type:

• median(L1)
• mean(L1)
• total(L1)
• quartile3(L1) - quartile1(L1) These give you the median, mean, sum, and interquartile range of the original data.

Create a new list where the lower half is decreased by 15 and the upper half is increased by 15. For example, if $L_1$ has 8 values, define:

• L2 = [L1[1]-15, L1[2]-15, L1[3]-15, L1[4]-15, L1[5]+15, L1[6]+15, L1[7]+15, L1[8]+15]. Then compute:
• median(L2)
• mean(L2)
• total(L2)
• quartile3(L2) - quartile1(L2) Compare each of these four outputs to the corresponding ones from $L_1$ and note which statistic is different between $L_1$ and $L_2$.

There are 48 numbers: 24 are less than the median and 24 are greater than the median.

In the new data set:

• Each of the 24 smaller numbers is decreased by 15.
• Each of the 24 larger numbers is increased by 15.

So the lower half is shifted down by 15, and the upper half is shifted up by 15.

Each of the 24 large numbers is increased by 15, which adds $24 \times 15$ to the total.

Each of the 24 small numbers is decreased by 15, which subtracts $24 \times 15$ from the total.

These changes cancel out:

So the sum of all numbers stays the same. Since the number of data points (48) does not change, and the sum stays the same, the mean also stays the same.

With 48 numbers, the median is the average of the 24th and 25th values when the data are ordered.

Call the original 24th value $m_1$ and the 25th value $m_2$. We are told the median is 100, so

In the transformation:

• $m_1$ is in the lower half, so it becomes $m_1 - 15$.
• $m_2$ is in the upper half, so it becomes $m_2 + 15$.

The new median is

So the median stays the same.

The interquartile range measures how far apart the middle half of the data are: it is $Q_3 - Q_1$, where $Q_1$ is the first quartile and $Q_3$ is the third quartile.

In this problem:

• All values in the lower half (including those around $Q_1$) were decreased by 15, so the new $Q_1$ is $Q_1 - 15$.
• All values in the upper half (including those around $Q_3$) were increased by 15, so the new $Q_3$ is $Q_3 + 15$.

The new interquartile range is

which is larger than before. Therefore, the one measure that does not have the same value in both data sets is the interquartile range.