What happens to the mean and median if you add the same number to every data point? What happens if you then multiply every point by the same positive number? What about the IQR, which measures spread?

Try a very simple data set R, like when all 80 numbers are the same, or when they are all very negative. After transforming to S, compare the mean, median, and IQR to those of R. Do they always become larger?

For a statement to be true under a “must” condition, it has to hold for every possible data set R, not just for some or most. A single counterexample is enough to show it is not guaranteed.

In Desmos, enter a sample list for R, such as R = [-40, -40, -40, -40, -40] (you can use 5 numbers instead of 80; the behavior of the statistics is the same).

Define S = 2*(R + 15) in Desmos. This applies the transformation $x \mapsto 2(x + 15)$ to every element of R.

Use Desmos’s list statistics: type mean(R), mean(S), median(R), and median(S) in separate lines and observe the numerical values. Check whether the mean and median of S are always larger than those of R for this and other test lists you try.

Compute the interquartile ranges with iqr(R) and iqr(S) (or by using quartile3 - quartile1 if needed). Try a data set where all values in R are the same and another where they are different, and observe whether the IQR in S is strictly greater in each case.

Each number $x$ in data set R is changed to a new number in S by:

So going from R to S means: add 15 to every value, then multiply every result by 2 (a shift, then a stretch).

For any data set, when you:

1. Add a constant $c$ to every value:

   • The mean increases by $c$.
   • The median increases by $c$.
   • The spread measures like IQR stay the same.

2. Multiply every value by a positive constant $k$:

   • The mean is multiplied by $k$.
   • The median is multiplied by $k$.
   • The IQR is multiplied by $k$.

Here, going from R to S is the combined effect: first add 15, then multiply by 2. If the mean of R is $m$ and the median of R is $d$, then for S:

• Mean of S $= 2(m + 15) = 2m + 30$.
• Median of S $= 2(d + 15) = 2d + 30$.

To see if the mean of S must be greater than the mean of R, compare $2m + 30$ and $m$.

We want to know when

Subtract $m$ from both sides:

So the mean of S is greater than the mean of R only when the mean of R is greater than $-30$.

But the problem does not restrict the numbers in R; the 80 real numbers could easily have a mean less than or equal to $-30$. For example, if all 80 numbers in R are $-40$, then:

• Mean of R $= -40$.
• Each value in S is $2(-40 + 15) = 2(-25) = -50$, so mean of S $= -50$.

Here the mean in S is actually smaller, so statement I (mean must be greater) is not always true.

The exact same algebra and example work for the median. If the median of R is $d$, then median of S is $2d + 30$, which is greater than $d$ only when $d > -30$. If all values are $-40$, the median in R is $-40$ and in S is $-50$, so the median does not have to be greater either. So statement II is also not always true.

Let the interquartile range (IQR) of R be $I$.

• Adding 15 to every value does not change distances between values, so IQR stays $I$.
• Then multiplying every value by 2 multiplies all distances by 2, so IQR becomes $2I$.

So the IQR of S is $2I$. To ask whether it must be greater than the IQR of R is to ask if we must have $2I > I$.

• If $I > 0$ (the data are not all the same), then $2I > I$.
• But if $I = 0$ (all 80 numbers in R are identical), then $2I = 2\cdot 0 = 0$, so IQR of S equals IQR of R, not greater.

Using the same example as before (all 80 values equal $-40$):

• In R, all values are the same, so IQR of R $= 0$.
• In S, all values are also the same ($-50$), so IQR of S $= 0$.

Thus statement III (IQR must be greater) is not always true either.

The question asks which measures must be greater for every possible data set R.

• I (mean) is not guaranteed to be greater.
• II (median) is not guaranteed to be greater.
• III (IQR) is not guaranteed to be greater.

Since none of I, II, or III must always be greater, the correct answer choice is D) None of the above.