Analyze changed data and outliers

These questions give you a data set, then change part of it and ask you to compare the two versions. The change might:

• add one or more values
• remove one or more values
• correct a value that was recorded wrong
• replace some values with new ones

Don’t reach for a whole-set shortcut unless the same rule hits every value, like adding $5$ to each one. Those shifts and scales are covered in Reason with range and standard deviation. And if a question gives two means and asks only for a missing or removed value, that’s total-over-count arithmetic, covered in Recover totals and weighted means.

When only some values change, ask four separate questions, one for each statistic. In short: total, middle, ends, distances.

Four questions for a changed data set

Think of the mean as a fair share: the total split evenly across all the values. So a change to some of the values can move the mean in only two ways, by changing the total or by changing the count.

Replace or correct a value

Say one value $a$ is replaced by $b$, and there are still $n$ values. Then

The mean changes by

So a larger replacement raises the mean, and a smaller one lowers it. The mean moves toward the new value, because that value changes the total everyone shares. In the opening example, $b-a=24-45=-21$ and $n=7$, so the mean dropped by $3$.

Add a value

Compare the new value with the old mean:

• A value above the old mean raises the mean.
• A value below the old mean lowers the mean.
• A value equal to the old mean leaves the mean the same.

The new mean lands somewhere between the old mean and the added value. A value far from the rest can pull the mean a lot, since its full size goes into the total.

Remove a value

Removing works the other way around:

• Removing a value above the old mean lowers the mean.
• Removing a value below the old mean raises the mean.
• Removing a value equal to the old mean leaves the mean the same.

The median depends on position in the ordered list, not on the total. An extreme value can change a lot and still stay on the same side of the middle. When that happens, the median doesn’t move.

That’s where the saying “outliers don’t affect the median” comes from, and it’s only half true. This is the part that trips people up. Adding or removing a value changes the count, so the middle spot itself moves. And a replacement that crosses to the other side of the middle can change which value sits there.

Here’s a crossing. Start with

The median is the fourth value, $8$. Now replace $30$ with $5$ and put the list back in order:

The new median is $6$. It went down because $5$ jumped from the far right to the left of the old middle.

Range and standard deviation both measure spread, but they look at different things.

• The range looks only at the two ends, so it changes only when the smallest value, the largest value, or both change.
• Standard deviation looks at how far the values typically sit from the mean, so it changes when those distances change.

The dot plots below show the same nine values before and after two swaps: $2$ becomes $7$, and $18$ becomes $13$.

Start with the mean. The two values that left add up to

and the two that replaced them add up to

Same total, same count, so the mean stays $10$. The median stays $10$ too.

Now the range. The ends change from $2$ and $18$ to $4$ and $16$, so the range shrinks:

Last, standard deviation. Check how far each swapped value sits from the mean, before and after:

Only those two values changed, and both moved closer to the same mean. So the standard deviation goes down.

This distance check is cleanest when the mean stays put. If the mean moves too, every value gets a new point to measure from, not just the swapped ones. So when some changes pull in and others push out, or the comparison is close, look at the whole new data set, or check it with a calculator list you’ve verified, instead of judging one value on its own.

You’ll usually do these by hand. The hard part isn’t the arithmetic. It’s knowing which feature controls each statistic. So mark exactly which values were added, removed, corrected or replaced, then ask the four questions: total, middle, ends, distances.

For a long, unordered list, a Desmos list can check your work:

• Type the original data once as A=[...].
• Type the complete revised data as B=[...].
• Check every entry against the question.
• Look at mean(A), median(A), mean(B) and median(B) as you need them.
• Use stdevp(A) and stdevp(B) only when the standard deviation comparison isn’t already clear.

The list checks your answer. It doesn’t replace the reasoning. Desmos can calculate two statistics, but it can’t tell you why an unchanged end keeps the range the same, or why crossing the middle moves the median.