One Variable Data: Distributions and Measures of Center and Spread Pattern - Analysis of Data Transformations

One-Variable Data: Analysis of Data Transformations

This pattern tests one specific concept: what happens to the mean (and sometimes the median) when you add a new data point to a data set? You do not always need to compute exact values. The SAT rewards conceptual understanding.

The Core Rule for the Mean

When you add a new value to a data set:

• If the new value equals the current mean → the mean stays the same.
• If the new value is greater than the current mean → the mean increases.
• If the new value is less than the current mean → the mean decreases.

That single rule solves nearly every question in this pattern.

The Core Rule for the Median

Adding a value to a data set may or may not change the median. It depends on where the new value falls relative to the current middle:

• If the new value is far above or below the existing data, the median usually shifts by at most one position (or not at all).
• The median is much more resistant to outliers than the mean.

On the SAT, when a question asks about both mean and median, the typical scenario is: an extreme value is added, which changes the mean but does not change the median.

Step-by-Step Method

1. Compute (or estimate) the mean of the original data set.
2. Compare the new value to the original mean.
3. Determine the direction of change: - New value $>$ mean → mean goes up → new mean $>$ old mean - New value $<$ mean → mean goes down → old mean $>$ new mean - New value $=$ mean → mean unchanged
4. If the question also asks about the median, find the original median and then figure out whether inserting the new value shifts the middle position.

Worked Example 1 — Table-Based

A basketball player scored 22, 28, 25, 30, and 20 points in five games. After a sixth game with 10 points, how does the original mean compare to the new mean?

A) Original mean $>$ new mean
B) Original mean $<$ new mean
C) Means are equal
D) Not enough information

SOLUTION

Original mean $= \dfrac{22 + 28 + 25 + 30 + 20}{5} = \dfrac{125}{5} = 25$
The new value is 10, which is less than 25.
Adding a value below the mean pulls the mean down, so the new mean is less than 25.
Therefore, the original mean is greater than the new mean.
Answer: A

Worked Example 2 — Graph-Based (Mean and Median)

The bar chart shows the number of laptops sold per day by a store over 11 days. On the 12th day, a special promotion results in 20 laptops sold. If this is added to the data, which measure(s) will be greater for the new data set?

I. The median number of laptops sold
II. The mean number of laptops sold

A) II only
B) I only
C) I and II
D) Neither I nor II

SOLUTION

From the bar chart, the 11 daily values are: 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5.
Original mean $= \dfrac{2(3) + 3(4) + 4(2) + 5(2)}{11} = \dfrac{6+12+8+10}{11} = \dfrac{36}{11} \approx 3.27$
Original median $=$ 6th value $= 3$

Adding 20 laptops (far above the mean of $\approx 3.27$): the mean increases.
For the median: with 12 values, the median is the average of the 6th and 7th values in the sorted list: 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 20. The 6th and 7th values are both 3, so the median stays at 3.

Only the mean increases. The median does not change.
Answer: A) II only

Worked Example 3 — Dot Plot (Mean and Median)

The dot plot shows an employee's commute times (in minutes) for 13 workdays. On the 14th day, a traffic accident causes a 90-minute commute. Which measure(s) will be greater for the new data set?

I. The median commute time
II. The mean commute time

A) I and II
B) II only
C) I only
D) Neither I nor II

SOLUTION

From the dot plot, the commute times cluster around 28–35 minutes. The original mean is in that range, and the original median (7th of 13 values) is about 32 minutes.

Adding 90 minutes — a massive outlier:
Mean: 90 is far above the original mean, so the mean increases.
Median: With 14 values, the new median is the average of the 7th and 8th values. Inserting 90 at the end of the sorted list shifts positions only slightly — the 7th and 8th values remain in the 30–32 range. The median does not change (or barely changes).

Only the mean increases.
Answer: B) II only

Common Gotchas

• Reversing the comparison. The question might ask "Is the original mean greater or less than the new mean?" Make sure you answer in the right direction.
• Thinking the mean cannot change. Adding any value that differs from the mean will change the mean. "The means are equal" is almost always wrong.
• Assuming the median always changes too. An outlier dramatically affects the mean but usually leaves the median unchanged. This is the SAT's favorite twist in this pattern.
• Choosing "not enough information." These questions always give you enough data to decide.

What to Do on Test Day

• Shortcut: Compare the new value to the original mean. That single comparison tells you the direction of change for the mean.
• If the question asks about both mean and median, the typical answer is "only the mean changes" — especially when the added value is an outlier.
• If new value $>$ old mean → mean goes up. If new value $<$ old mean → mean goes down.
• "Not enough information" and "means are equal" are almost always traps.
• For graph-based questions, read the data values from the chart carefully before reasoning about mean and median.