Effect of Adding a Constant on Summary Statistics

Shifting Every Observation by the Same Amount

In The Empirical Rule for Mound-Shaped Data, you used the mean and standard deviation to describe how observations are centered and spread out. Now consider what happens if every observation is increased or decreased by the same amount. This operation shifts the data values along the number line, changing their location but not their distances from one another.

For instance, if every recorded wait time is adjusted upward by 4 minutes, each individual wait time is 4 minutes larger. The mean and median should also move upward by 4 minutes. But the gap between any two wait times remains the same, so measures of spread do not change. This predictable pattern lets you update summaries without recalculating them from scratch.

Which Statistics Change?

The mean shifts by the added constant. If the original observations have mean \(\bar{x}\), then adding \(c\) to each of the \(n\) observations adds \(nc\) to their sum. Dividing that new sum by \(n\) shows that the new mean is \(\bar{x}+c\). The same idea applies to the population mean \(\mu\).

The median shifts by \(c\) as well. Adding the same number to every observation does not change their order, so the middle observation—or the average of the two middle observations—moves by exactly \(c\). Quartiles and other percentiles shift by \(c\) for the same reason: their positions in the ordered data stay the same, while the values at those positions increase or decrease.

The minimum and maximum each shift by \(c\). However, the range, which is maximum minus minimum, does not change: the \(+c\) added to the maximum cancels the \(+c\) added to the minimum. The IQR is also unchanged because both \(Q_1\) and \(Q_3\) shift by \(c\), so their difference stays constant.

Standard deviation stays the same because each observation remains the same distance from the mean as before. For example, after shifting the data, the difference between a value \(x+c\) and the new mean \(\bar{x}+c\) is \((x+c)-(\bar{x}+c)=x-\bar{x}\). The deviations—and therefore the squared deviations used to calculate variance and standard deviation—are unchanged.

A Consistent Method for Updating Summaries

The main question is whether a statistic describes location or spread. A location value, such as a mean or quartile, follows the shift. A spread value, such as a range or standard deviation, describes distances and is unaffected. For an unfamiliar summary, ask whether it depends on the absolute values themselves or on distances and ordering.

A shift also preserves the ordering and shape of a distribution: each observation moves the same distance in the same direction. A dotplot or boxplot moves along the number line, but the distances between values and the overall pattern are unchanged. The 1.5 IQR rule flags the same observations as potential outliers, since the quartiles shift while the IQR remains constant.

Worked Examples: Updating Summaries After a Shift

Common Mistakes and AP Exam Tips

• Changing spread statistics by the constant. A common error is to add 8 to the IQR or standard deviation when every value increases by 8. These statistics measure distances, and all those distances are preserved. A full-credit answer leaves them unchanged.
• Forgetting the sign of a negative shift. Adding \(-5\) decreases location summaries by 5; it does not increase them. Write the signed constant before updating values.
• Updating the mean but not the median or quartiles. Every observation moves by the same amount without changing its rank. Therefore, the median, quartiles, and percentiles all move by that amount too.
• Confusing a statistic’s value with its interpretation. When a mean changes from 9.5 to 17.5 minutes, say that the adjusted mean delivery time is 17.5 minutes. Do not say the standard deviation increased unless it actually changed.
• Leaving out context and units. A full-credit explanation identifies the variable and group, reports units, and distinguishes center from spread. For example: “The correction raises the median sensor reading by 5 units, while the IQR remains 9 units.”
• Assuming a shift changes shape or relative standing. The values move, but their order and pairwise distances stay the same. A particular observation remains above, below, or tied with the same observations as before.

When explaining your reasoning, connect the result to what each statistic measures: center statistics locate the distribution and follow the shift; spread statistics describe distances and are unchanged. If asked to verify, show that a pair of shifted values has the same difference as before, or compare deviations from the original and shifted means.