This topic emphasizes two practical ideas: converting units changes numerical summaries in predictable ways, and unusually large or small observations should be identified with a stated statistical rule.

How unit changes affect statistics

A change of units happens when every observation is converted by the same rule, such as inches to centimeters or pounds to kilograms. In AP Statistics, the key question is not just whether the numbers change, but which statistics change and how.

Shifting and rescaling data

When the same constant is added to or subtracted from every value, measures of center move by that constant. The mean, median, and quartiles all shift, because the whole distribution is translated left or right. However, measures of spread do not change under a pure shift. The range, IQR, and standard deviation stay the same because the distances between observations are unchanged.

When every value is multiplied or divided by the same positive constant, both center and spread are rescaled. The mean, median, quartiles, range, IQR, and standard deviation are all multiplied or divided by that factor. Because variance is based on squared distances, it changes by the square of the scale factor. A unit conversion like inches to centimeters is this kind of change.

Some conversions use both operations. For example, a temperature conversion can multiply and then add. In that case, the change in center comes from both steps, while the change in spread comes only from the multiplication step. The overall shape of the distribution and the order of the data values stay the same after a consistent unit conversion.

This matters because numerical summaries must always be reported with the correct units. A standard deviation measured in centimeters is not numerically equal to a standard deviation measured in meters, even though both describe the same data set.

Detecting outliers

An outlier is a value that stands unusually far from the rest of the observations.

You should not label a value as an outlier just because it “looks far away.” In AP Statistics, you are expected to support the claim with a clear rule and appropriate numerical evidence.

The $1.5 \times IQR$ rule

The interquartile range, or IQR, is the distance from $Q_1$ to $Q_3$, so $IQR = Q_3 - Q_1$. The $1.5 \times IQR$ rule creates two cutoffs, sometimes called fences. Values beyond those cutoffs are flagged as outliers.

This rule is often preferred when the distribution is skewed or already contains extreme values, because quartiles are not pulled strongly by unusually large or small observations. To use the rule correctly, find $Q_1$, $Q_3$, and the $IQR$, calculate the lower and upper fences, and then compare each data value with those boundaries. Any observation below the lower fence or above the upper fence is identified as an outlier.

The $2$ standard deviation rule

Another possible rule is to flag observations that are at least 2 standard deviations from the mean.

A normal curve diagram marks positions relative to the center using standard deviations, with labeled reference points at $-\sigma$ and $+\sigma$. This helps interpret rules like $\bar{x} \pm 2s$ as fixed “distance-from-the-mean” cutoffs measured in standard deviation units, which is most appropriate when the distribution is roughly symmetric. Source

This creates a lower cutoff and an upper cutoff centered at the sample mean.

This method depends on the mean and standard deviation, so its cutoffs can be influenced by the very outliers you are trying to detect. For that reason, it is usually more appropriate when the distribution is roughly symmetric and not strongly affected by extreme values. If an observation is less than the lower cutoff or greater than the upper cutoff, it is flagged by this rule.

Linking unit changes and outlier detection

A consistent unit conversion changes the numerical values of the outlier boundaries, but it does not change which observations are unusually far from the rest. If every measurement is converted from one unit to another, the same cases remain outliers because the data values and the cutoffs are transformed together.

For example, if a data set is converted from inches to centimeters, all values are multiplied by the same positive constant. The mean, quartiles, IQR, and standard deviation all change to the new unit. The lower and upper fences, or the $\bar{x} \pm 2s$ cutoffs, also change to that same unit. As a result, an observation that was beyond a cutoff before the conversion remains beyond the corresponding cutoff after the conversion.

The main danger is mixing units or forgetting to update the units when reporting a statistic. An outlier rule only makes sense if every value in the data set is expressed in the same measurement system.

What to write on an AP response

When you are asked about unit changes or outliers, strong responses usually do the following:

• Name the rule being used, such as the $1.5 \times IQR$ rule or the $2$ standard deviation rule.

• State the relevant cutoffs before deciding whether a value is an outlier.

• Compare the observation to the cutoff numerically, not just visually.

• Include units when reporting transformed statistics.

• Describe the effect of the conversion on the statistic, rather than saying only that it “changes.”