Comparing Group Summaries Before and After Transformation

Why Compare Both Scales?

A transformation can make a group’s plot easier to read, but it also changes the scale on which its summaries and gaps are expressed. Comparing displays before and after transformation helps answer two different questions: what features of the data are easier to see on the transformed scale, and what do the transformed summaries mean in the original units?

For example, a right-skewed group may look less skewed after taking logarithms, and a large observation may sit less far from the rest of the group. Neither change means that the original observation has disappeared or that the original-scale summaries have changed. The data are the same observations represented on a different scale.

Definition: An original-scale summary describes data in their recorded units. A transformed-scale summary describes the transformed values, in the units of the transformation. To compare groups responsibly, state which scale each plot, summary, gap, or outlier flag refers to.

A transformation is applied to each observation before making the transformed plot or calculating its summaries. For a natural-log transformation, each positive observation \(x\) becomes \(\ln(x)\). A logarithm requires positive values. Earlier material on log and square-root transformations describes when those transformations may be useful; the task here is to compare the resulting descriptions, not to treat either scale as the only valid one.

Plots on different scales do not have directly comparable axis distances. A horizontal distance of 1 on a log scale does not mean an increase of 1 in the original units. Likewise, a transformed mean or standard deviation is not an original-scale mean or standard deviation. Labeling the scale is therefore part of the statistical description, not a cosmetic detail.

A Consistent Comparison

For a useful comparison, first describe each group on its original scale, then describe the same group after transformation. Look at the overall shape, the center and spread, any observations flagged by the 1.5-IQR rule, and the gaps between groups. Keep the units attached to every numerical conclusion.

Shape: Does a long tail appear shorter or less dominant on the transformed plot? A transformation may reduce visible skewness, but the data remain the same observations.
Center and spread: Report summaries on the scale where they were calculated. Do not compare numerical standard deviations from different scales as though they measured the same kind of distance.
Outlier flags: Recalculate quartiles, the IQR, and fences on the transformed values if applying the 1.5-IQR rule there. A flag depends on the scale and rule; it is not proof that an observation is erroneous.
Group gaps: State whether a gap is an additive difference in original units or a difference on the transformed scale. For log-transformed values, a difference corresponds to a ratio on the original scale.

For the examples below, quartiles are calculated by splitting an ordered data set into lower and upper halves and taking the median of each half. This convention makes the calculations explicit. Other software may use a different quartile convention, especially with small samples, so reports should identify the convention when exact quartiles matter.

Worked Example: A Right Tail on Two Scales

Suppose a hypothetical set of five service times, in minutes, is \(2, 3, 4, 5, 20\). Compare its center, spread, shape, and 1.5-IQR flag on the original scale and after taking natural logarithms. The values are positive, so the log transformation is defined.

Original scale. The mean is

\bar{x}=\frac{2+3+4+5+20}{5}=\frac{34}{5}=6.8\text{ minutes}.

The median is 4 minutes. The lower and upper halves are \((2,3)\) and \((5,20)\), so \(Q_1=(2+3)/2=2.5\) and \(Q_3=(5+20)/2=12.5\) minutes. Therefore,

IQR=12.5-2.5=10,\qquad \text{upper fence}=12.5+1.5(10)=27.5\text{ minutes}.

The largest value, 20 minutes, is below the upper fence, so no observation is flagged by this rule. Still, the plot would show a long upper tail: 20 is much larger than the other four times, and the mean of 6.8 minutes is above the median of 4 minutes.

Natural-log scale. The transformed values, rounded to four decimal places, are \(0.6931, 1.0986, 1.3863, 1.6094, 2.9957\). Their mean and median are

\overline{\ln(x)} =\frac{0.6931+1.0986+1.3863+1.6094+2.9957}{5} \approx 1.5566,\qquad \operatorname{median}(\ln(x))=1.3863.

The transformed quartiles are \(Q_1=(0.6931+1.0986)/2\approx0.8959\) and \(Q_3=(1.6094+2.9957)/2\approx2.3026\). Thus \(IQR\approx1.4067\) and the upper fence is \(2.3026+1.5(1.4067)\approx4.4126\). The largest transformed value, 2.9957, is also below its fence: it is not flagged.

The high value is closer to the rest of the data in its position on the log scale, so the right tail is less stretched in the transformed plot. But that visual change does not make the 20-minute service time small in minutes. The transformed mean of about 1.5566 is in log units, not minutes.

Outlier Flags Depend on the Scale

The 1.5-IQR rule identifies values beyond fences calculated from a particular distribution’s quartiles. Transforming data can change both the quartiles and the IQR, so a value can be flagged on one scale but not another. The rule must be applied separately to each set of values; a fence calculated in minutes is not a fence for log-minutes.

A monotone increasing transformation such as the natural logarithm preserves the ordering of observations: if one original value is larger than another, its transformed value is also larger. Thus the transformation does not change which observation is largest. It can change how far apart observations appear, however, and that can change the quartiles, fences, or the flag. A flag on either scale describes relative position under that scale’s rule; it does not diagnose a data error.

Worked Example: Checking an Outlier Flag After Transformation

Consider hypothetical measurements in positive units: \(2, 3, 3, 4, 4, 5, 6, 30\). Compare the 1.5-IQR rule before and after taking natural logarithms. These observations are positive, so all eight can be transformed.

Original scale. The sum is 57, so the mean is \(57/8=7.125\) units. The median is \((4+4)/2=4\) units. The lower half is \(2,3,3,4\), giving \(Q_1=(3+3)/2=3\). The upper half is \(4,5,6,30\), giving \(Q_3=(5+6)/2=5.5\). Therefore,

IQR=5.5-3=2.5,\qquad \text{upper fence}=5.5+1.5(2.5)=9.25.

Since \(30>9.25\), the observation 30 is flagged on the original scale.

Natural-log scale. The transformed data, in the same order and rounded to four decimal places, are \(0.6931, 1.0986, 1.0986, 1.3863, 1.3863, 1.6094, 1.7918, 3.4012\). The lower-half median is \(Q_1=1.0986\), and the upper-half median is \(Q_3=(1.6094+1.7918)/2\approx1.7006\). Hence,

IQR=1.7006-1.0986=0.6020,\qquad \text{upper fence}=1.7006+1.5(0.6020)=2.6036.

Because \(\ln(30)\approx3.4012>2.6036\), the largest observation is flagged on the log scale too. Its distance from the other values is compressed by the transformation, but it remains unusually large relative to the transformed quartiles. Nothing in this calculation establishes that 30 is a mistake; it should be checked in context and reported accurately.

Group Gaps Must Be Named by Scale

The scale matters especially when comparing centers across groups. A difference between two original-scale medians is an additive gap in the recorded units. With natural logarithms, a difference between transformed medians has a ratio interpretation: if the transformed median for group B exceeds that for group A by \(d\), the corresponding ratio of original-scale medians is \(e^d\), provided the transformed medians correspond to the original medians. For an odd sample size, the median observation transforms to the median of the transformed observations because the ordering is preserved.

This distinction prevents a common error: describing a log-scale difference as though it were a difference in the original units. The transformed gap can be useful when relative differences are the focus, but it should be translated or labeled rather than presented as an original-unit increase.

Worked Example: Reporting a Gap on Both Scales

Suppose two hypothetical groups have measurements in the same positive units. Group A has \(10,12,14,16,18\), and Group B has \(20,24,28,32,36\). Compare the group medians before and after taking natural logarithms.

On the original scale, the medians are 14 units for A and 28 units for B. The additive gap is

28-14=14\text{ units}.

A suitable original-scale statement is: Group B’s median is 14 units higher than Group A’s median. The ratio is \(28/14=2\), so B’s median is also twice A’s median.

On the log scale, the medians are \(\ln(14)\approx2.6391\) and \(\ln(28)\approx3.3322\). Their gap is

\ln(28)-\ln(14) =\ln\left(\frac{28}{14}\right) =\ln(2) \approx0.6931.

The log-scale gap is about 0.6931 log units, not 0.6931 original units. Exponentiating it gives \(e^{0.6931}\approx2\), the ratio of the original-scale medians. Because every value in B is twice the corresponding value in A, their log-transformed values differ by the same \(\ln(2)\) at each position. The two log-scale plots therefore have the same shape and spread, shifted by about 0.6931.

Why the Log Gap Represents a Ratio

The ratio interpretation follows directly from the logarithm rule for a difference. If \(m_A\) and \(m_B\) are positive group medians, then

\ln(m_B)-\ln(m_A) =\ln\left(\frac{m_B}{m_A}\right).

Exponentiating both sides gives

e^{\,\ln(m_B)-\ln(m_A)}=\frac{m_B}{m_A}.

Thus, a log-median gap of \(d\) corresponds to a ratio \(e^d\). This calculation does not say that the groups differ by \(d\) original units. It describes a multiplicative comparison of their medians. For other transformations, the interpretation differs; do not apply the log-ratio interpretation to a square-root-scale gap.

Common Mistakes and a Reporting Checklist

The most common mistake is to compare numerical summaries from different scales without naming their units. For example, seeing a transformed mean that is smaller than the original mean does not show that the group’s measurements decreased. Those means summarize different values and have different meanings.

Common mistake: “The transformed group mean is 1.6, so the typical measurement is 1.6 units.”

Correction: If the values were natural-log transformed, 1.6 is a mean in log units. It is not the original-scale mean or a direct statement that a typical observation is 1.6 original units. Name the transformed scale, and use an appropriate back-transformation only when its interpretation is justified.

A clear comparison can follow this sequence:

Identify the original units and name the transformation.
Describe original-scale shape, center, spread, and any outlier flags.
Describe the same features on the transformed scale, with transformed units where relevant.
State whether group gaps are additive original-unit differences or transformed-scale differences.
For a log-scale gap, give its ratio interpretation when appropriate, and do not treat an outlier flag as proof of bad data.

The purpose of comparing scales is not to declare one universally correct. The original scale answers questions about actual measured amounts. A transformed scale may make relative patterns, tails, or group differences easier to describe. A careful conclusion keeps both meanings visible and ties each statement to the scale that supports it.

Check Your Understanding

Use the scale explicitly in each answer.

A group has a long right tail on the original plot but a shorter-looking tail after taking logarithms. What has changed, and what has not changed?
If an observation is above the original-scale upper fence, what must you do before deciding whether it is flagged on the log scale?
Two positive group medians are 8 and 12 units. What is their original-scale additive gap, and what is their ratio?
A log-scale median gap is \(\ln(3)\). What ratio of original-scale medians does this represent?
Why is it incorrect to describe a natural-log-scale gap of 0.5 as an increase of 0.5 original units?