AP Syllabus focus:
‘Discussing how changes in units of measurement affect the values of mean, median, range, IQR, and standard deviation.
- Analyzing the implications of unit conversion on summary statistics and data interpretation.
- Skill 2.C: Gaining insight into the importance of consistent units when calculating and reporting statistical measures.’
Changing units of measurement influences how numerical summaries behave, making it essential to understand how transformations such as scaling or shifting affect center and variability statistics.
Understanding How Unit Changes Affect Summary Statistics
When data are converted from one measurement unit to another—such as inches to centimeters or dollars to euros—each quantitative summary measure responds differently. This subsubtopic examines the effects of adding, subtracting, multiplying, or dividing all data values by a constant. Because statistical summaries serve as the foundation for describing distributions, clarity about how they change ensures accurate interpretation and valid comparisons.
Additive Transformations: Shifting All Data Values
An additive transformation occurs when the same constant is added to or subtracted from every data value, such as converting temperatures from Celsius to Fahrenheit.
The image shows an original distribution and a horizontally shifted version, illustrating how adding a constant moves the entire dataset without altering its spread or shape. Source.
This process changes the location of the distribution but does not alter its spread.
Measures Affected by Additive Changes
Mean — shifts by the same constant added to each data value.
Median — shifts by the same constant because its position in the ordered dataset changes in parallel.
Minimum and Maximum — each increases or decreases by the added constant.
Range, Interquartile Range (IQR), and Standard Deviation — remain unchanged because the differences between data points are preserved.
Additive Transformation: A change applied by adding or subtracting the same constant to every data point in a dataset.
The stability of measures of variability reflects that spread depends on differences among values, not their absolute positions on a numerical scale.
Multiplicative Transformations: Scaling All Data Values
A multiplicative transformation occurs when all data values are multiplied or divided by the same positive constant, such as converting height from feet to centimeters or time from hours to minutes.
The figure compares an original distribution with a scaled version, demonstrating how multiplying data expands or compresses the spread while retaining the overall distribution shape. Source.
Scaling affects both the center and the spread of a distribution.
Measures Affected by Multiplicative Changes
Mean — is multiplied or divided by the same scaling constant.
Median — is also multiplied or divided by the scaling constant.
Minimum and Maximum — scale proportionally with the applied constant.
Range — scales by the constant because the distance between extremes changes proportionally.
IQR — is multiplied or divided by the constant, as quartile differences expand or contract.
Standard Deviation — changes by the absolute value of the scaling constant because deviations from the mean grow or shrink in proportion.
Multiplicative Transformation: A change applied by multiplying or dividing every data value by the same nonzero constant.
These transformations alter the overall size of the distribution, making unit consistency critically important when comparing statistics across different contexts.
Why Center and Variability Respond Differently
Measures of center describe where data cluster, while measures of variability describe how widely the data are spread. Additive transformations shift a distribution without altering relationships between data values, leaving variability intact. In contrast, multiplicative transformations enlarge or compress the distances between data points, directly affecting measures of spread.
Key Insights for AP Statistics
Center measures (mean, median) shift with additive changes and scale with multiplicative changes.
Variability measures (range, IQR, standard deviation) remain unchanged under additive transformations but scale under multiplicative transformations.
Shape of the distribution remains unchanged under either type of transformation. Skewness, modality, and the presence of outliers are preserved.
EQUATION
= additive constant shifting the data
= multiplicative constant scaling the data
= original data value in the initial unit
Because the same transformation applies to every data point, the distribution maintains its structural properties even as its scale or location changes.
Implications for Data Interpretation
Unit consistency is essential for both clarity and comparability. When datasets differ in units, summaries such as mean or standard deviation cannot be directly compared without converting them to a common scale. For example, differences in measurement units may exaggerate or diminish the appearance of variability depending on the scaling constant. Careful attention to whether statistics are resistant or nonresistant ensures accurate interpretation after transformation.
Considerations When Reporting Summary Measures
Always report the unit accompanying any statistic.
Ensure that comparisons are made only between values expressed in the same unit.
Understand that transformed datasets preserve the relative standing of data points, even when numerical summary values change.
Recognize that large multiplicative constants may inflate variability metrics, potentially affecting visual representations such as boxplots or histograms.
Unit transformations do not change the underlying story told by the data; instead, they modify how that story is expressed numerically. This makes an understanding of transformation effects essential for accurate and meaningful statistical communication.
FAQ
Z-scores remain unchanged when a dataset undergoes unit transformations. This is because both the numerator (the distance from the mean) and the denominator (the standard deviation) shift or scale proportionally.
A shift cancels out entirely when subtracting the mean, and a multiplicative change is divided out during standardisation.
Therefore, z-scores preserve relative standing regardless of the units used.
The IQR changes only under multiplicative transformations, and all distances from quartiles scale by the same constant.
This means the threshold for defining outliers (Q1 − 1.5×IQR and Q3 + 1.5×IQR) scales perfectly with the data.
Because both the data points and the cut-off boundaries scale simultaneously, no new outliers appear and none disappear.
Yes. Although the statistical properties scale predictably, the practical meaning of spread can feel different in a new unit.
For example:
• A standard deviation of 0.5 metres may feel small, but 50 centimetres may appear comparatively large.
• Communicating in an inappropriate unit can exaggerate or minimise perceived variation.
Selecting units appropriate to context helps avoid misinterpretation.
Yes. While statistical relationships remain intact, the visual interpretation can shift depending on the scale used.
For example:
• Scaling can make small fluctuations appear large when graphed.
• Shifting can reposition a distribution so that key features fall outside a standard axis range.
Careful axis labelling and consistent units help prevent misrepresentation in graphical summaries.
Shape depends on relative spacing, not absolute values. Additive transformations move every point equally, and multiplicative ones stretch every distance proportionally.
Because the ratios between distances remain constant, patterns such as skewness, modality, and clustering are preserved.
Even when visually stretched, the underlying structure of the distribution remains identical in form.
Practice Questions
Question 1 (1–3 marks)
A dataset measuring the heights of plants is recorded in centimetres. A researcher decides to convert all measurements into millimetres by multiplying each value by 10.
a) State the effect this transformation has on the mean.
b) State the effect this transformation has on the standard deviation.
c) State the effect this transformation has on the shape of the distribution.
Question 1
a) 1 mark: Mean is multiplied by 10.
b) 1 mark: Standard deviation is multiplied by 10.
c) 1 mark: Shape remains unchanged (no effect on skewness, modality, or overall form).
Question 2 (4–6 marks)
A teacher records the test scores (out of 50) of a group of students. To adjust for a particularly difficult paper, the teacher adds 5 points to every student’s score. Later, the teacher converts the adjusted scores to percentages by multiplying all values by 2.
a) Describe the overall transformation applied to the original scores.
b) Explain how the median and interquartile range (IQR) of the adjusted percentage scores compare with those of the original scores.
c) The teacher claims that these transformations do not affect the presence or absence of outliers. Evaluate this claim and justify your reasoning.
Question 2
a) 1 mark: Correctly identifies the linear transformation as multiplying by 2 and adding 10 (or equivalent explanation combining the additive and multiplicative steps).
b) 2 marks:
• 1 mark for stating the median is shifted and then scaled (final median = original median + 5, then multiplied by 2).
• 1 mark for stating the IQR is unchanged by the addition of 5 but is then multiplied by 2 due to scaling.
c) 2–3 marks:
• 1 mark for stating that outliers remain the same when expressed relative to the transformed scale.
• 1 mark for explaining that both additive and multiplicative transformations preserve the relative spacing between data points.
• 1 mark for clearly justifying that rules for detecting outliers (such as the 1.5×IQR rule) scale consistently with the transformation.
