Effect of Multiplying by a Constant on Summary Statistics

Rescaling Every Observation

In Effect of Adding a Constant on Summary Statistics, you saw that adding the same amount to every observation changes location summaries but leaves distances unchanged. Multiplying every observation by a positive constant behaves differently: it changes both the location and the distances. This is what happens when a measurement is reported in a different unit, such as converting inches to centimeters.

For example, one inch is 2.54 centimeters. If a set of lengths is recorded in inches, converting each value to centimeters means multiplying each value by 2.54. A length of 4 inches becomes 10.16 centimeters, and every distance between two lengths also becomes 2.54 times as large numerically. The physical lengths have not changed; only the unit used to express them has changed.

Definition: Multiplying a data set by a constant \(k\) means replacing every observation \(x\) with \(kx\). For a positive scale factor \(k\), every value and every distance between values is multiplied by \(k\).

How Center and Spread Summaries Change

When \(k\) is positive, multiplying all observations by \(k\) preserves their order. The mean is multiplied by \(k\), because the sum of the values is multiplied by \(k\) while the number of observations stays the same. The median, quartiles, and percentiles are also multiplied by \(k\): the observations stay in the same order, and the value at each ranked position is scaled by \(k\).

\text{new mean}=k\bar{x} \qquad\text{and}\qquad \text{new median}=k(\text{original median})

The minimum and maximum are multiplied by \(k\) as well. The range is the maximum minus the minimum, so it is multiplied by \(k\). The IQR is \(Q_3-Q_1\); because both quartiles are multiplied by \(k\), the IQR is multiplied by \(k\) too. Standard deviation also scales by \(k\): every deviation from the mean becomes \(k\) times as large.

Variance needs special attention. It is based on squared deviations from the mean. Multiplying each deviation by \(k\) multiplies each squared deviation by \(k^2\). Thus, variance is multiplied by \(k^2\), while standard deviation is multiplied by \(k\). This also means that the units of variance are squared units: for example, square inches become square centimeters.

\begin{aligned} \text{new range} &= k(\text{original range})\\ \text{new IQR} &= k(\text{original IQR})\\ \text{new standard deviation} &= k(\text{original standard deviation})\\ \text{new variance} &= k^2(\text{original variance}) \end{aligned}

Key distinction: For a positive scale factor, location summaries and distance-based spread summaries are multiplied by \(k\). Variance is multiplied by \(k^2\), because it is based on squared distances.

A positive rescaling does not change the distribution’s shape or the observations’ relative order. A graph’s horizontal axis is expressed on a different numerical scale, but clusters and gaps correspond to the same measurements. The z-score of each observation is also unchanged: the difference from the mean and the standard deviation are both multiplied by \(k\), so their ratio stays the same.

\frac{kx-k\bar{x}}{ks} = \frac{x-\bar{x}}{s}

These rules apply directly to ordinary unit conversions, whose conversion factors are positive. If a data set is multiplied by a negative constant, the order reverses, so the quartiles and minimum and maximum need extra care. That is not what happens when converting between units such as inches and centimeters.

A Method for Rescaling Summaries

Start by writing the conversion as a multiplication and identify the factor \(k\). Then classify each requested statistic. Multiply location measures and distance-based spread measures by \(k\), but multiply variance by \(k^2\). Finally, attach the new units and check that the result makes sense: a conversion to smaller units gives larger numerical values for the same measurement.

Identify the scale factor.
Write the conversion in the direction needed. For inches to centimeters, multiply by \(2.54\); for inches to feet, multiply by \(1/12\).

Update location summaries.
Multiply the mean, median, quartiles, percentiles, minimum, and maximum by the positive factor \(k\).

Update spread summaries.
Multiply the range, IQR, and standard deviation by \(k\). Multiply variance by \(k^2\).

Report the new units.
Use the converted measurement unit for values such as the mean and standard deviation, and its square for variance.

Worked Examples: Changing the Measurement Scale

Worked Example: Converting Inch Measurements to Centimeters

A fictional sample of five object lengths is \(2, 4, 6, 8, 10\) inches. Convert the data to centimeters and find how the mean, median, quartiles, range, IQR, variance, and sample standard deviation change. Use \(1\) inch \(=2.54\) centimeters and the median-of-halves convention for quartiles.

Find the original summaries. The sum is \(2+4+6+8+10=30\), so the mean is \(30/5=6\) inches. The median is 6 inches. The lower half is \(2,4\), giving \(Q_1=(2+4)/2=3\) inches; the upper half is \(8,10\), giving \(Q_3=(8+10)/2=9\) inches. The range is \(10-2=8\) inches, and the IQR is \(9-3=6\) inches.

The deviations from the mean are \(-4,-2,0,2,4\) inches. Their squares sum to \(16+4+0+4+16=40\) square inches. The sample variance is \(40/(5-1)=10\) square inches, so the sample standard deviation is \(\sqrt{10}\approx3.1623\) inches.

Apply the conversion factor. Here \(k=2.54\). Multiply the location and distance summaries by \(2.54\), and multiply the variance by \(2.54^2=6.4516\).

\begin{aligned} \text{mean} &= 6(2.54)=15.24\text{ cm}\\ \text{median} &= 6(2.54)=15.24\text{ cm}\\ Q_1 &= 3(2.54)=7.62\text{ cm},\qquad Q_3=9(2.54)=22.86\text{ cm}\\ \text{range} &= 8(2.54)=20.32\text{ cm}\\ \text{IQR} &= 6(2.54)=15.24\text{ cm}\\ s^2 &= 10(2.54^2)=64.516\text{ cm}^2\\ s &= \sqrt{10}(2.54)\approx8.0322\text{ cm} \end{aligned}

Check with the converted data. The lengths become \(5.08, 10.16, 15.24, 20.32, 25.40\) centimeters. Their sum is \(76.20\), and \(76.20/5=15.24\) centimeters, confirming the converted mean. The endpoints give a range of \(25.40-5.08=20.32\) centimeters. The converted deviations are \(-10.16,-5.08,0,5.08,10.16\); their squared sum is \(258.064\), and \(258.064/4=64.516\) square centimeters. Taking the square root gives approximately \(8.0322\) centimeters, as predicted.

Conclude in context. The sample’s mean object length is 15.24 centimeters, and its sample standard deviation is about 8.0322 centimeters. The physical lengths have not changed; their numerical summaries have been rescaled from inches to centimeters.

Worked Example: Converting a Summary From Centimeters to Meters

A fictional group of measurements in centimeters has mean 168, median 165, \(Q_1=158\), \(Q_3=174\), range 32, and sample standard deviation 8.5. Convert the summaries to meters. Also find the IQR and sample variance in the new units.

Identify the factor. Since \(100\) centimeters equal \(1\) meter, convert centimeters to meters by multiplying by \(k=0.01\). First, the original IQR is \(174-158=16\) centimeters. The original sample variance is \(8.5^2=72.25\) square centimeters.

Update the summaries. Multiply the mean, median, quartiles, range, IQR, and standard deviation by \(0.01\). Multiply the variance by \(0.01^2=0.0001\).

\begin{aligned} \text{mean} &= 168(0.01)=1.68\text{ m}\\ \text{median} &= 165(0.01)=1.65\text{ m}\\ Q_1 &= 158(0.01)=1.58\text{ m},\qquad Q_3=174(0.01)=1.74\text{ m}\\ \text{range} &= 32(0.01)=0.32\text{ m}\\ \text{IQR} &= 16(0.01)=0.16\text{ m}\\ s &= 8.5(0.01)=0.085\text{ m}\\ s^2 &= 72.25(0.0001)=0.007225\text{ m}^2 \end{aligned}

Check the spread calculations. The new quartiles give \(1.74-1.58=0.16\) meters for the IQR. Squaring the new standard deviation gives \(0.085^2=0.007225\) square meters, which matches the converted variance. These checks confirm both the scaling and the units.

Conclude in context. In meters, the group’s median measurement is 1.65 meters, the middle half extends from 1.58 to 1.74 meters, and the sample standard deviation is 0.085 meters. The variance is \(0.007225\) square meters, not meters.

Worked Example: Converting Inches to Feet

A fictional set of four lengths is \(12,18,24,30\) inches. Convert the summaries to feet. Use the median-of-halves convention for quartiles.

Find the original summaries. The mean is \((12+18+24+30)/4=84/4=21\) inches. The median is \((18+24)/2=21\) inches. The lower half \(12,18\) gives \(Q_1=(12+18)/2=15\) inches, and the upper half \(24,30\) gives \(Q_3=(24+30)/2=27\) inches. Thus, the IQR is \(27-15=12\) inches and the range is \(30-12=18\) inches.

The deviations from 21 are \(-9,-3,3,9\); their squared sum is \(81+9+9+81=180\). The sample variance is \(180/(4-1)=60\) square inches, and the sample standard deviation is \(\sqrt{60}\approx7.7460\) inches.

Convert to feet. Since one foot is 12 inches, the factor is \(k=1/12\). The mean, median, quartiles, and distance-based spreads are divided by 12. Variance is divided by \(12^2=144\).

\begin{aligned} \text{mean} &= 21/12=1.75\text{ ft}\\ \text{median} &= 21/12=1.75\text{ ft}\\ Q_1 &= 15/12=1.25\text{ ft},\qquad Q_3=27/12=2.25\text{ ft}\\ \text{IQR} &= 12/12=1\text{ ft},\qquad \text{range}=18/12=1.5\text{ ft}\\ s^2 &= 60/144\approx0.4167\text{ ft}^2\\ s &= \sqrt{60}/12\approx0.6455\text{ ft} \end{aligned}

Check with the converted observations. The lengths in feet are \(1,1.5,2,2.5\). Their mean is \(7/4=1.75\) feet, and their range is \(2.5-1=1.5\) feet. Their deviations from 1.75 are \(-0.75,-0.25,0.25,0.75\), whose squared sum is \(1.25\). Therefore, the sample variance is \(1.25/3\approx0.4167\) square feet, and the sample standard deviation is \(\sqrt{1.25/3}\approx0.6455\) feet.

Conclude in context. Expressed in feet, the lengths have a mean and median of 1.75 feet, an IQR of 1 foot, and a sample standard deviation of about 0.6455 feet. The smaller numerical values result from measuring the same lengths in a larger unit.

Common Mistakes and AP Exam Tips

Leaving the spread unchanged. Unlike adding a constant, multiplying by a positive factor changes distances. If every measurement is multiplied by \(2.54\), the range, IQR, and standard deviation are also multiplied by \(2.54\).
Multiplying variance by the wrong amount. Variance uses squared deviations, so it is multiplied by \(k^2\), not \(k\). A full-credit answer also gives variance in squared units.
Using the conversion factor in the wrong direction. Inches to centimeters requires multiplication by \(2.54\); centimeters to inches requires division by \(2.54\). Check whether the numerical values should get larger or smaller.
Forgetting to convert the units in the explanation. If a converted mean is 15.24, identify it as 15.24 centimeters, not 15.24 inches. State variance in square units, such as square centimeters.
Claiming that the data or physical measurements changed. A unit conversion changes how values are written, not the underlying objects or their pattern. The relative order and shape remain the same for a positive factor.

For full credit, show the conversion factor, state which rule applies to each statistic, and report the results with appropriate units. For example: “The sample standard deviation is multiplied by \(2.54\), giving about 8.0322 centimeters; the sample variance is multiplied by \(2.54^2\), giving 64.516 square centimeters.”

Key takeaway: Multiplying every observation by a positive factor \(k\) multiplies the mean, median, quartiles, range, IQR, and standard deviation by \(k\). Variance is multiplied by \(k^2\). For a unit conversion, the values change scale, but the underlying measurements and distribution pattern do not.

Check Your Understanding

Use the positive scale-factor rules to answer each question. Include units where appropriate.

A data set has mean 7 inches, median 6 inches, and IQR 4 inches. Convert these summaries to centimeters using \(1\) inch \(=2.54\) centimeters.
A group’s range is 45 centimeters. What is the range in meters? State the scale factor you use.
A sample standard deviation is 3 feet. What is the standard deviation in inches? What is the scale factor?
A sample variance is \(25\) square meters. Convert it to square centimeters using \(1\) meter \(=100\) centimeters.
Why does converting measurements from one positive unit scale to another preserve their order and distribution shape?