Multiplying Observations by a Positive Constant

Why Positive Rescaling Matters

A measurement can be recorded in different units without changing what is being measured. For example, a length in inches can be converted to centimeters by multiplying its numerical value by 2.54. If a data set contains many lengths, converting each observation is usually unnecessary: the mean, median, quartiles, range, interquartile range, and standard deviation can all be converted directly. The key is to know which summaries scale by the conversion factor and why.

This tutorial considers a particular operation: every observation is multiplied by the same positive constant. It applies to unit conversions that are pure multiplication, such as inches to centimeters or liters to milliliters. The constant must be positive because that preserves the order of the observations, which matters for medians and quartiles.

Definition: Given observations \(x_1,x_2,\ldots,x_n\) and a positive constant \(c\), the positively rescaled observations are \(y_i=cx_i\) for every \(i\). The constant \(c\) is the scale factor. A summary is calculated from the \(y_i\) values after rescaling, or equivalently obtained by applying the corresponding scaling rule to the original summary.

How the Common Summaries Change

The mean is a sum divided by the sample size. Multiplying each observation by \(c\) therefore multiplies the sum, and hence the mean, by \(c\). Since \(c\) is positive, the order of the observations is unchanged: if \(x_i<x_j\), then \(cx_i<cx_j\). Thus the median and quartiles, which are determined by the ordered values, also multiply by \(c\). This is true whether a quartile convention selects an observed value or interpolates between ordered values, provided the same convention is used before and after scaling.

Theorem: If \(y_i=cx_i\) for every observation, with \(c>0\), then the sample mean, median, and quartiles are each multiplied by \(c\). The range and interquartile range are multiplied by \(c\), and the sample standard deviation is multiplied by \(c\). All these transformed summaries retain the corresponding transformed measurement units.

For the range, subtract the smallest observation from the largest. Positive multiplication preserves which values are smallest and largest, so the difference is multiplied by \(c\). The interquartile range is \(Q_3-Q_1\), where \(Q_1\) and \(Q_3\) are the first and third quartiles. Because both quartiles are multiplied by \(c\), their difference is also multiplied by \(c\).

The standard deviation measures spread around the mean. Its scaling rule follows from how deviations and squared deviations change. For the sample standard deviation, the definition is

s_x=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}.

Here \(\bar{x}\) is the sample mean and \(n\) is the sample size. After rescaling, \(\bar{y}=c\bar{x}\), so each deviation from the mean becomes \(y_i-\bar{y}=c(x_i-\bar{x})\). Squaring multiplies each squared deviation by \(c^2\). The denominator \(n-1\) is unchanged. Taking the square root multiplies the standard deviation by \(c\), because \(c>0\).

A Short Derivation of the Standard Deviation Rule

Substitute \(y_i=cx_i\) and \(\bar{y}=c\bar{x}\) into the definition of the sample standard deviation:

s_y =\sqrt{\frac{\sum_{i=1}^{n}(y_i-\bar{y})^2}{n-1}} =\sqrt{\frac{\sum_{i=1}^{n}[c(x_i-\bar{x})]^2}{n-1}} =\sqrt{c^2\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}} =c s_x.

The last equality uses \(\sqrt{c^2}=c\), which holds because \(c\) is positive. If the population standard deviation is used instead, the same reasoning applies: its denominator is the population size rather than \(n-1\), and that denominator does not change. Population standard deviation is also multiplied by \(c\).

This derivation also explains why standard deviation has the same units as the original measurements. Squared deviations temporarily have squared units, but taking the square root returns to the original scale. The variance, by contrast, is multiplied by \(c^2\) and has squared units.

Worked Example: Converting Inches to Centimeters

Suppose eight hypothetical object lengths, measured in inches, are \(6, 7, 8, 9, 11, 12, 13,\) and \(14\). Convert the five-number summary, range, interquartile range, mean, and sample standard deviation to centimeters. Use the median-of-halves convention for quartiles: split the ordered data into lower and upper halves and take each half’s median.

The conversion is \(y=2.54x\), so the positive scale factor is \(c=2.54\) centimeters per inch. The original mean is \(\bar{x}=80/8=10\) inches. The median is \((9+11)/2=10\) inches. The lower half is \(6,7,8,9\), giving \(Q_1=(7+8)/2=7.5\) inches; the upper half is \(11,12,13,14\), giving \(Q_3=(12+13)/2=12.5\) inches. The minimum and maximum are 6 and 14 inches, so the range is \(14-6=8\) inches and the interquartile range is \(12.5-7.5=5\) inches.

The sum of squared deviations from the mean is \(({-4})^2+({-3})^2+({-2})^2+({-1})^2+1^2+2^2+3^2+4^2=60\) square inches. Thus

s_x=\sqrt{\frac{60}{8-1}}=\sqrt{\frac{60}{7}}\text{ in}.

Now multiply each summary by 2.54. The converted mean and median are both \(2.54(10)=25.4\) cm. The quartiles are \(Q_1=2.54(7.5)=19.05\) cm and \(Q_3=2.54(12.5)=31.75\) cm. The minimum and maximum are 15.24 cm and 35.56 cm, so the range is \(2.54(8)=20.32\) cm and the interquartile range is \(2.54(5)=12.70\) cm. Finally,

s_y=2.54\sqrt{\frac{60}{7}}\text{ cm}\approx 7.436\text{ cm}.

The converted lengths have mean 25.4 cm, median 25.4 cm, quartiles 19.05 cm and 31.75 cm, range 20.32 cm, interquartile range 12.70 cm, and sample standard deviation about 7.436 cm. The calculations use the unrounded original standard deviation until the final step.

Checking the Rules in Other Units

A scale factor can be much larger or smaller than 1. The same rules apply either way. In particular, multiplying by a factor greater than 1 makes all these summaries larger by that factor; multiplying by a positive factor less than 1 makes them smaller. These changes reflect the choice of measurement units, not a change in the relative pattern of the data.

Worked Example: Converting Completion Times to Seconds

Seven hypothetical task times, recorded in minutes, are \(4,5,7,8,10,12,\) and \(14\). Convert the mean, median, quartiles, range, interquartile range, and sample standard deviation to seconds. Use the median-of-halves convention, excluding the overall median when forming the two halves.

The scale factor is \(c=60\) seconds per minute. The mean is \(\bar{x}=60/7\approx8.571\) minutes, and the median is 8 minutes. The lower half is \(4,5,7\), so \(Q_1=5\) minutes; the upper half is \(10,12,14\), so \(Q_3=12\) minutes. The range is \(14-4=10\) minutes, and the interquartile range is \(12-5=7\) minutes.

For the standard deviation, the sum of the observations is 60 and the sum of their squares is 594. The sum of squared deviations is therefore \(594-60^2/7=558/7\). Dividing by \(n-1=6\) gives a sample variance of \(93/7\) square minutes, so \(s_x=\sqrt{93/7}\approx3.645\) minutes.

Multiplying the summaries by 60 gives a mean of \(60(60/7)=3600/7\approx514.3\) seconds and a median of \(60(8)=480\) seconds. The quartiles are \(60(5)=300\) seconds and \(60(12)=720\) seconds. The range is \(60(10)=600\) seconds, the interquartile range is \(60(7)=420\) seconds, and the sample standard deviation is

s_y=60\sqrt{\frac{93}{7}}\text{ s}\approx218.7\text{ s}.

Each converted quantity is expressed in seconds. For instance, \(218.7\) seconds is the typical sample spread around the mean, expressed in the new unit; it is not a change in the underlying times.

Worked Example: Converting Liters to Milliliters

Five hypothetical liquid volumes are \(0.8,1.0,1.2,1.5,\) and \(2.0\) liters. Use the median-of-halves convention, excluding the middle observation when finding the quartiles, and convert the summaries to milliliters.

Since \(1\) liter is \(1000\) milliliters, the scale factor is \(c=1000\). The mean is \((0.8+1.0+1.2+1.5+2.0)/5=1.3\) L, and the median is 1.2 L. The lower half is \(0.8,1.0\), so \(Q_1=0.9\) L; the upper half is \(1.5,2.0\), so \(Q_3=1.75\) L. The range is \(2.0-0.8=1.2\) L, and the interquartile range is \(1.75-0.9=0.85\) L.

For the sample standard deviation, the deviations from the mean 1.3 L are \(-0.5,-0.3,-0.1,0.2,\) and \(0.7\) L. Their squared values sum to \(0.25+0.09+0.01+0.04+0.49=0.88\) square liters. Therefore,

s_x=\sqrt{\frac{0.88}{5-1}}=\sqrt{0.22}\text{ L}\approx0.4690\text{ L}.

Multiplying the mean, median, quartiles, range, interquartile range, and standard deviation by 1000 gives 1300 mL, 1200 mL, 900 mL and 1750 mL, 1200 mL, 850 mL, and approximately 469.0 mL, respectively. The arithmetic is consistent with converting every original volume to milliliters and then calculating its summaries.

What the Rule Does—and Does Not—Say

Positive rescaling preserves the shape of a distribution in the sense that the relative ordering and spacing pattern are retained, with all distances stretched or compressed by the same factor. For example, if one observation is 3 units above another before rescaling, it is \(3c\) units above it afterward. The center and spread summaries change in a predictable, consistent way. This is useful when comparing reports made in different units: first express the summaries in the same units, then make the comparison.

A crucial qualification is that not every unit conversion is pure multiplication. Converting Celsius to Fahrenheit, for example, involves both multiplication and adding a constant. The rules in this tutorial apply directly only when the conversion has the form \(y=cx\) with \(c>0\); applying them to a conversion that adds an offset can give incorrect location summaries. In addition, \(c\) must be positive. A negative multiplier reverses the order, while multiplying by zero makes every transformed observation identical.

Common Mistakes

Scaling only the mean. Every location summary listed here—including the median and quartiles—must be converted. The range, interquartile range, and standard deviation must also be scaled.
Multiplying the standard deviation by \(c^2\). Squared deviations are multiplied by \(c^2\), but the square root in the standard deviation definition leaves a factor of \(c\). It is the variance, not the standard deviation, that scales by \(c^2\).
Subtracting after converting only one endpoint. To find a range or interquartile range in new units, convert both endpoints or multiply the original difference by \(c\). Both methods give the same result.
Rounding too early. Keep the exact or unrounded statistic through multiplication, then round the final answer. For example, use \(2.54\sqrt{60/7}\), rather than a prematurely rounded standard deviation, when converting the sample standard deviation in the first example.
Using a multiplication rule for a conversion with an offset. Check the conversion formula first. These rules require \(y=cx\), not a formula that also adds a constant.

Key takeaway: When every observation is multiplied by the same positive factor \(c\), the mean, median, quartiles, range, interquartile range, and standard deviation are all multiplied by \(c\). The variance is multiplied by \(c^2\). Preserve precision during calculations and report each result in its correct converted units.

Check Your Understanding

For each question, assume the same positive factor is applied to every observation unless stated otherwise.

A sample has mean 12 kg, median 11 kg, range 8 kg, and sample standard deviation 3 kg. Each value is converted to grams by multiplying by 1000. What are the converted summaries, with units?
A data set has \(Q_1=4\) and \(Q_3=10\). If every observation is multiplied by 0.5, what are the new quartiles and interquartile range? Explain why the order of the observations does not change.
The sample standard deviation of measurements is 2.4 meters. The measurements are converted to centimeters. Write the scaling calculation for the new standard deviation and state its units.
Why is the variance multiplied by \(c^2\) while the standard deviation is multiplied by \(c\)?
Does the rule in this tutorial apply directly to converting Celsius temperatures to Fahrenheit? Explain which feature of the conversion formula matters.