Multiplying Observations by a Constant

Introduction

Multiplication changes the scale of a data set. If every recorded time is converted from minutes to seconds, for instance, each value is multiplied by 60. The observations become larger, but their relative pattern does not change: values that were farther above the center remain farther above it, now in a different unit. Understanding how summaries respond to this transformation lets us update them without recalculating every summary from scratch.

The sign of the multiplier matters. Multiplying by a positive constant stretches or compresses the scale while preserving the order of observations. Multiplying by a negative constant also reverses their order. The mean and median follow the signed multiplier; nonnegative measures of spread follow its absolute value. Variance is the exception in form: it follows the square of the multiplier.

Definition: Suppose the original observations are \(x_1,x_2,\ldots,x_n\), and every observation is transformed by the same constant \(c\), so that \(y_i=cx_i\) for each \(i\). The transformed data set is \(y_1,y_2,\ldots,y_n\). The value of \(c\) may be positive, negative, or zero.

Centers and Order-Based Summaries

The sample mean is the sum of the observations divided by \(n\). After multiplication by \(c\), each term in the sum is multiplied by \(c\), so the transformed mean is \(c\bar{x}\). The median also scales by \(c\), although the reason depends on whether the multiplier preserves or reverses the order.

Theorem: Multiplying every observation by \(c\) changes the sample mean from \(\bar{x}\) to \(c\bar{x}\), and changes the median from \(\operatorname{median}(x)\) to \(c\operatorname{median}(x)\). For \(c=0\), every transformed observation, and therefore every center, is zero.

When \(c>0\), the observations remain in the same order. Each ordered value and each central position is multiplied by \(c\). When \(c<0\), larger original values become smaller transformed values. The order reverses, but the middle observation—or the average of the two middle observations—still becomes \(c\) times the original median. Thus the median rule applies for either sign.

Quartiles require attention to this order reversal. Using the median-of-halves convention, a positive multiplier sends \(Q_1\) to \(cQ_1\) and \(Q_3\) to \(cQ_3\). A negative multiplier reverses the two halves, so the transformed lower quartile is \(cQ_3\), while the transformed upper quartile is \(cQ_1\). The labels “lower” and “upper” refer to positions in the sorted transformed data, not to the numerical sign of a quartile.

Worked Example: A Negative Multiplier Reverses the Order

Consider the hypothetical data \(2,4,6,8,10\), and multiply every value by \(c=-2\). Find the transformed mean, median, quartiles, range, and interquartile range. Use the median-of-halves convention for quartiles.

The original mean is \( \bar{x}=(2+4+6+8+10)/5=30/5=6 \). The transformed observations are \(-4,-8,-12,-16,-20\); in increasing order they are \(-20,-16,-12,-8,-4\). Their mean is \(c\bar{x}=(-2)(6)=-12\), which also agrees with their sum, \(-60\), divided by 5.

The original median is 6, so the transformed median is \((-2)(6)=-12\). In the sorted transformed data, \(-12\) is indeed the middle value. For the original data, \(Q_1=3\) and \(Q_3=9\). Since the multiplier is negative, the quartiles switch positions: the transformed \(Q_1=(-2)(9)=-18\), and the transformed \(Q_3=(-2)(3)=-6\). The sorted transformed data confirm these values: the lower half is \(-20,-16\), with median \(-18\), and the upper half is \(-8,-4\), with median \(-6\).

The original range is \(10-2=8\), and the transformed range is \((-4)-(-20)=16\), equal to \(|-2|(8)\). The original interquartile range is \(9-3=6\); the transformed value is \((-6)-(-18)=12=|-2|(6)\). The transformed spread is positive even though the multiplier is negative.

Spread and the Squared Scaling Rule

Spread measures distances from observations to other values, such as the center or quartiles. Multiplying a distance by \(c\) changes its magnitude by \(|c|\): a negative multiplier reverses direction but cannot make a distance negative. This explains why the range, interquartile range, and standard deviation scale by the absolute value of \(c\).

Variance is based on squared deviations from the mean. The sample variance uses \(n-1\) in the denominator, while population variance divides by the number of population values. In either case, subtracting the transformed mean from a transformed observation produces \(c\) times the original deviation. Squaring that deviation introduces \(c^2\).

Theorem: If \(y_i=cx_i\), then the sample variance and standard deviation transform as \(s_y^2=c^2s_x^2\) and \(s_y=|c|s_x\). For a population, the corresponding rules are \(\sigma_y^2=c^2\sigma_x^2\) and \(\sigma_y=|c|\sigma_x\). The range and interquartile range also multiply by \(|c|\).

Here is the sample-variance justification in full. By definition, the sample variance is the sum of squared deviations from the sample mean divided by \(n-1\). The transformed mean is \(c\bar{x}\), so each transformed deviation is \(y_i-\bar{y}=cx_i-c\bar{x}=c(x_i-\bar{x})\). Substituting into the variance formula gives:

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

Taking the nonnegative square root gives \(s_y=\sqrt{c^2s_x^2}=|c|s_x\). The population proof is the same substitution, with the population mean and the population-size denominator in the variance formula. For range, subtract the smallest transformed value from the largest: the distance between the endpoints is multiplied by \(|c|\), whether their order is preserved or reversed. The same distance argument applies to the gap between quartiles, so the interquartile range also scales by \(|c|\).

Worked Example: Checking Variance and Standard Deviation

For the hypothetical sample \(2,4,6,8,10\), multiply each observation by \(-2\). Calculate the original and transformed sample variance and standard deviation to check the scaling rules.

The original mean is 6. The deviations from the mean are \(-4,-2,0,2,4\), whose squares sum to \(16+4+0+4+16=40\). With \(n=5\), the sample variance and standard deviation are \(s_x^2=40/(5-1)=10\) and \(s_x=\sqrt{10}\approx3.162\).

The transformed values are \(-4,-8,-12,-16,-20\), with mean \(-12\). Their deviations, in the same order as these observations, are \(8,4,0,-4,-8\); their squares also sum to \(64+16+0+16+64=160\). Therefore \(s_y^2=160/(5-1)=40\) and \(s_y=\sqrt{40}\approx6.325\).

The rules give the same results directly: \(c^2s_x^2=(-2)^2(10)=40\), and \(|c|s_x=2\sqrt{10}\approx6.325\). Variance is multiplied by 4, not by \(-2\); standard deviation is multiplied by 2.

Units and Zero Multipliers

A constant may represent a unit conversion. If measurements in centimeters are converted to meters, every observation is multiplied by \(0.01\). A center and a standard deviation then have units of meters rather than centimeters. A variance has squared units, so its numerical value is multiplied by \(0.01^2\), and its units change from square centimeters to square meters. Keeping track of squared units is a useful check on the variance rule.

If \(c=0\), all transformed observations equal zero. Every center, quartile, range, interquartile range, variance, and standard deviation is zero. This case also agrees with the formulas: \(c\bar{x}=0\), \(|c|s_x=0\), and \(c^2s_x^2=0\).

Worked Example: Converting Measurements From Centimeters to Meters

Suppose a hypothetical sample of three lengths is \(120\), \(150\), and \(180\) centimeters. Convert the data to meters and compare the mean, sample variance, and sample standard deviation.

The conversion multiplier is \(c=0.01\), so the transformed measurements are \(1.20\), \(1.50\), and \(1.80\) meters. The original mean is \((120+150+180)/3=150\) cm. The transformed mean is \(0.01(150)=1.50\) m.

In centimeters, the deviations from the mean are \(-30,0,30\), with squared deviations totaling \(1{,}800\) cm\(^2\). Since \(n=3\), the sample variance is \(s_x^2=1{,}800/(3-1)=900\) cm\(^2\), and the sample standard deviation is \(s_x=\sqrt{900}=30\) cm.

In meters, the deviations are \(-0.30,0,0.30\), with squared deviations totaling \(0.09+0+0.09=0.18\) m\(^2\). Thus \(s_y^2=0.18/(3-1)=0.09\) m\(^2\), and \(s_y=\sqrt{0.09}=0.30\) m. These match the transformation rules: \(0.01^2(900)=0.09\) m\(^2\) and \(|0.01|(30)=0.30\) m.

Common Mistakes and Why the Rules Matter

These rules are useful when data are rescaled, such as when changing units, multiplying prices by an exchange rate, or expressing every measurement in a different scale. They also help check calculations: if all observations are doubled, the standard deviation must double and the variance must quadruple. A result that does not follow this pattern indicates an arithmetic error or inconsistent transformation.

Multiplying standard deviation by a negative number. A negative multiplier reverses the observations’ order, not the sign of their spread. Use \(|c|s\), so standard deviation remains nonnegative.
Multiplying variance by \(c\) instead of \(c^2\). Variance uses squared deviations. Since each deviation gains a factor \(c\), its square gains a factor \(c^2\).
Forgetting that negative multiplication swaps quartiles. For \(c<0\), transformed \(Q_1\) comes from the original \(Q_3\), and transformed \(Q_3\) comes from the original \(Q_1\). The interquartile range still scales by \(|c|\).
Ignoring units. Standard deviation has the same units as the data; variance has squared units. A unit conversion therefore applies the square of its multiplier to variance.

Key takeaway: Under \(y_i=cx_i\), signed centers are multiplied by \(c\), nonnegative measures of spread such as range, interquartile range, and standard deviation are multiplied by \(|c|\), and variance is multiplied by \(c^2\).

Check Your Understanding

Use the scaling rules to answer these questions. For quartiles, use the median-of-halves convention.

A data set has mean 12 and sample standard deviation 3. If every value is multiplied by 4, what are the transformed mean and standard deviation?
A sample has variance 5. If every observation is multiplied by \(-3\), what is the transformed sample variance?
Why does multiplying every value by \(-2\) not make the standard deviation negative?
A data set has \(Q_1=4\) and \(Q_3=10\). What are the transformed quartiles and interquartile range after multiplication by \(-2\)?
Measurements are converted by multiplying each value by \(0.1\). By what factor does the variance change, and why?