A New Way to View the Magnitude of the Difference between the Arithmetic Mean and the Geometric Mean and the Difference between the Slopes When a Continuous Dependent Variable Is Expressed in Raw Form Versus Logged Form

Abstract

In Trond Petersen’s “Multiplicative Models for Continuous Dependent Variables: Estimation on Unlogged versus Logged Form” (this volume, pp. 113–164), the following phenomenon is noted: With the arithmetic and geometric means of a nonnegative quantitative variable measured for two groups (say, groups 1 and 2), it is possible for the arithmetic mean $(A_{1})$ for group 1 to be larger than the arithmetic mean $(A_{2})$ for group 2 and the geometric mean $(G_{1})$ for group 1 to be smaller than the geometric mean $(G_{2})$ for group 2. In the present note, some new formulas are introduced that will help to make clear when this phenomenon will occur and when it will not occur. The phenomenon considered here is of interest in situations where, for example, as noted in Petersen’s paper, results obtained using the raw form for a continuous dependent variable are compared with the corresponding results obtained using the logged form for the dependent variable. In addition, the formulas that are introduced in the present note also provide a new way to view the magnitude of the difference between the arithmetic mean and the geometric mean.

Keywords

multiplicative models semi-logarithmic model transformation of dependent variables arithmetic versus geometric means estimation on unlogged versus logged form

1. Introduction

We begin this note by considering first the magnitude of the difference between the arithmetic mean $(A)$ and the geometric mean $(G)$ . Some new explicit formulas for the magnitude of this difference will be introduced here.

For the sake of simplicity, we shall focus our attention in this introduction on the special case in which the group (or the sample) under investigation is of size $N = 2$ . For this special case, we let $x_{1}$ and $x_{2}$ denote the scores for a nonnegative quantitative variable measured for individuals 1 and 2, respectively, in the group (or sample). The difference between the arithmetic mean $(A)$ and the geometric mean $(G)$ of these scores can, of course, be expressed simply as

A - G = (x_{1} + x_{2}) / 2 - \sqrt{x_{1} x_{2}} .

Now it is interesting to note that the variance $(V)$ of the square root of these scores (i.e., the variance of $\sqrt{x_{1}}$ and $\sqrt{x_{2}}$ ) can be written as

V = (\sqrt{x_{1}} - \sqrt{x_{2}})^{2} / 2 = (x_{1} + x_{2}) / 2 - \sqrt{x_{1} x_{2}} = A - G .

Thus, from formula (2) we see that

A = G + V .

Note that $V$ in formulas (2) and (3) is not the variance of the $x$ scores but rather the variance of the square root of these scores. From formula (3), we note that

A \geq G .

With the introduction of formulas (2) and (3), we obtain a simple explanation for the inequality (4) in the special case considered in this paragraph.

The inequality (4) is well known, and it is called the arithmetic mean/geometric mean (AM/GM) inequality. (A more general version of the AM/GM inequality is called Jensen’s inequality; e.g., see Roberts and Varberg 1973.)

We next consider two groups (or two samples, one sample from each of two populations) in the special case in which each group is of size $N = 2$ . For expository purposes, we let $A_{1}$ , $G_{1}$ , and $V_{1}$ denote the arithmetic mean of the $x$ scores, the geometric mean of the $x$ scores, and the variance of the square root of these scores, respectively, for group 1, and we let $A_{2}$ , $G_{2}$ , and $V_{2}$ denote the corresponding quantities for group 2. From formula (3), we obtain

A_{1} = G_{1} + V_{1},

and

A_{2} = G_{2} + V_{2};

and thus

A_{1} - A_{2} = (G_{1} - G_{2}) + (V_{1} - V_{2}) .

From formula (5) we see that the difference between $(A_{1} - A_{2})$ and $(G_{1} - G_{2})$ is determined completely by $(V_{1} - V_{2})$ —that is, by the difference between the variance of the square root of the scores in group 1 and the variance of the square root of the scores in group 2.

We also see from formula (5) that

A_{1} - A_{2} ≷ 0

whenever

G_{1} - G_{2} ≷ V_{2} - V_{1};

and that

G_{1} - G_{2} ≷ 0

whenever

A_{1} - A_{2} ≷ V_{1} - V_{2} .

Thus, we find that $A_{1} - A_{2}$ and $G_{1} - G_{2}$ will have the same sign whenever

A_{1} - A_{2} ≷ V_{1} - V_{2} ≷ G_{2} - G_{1};

and $A_{1} - A_{2}$ and $G_{1} - G_{2}$ will have different signs (i.e., $A_{1} - A_{2} > 0$ and $G_{1} - G_{2} < 0$ , or $A_{1} - A_{2} < 0$ and $G_{1} - G_{2} > 0$ ) whenever

A_{1} - A_{2} ≶ V_{1} - V_{2} ≷ G_{2} - G_{1} .

Here now is a simple example in which $A_{1} - A_{2} > 0$ and $G_{1} - G_{2} < 0$ . This is the sign-switching phenomenon noted in Trond Petersen’s paper and cited in the abstract of the present note: For sample 1, we let $(x_{1}, x_{2}) = (4, 25)$ , so $A_{1} = 14 \frac{1}{2}$ and $G_{1} = 10$ ; and for sample 2, we let $(x_{1}, x_{2}) = (9, 16)$ , so $A_{2} = 12 \frac{1}{2}$ and $G_{2} = 12$ . Thus, $A_{1} - A_{2} = 2$ , and $G_{1} - G_{2} = - 2$ . For further insight into this example of the sign-switching phenomenon, see Appendix A1 in this note.

2. Some More General Formulas

For expository purposes, we next consider the special case in which the group (or the sample) under investigation is of size $N = 4$ . For this case, we let $x_{1}, x_{2}, x_{3}, x_{4}$ denote the scores for a nonnegative quantitative variable measured for individuals 1, 2, 3, and 4, respectively, in the group (or sample). The difference between the arithmetic mean and the geometric mean can, of course, be expressed simply as

A - G = (x_{1} + x_{2} + x_{3} + x_{4}) / 4 - (x_{1} x_{2} x_{3} x_{4})^{\frac{1}{4}} .

Now, for this special case, we let

V [\sqrt{x_{1}}, \sqrt{x_{2}}], V [\sqrt{x_{3}}, \sqrt{x_{4}}], and V [(x_{1} x_{2})^{1 / 4}, (x_{3} x_{4})^{1 / 4}]

denote the variance of the square root of the scores $x_{1}$ and $x_{2}$ (i.e., the variance of $\sqrt{x_{1}}$ and $\sqrt{x_{2}}$ ), the variance of the square root of the scores $x_{3}$ and $x_{4}$ (i.e., the variance of $\sqrt{x_{3}}$ and $\sqrt{x_{4}}$ ), and the variance of $(x_{1} x_{2})^{1 / 4}$ and $(x_{3} x_{4})^{1 / 4}$ , respectively. With the variances as defined previously, we now introduce $V^{*}$ to denote the following “variance” formula:

V^{*} = \frac{1}{2} {V [\sqrt{x_{1}}, \sqrt{x_{2}}] + V [\sqrt{x_{3}}, \sqrt{x_{4}}]} + V [(x_{1} x_{2})^{1 / 4}, (x_{3} x_{4})^{1 / 4}] .

Thus, with $V^{*}$ defined by (8), we see that $V^{*}$ can also be expressed as follows:

\begin{matrix} V^{*} = \frac{1}{2} {(\sqrt{x_{1}} - \sqrt{x_{2}})^{2} / 2 + {(\sqrt{x_{3}} - \sqrt{x_{4}})}^{2} / 2} + {[(x_{1} x_{2})^{1 / 4} - {(x_{3} x_{4})}^{1 / 4}]^{2} / 2} \\ = {x_{1} + x_{2} + x_{3} + x_{4}} / 4 - {\sqrt{x_{1} x_{2}} + \sqrt{x_{3} x_{4}}} / 2 + {\sqrt{x_{1} x_{2}} + \sqrt{x_{3} x_{4}}} / 2 - {(x_{1} x_{2} x_{3} x_{4})}^{1 / 4} \\ = A - G, \end{matrix}

where $A$ is the arithmetic mean of the $x$ scores (namely, $x_{1}, \dots, x_{4}$ ) and $G$ is the geometric mean of these scores. Thus, in the special case considered here, we see that

A = G + V^{*} .

As we did earlier with formula (3), we can now use formula (10) to provide an explicit explanation of the inequality $A \geq G$ in the special case considered here. In addition, we can also use formula (10) in this case to obtain a formula corresponding to formula (5) with the $V_{1}$ and $V_{2}$ in formula (5) replaced now by the corresponding “variance” formulas—that is, $V_{1}^{*}$ and $V_{2}^{*}$ , respectively—which are obtained by applying formula (8), using the $x$ scores in group 1 to obtain $V_{1}^{*}$ and the corresponding $x$ scores in group 2 to obtain $V_{2}^{*}$ . Similarly, using formula (10) and the formula corresponding to formula (5) with $V_{1}$ and $V_{2}$ in formula (5) replaced for $V_{1}^{*}$ and $V_{2}^{*}$ , respectively, we can also obtain the inequality corresponding to the inequalities in (6a) and (6b) and the inequalities presented after formula (5) and before (6a) and (6b).

The results presented previously—for example, in formulas (3) and (10)—can be directly generalized to the case in which the group (or the sample) under investigation is of size $N = 2^{n}$ (for $n = 1, 2, 3, \dots$ ). We have presented the results obtained for the case where $n = 1$ and $n = 2$ , and for expository purposes, instead of presenting now the general formula, we provide more insight into how the general formula can be obtained by considering next the result for the case where $n = 3$ —that is, where $N = 2^{3} = 8$ . For expository purposes, we let $A [x_{1}, \dots, x_{4}]$ , $G [x_{1}, \dots, x_{4}]$ , and $V^{*} [x_{1}, \dots, x_{4}]$ denote the arithmetic mean $A$ , the geometric mean $G$ , and the corresponding “variance” formula $V^{*}$ —namely, formula (8)—respectively, obtained using the scores $x_{1}, \dots, x_{4}$ , and we let $A [x_{5}, \dots, x_{8}]$ , $G [x_{5}, \dots, x_{8}]$ , and $V^{*} [x_{5}, \dots, x_{8}]$ denote the corresponding quantities obtained using the corresponding scores $x_{5}, \dots, x_{8}$ . From formula (10), we see that

V^{*} [x_{1}, \dots, x_{4}] = A [x_{1}, \dots, x_{4}] - G [x_{1}, \dots, x_{4}],

and

V^{*} [x_{5}, \dots, x_{8}] = A [x_{5}, \dots, x_{8}] - G [x_{5}, \dots, x_{8}] .

We now introduce $V^{* *}$ to denote the following “variance” formula:

V^{* *} = \frac{1}{2} {V^{*} [x_{1}, \dots, x_{4}] + V^{*} [x_{5}, \dots, x_{8}]} + V [(x_{1} x_{2} x_{3} x_{4})^{1 / 8}, (x_{5} x_{6} x_{7} x_{8})^{1 / 8}],

where $V [(x_{1} x_{2} x_{3} x_{4})^{1 / 8}, (x_{5} x_{6} x_{7} x_{8})^{1 / 8}]$ denotes the variance of $(x_{1} x_{2} x_{3} x_{4})^{1 / 8}$ and $(x_{5} x_{6} x_{7} x_{8})^{1 / 8}$ . Applying the formulas in (11) to formula (12), we find that

\begin{matrix} V^{* *} = \frac{1}{2} {A [x_{1}, \dots, x_{4}] + A [x_{5}, \dots, x_{8}] - G [x_{1}, \dots, x_{4}] - G [x_{5}, \dots, x_{8}]} \\ + {[(x_{1} x_{2} x_{3} x_{4})^{1 / 8} - {(x_{5} x_{6} x_{7} x_{8})}^{1 / 8}]^{2} / 2} \\ = A - \frac{1}{2} {(x_{1} x_{2} x_{3} x_{4})^{1 / 4} + {(x_{5} x_{6} x_{7} x_{8})}^{1 / 4}} \\ + \frac{1}{2} [(x_{1} x_{2} x_{3} x_{4})^{1 / 4} + {(x_{5} x_{6} x_{7} x_{8})}^{1 / 4} - 2 (x_{1} \dots x_{8})^{1 / 8}] \\ = A - G, \end{matrix}

where $A$ is the arithmetic mean of the $x$ scores (namely, $x_{1}, \dots, x_{8}$ ) and $G$ is the geometric mean of these scores. Thus, in the case considered here too, we again see that

A = G + V^{* *},

where $V^{* *}$ is now defined by formula (12). Considering the definition of the “variance” terms in formula (12), we find that $V^{* *}$ can also be rewritten as

\begin{matrix} V^{* *} = \frac{1}{4} {\sum_{i = 1}^{4} V [\sqrt{x_{2 i - 1}}, \sqrt{x_{2 i}}]} + \frac{1}{2} {\sum_{i = 1}^{2} V [(x_{4 i - 3} x_{4 i - 2})^{1 / 4}, (x_{4 i - 1} x_{4 i})^{1 / 4}]} \\ + V [{(\prod_{i = 1}^{4} x_{i})}^{1 / 8}, {(\prod_{i = 5}^{8} x_{i})}^{1 / 8}] . \end{matrix}

As noted earlier in this section, the results presented previously (e.g., formulas [3], [10], and [14] and the related formulas [2], [8], [12], and [15]) can be directly generalized to the case in which the group (or sample) under investigation is of size $N = 2^{n}$ (for $n = 1, 2, 3, \dots$ ). In addition, it is also possible to obtain somewhat similar (but more complicated) results when the size $N$ of the group (or sample) cannot be expressed as $N = 2^{n}$ (for $n = 1, 2, 3, \dots$ ). These cases will be considered further in Appendixes A2 and A3.

3. A Different Perspective

We now present an approach to the examination of the magnitude of the difference between the arithmetic mean and the geometric mean that is quite different from the approach presented earlier in this note. For a group (or a sample) of any given size $N$ , we again let $A$ and $G$ denote the arithmetic mean and the geometric mean, respectively, of the $x$ scores—that is,

A = \sum_{i = 1}^{N} x_{i} / N, G = {(Π_{i = 1}^{N} x_{i})}^{1 / N} .

From (16) we see that

\log A = \log [\sum_{i = 1}^{N} x_{i} / N], \log G = \frac{1}{N} \sum_{i = 1}^{N} \log x_{i},

where log denotes the natural logarithm. The logarithmic function, $\log x$ , is a concave function, defined for all values of $x > 0$ (e.g., see Roberts and Varberg 1973). Because of this concavity, we find that

\log A \geq \log G;

we thus obtain

A \geq G .

As noted in Section 1, inequality (19) is well known. To gain more insight into the magnitude of the difference between the left side and right side of these inequalities, we next consider inequality (18) in more detail.

For the sake of simplicity, we begin here by considering again the special case in which $N = 2$ . In this case, for specified $x$ scores (say, $x_{1}$ and $x_{2}$ ), inequality (18) can be expressed as

\log [(x_{1} + x_{2}) / 2] \geq (\log x_{1} + \log x_{2}) / 2;

we also find, more generally, that

\log [(a_{1} x_{1} + a_{2} x_{2}) / (a_{1} + a_{2})] \geq (a_{1} \log x_{1} + a_{2} \log x_{2}) / (a_{1} + a_{2}),

for $a_{1} \geq 0$ and $a_{2} \geq 0$ . Inequality (20) states that the logarithm of the midpoint between $x_{1}$ and $x_{2}$ is greater than or equal to the midpoint between $\log x_{1}$ and $\log x_{2}$ ; in addition, inequality (21) states, more generally, that the logarithm of each point in the interval between $x_{1}$ and $x_{2}$ is greater than or equal to the height of the corresponding point on the line segment (or chord) between $(x_{1}, \log x_{1})$ and $(x_{2}, \log x_{2})$ . Note that $(a_{1} x_{1} + a_{2} x_{2}) / (a_{1} + a_{2})$ is a weighted average of $x_{1}$ and $x_{2}$ , and the term on the right side of (21) is the corresponding weighted average of $\log x_{1}$ and $\log x_{2}$ . For specified values of $x_{1}$ and $x_{2}$ , a simple examination of the graph of the logarithmic function will make clear what is the magnitude of the difference between the term on the left side of inequality (20) and the term on the right side of this inequality; a similar simple examination of the graph of the logarithmic function can also be applied with respect to the terms on each side of inequality (21).

Next we consider the special case in which $N = 3$ . In this example, for specified $x$ scores (say, $x_{1}$ , $x_{2}$ , and $x_{3}$ ), inequality (18) can be expressed as

\log [(x_{1} + x_{2} + x_{3}) / 3] \geq (\log x_{1} + \log x_{2} + \log x_{3}) / 3 .

Inequality (22) states that the logarithm of the arithmetic average of $x_{1}$ , $x_{2}$ , and $x_{3}$ is greater than or equal to the arithmetic average of the corresponding logarithms. To gain more insight into this inequality, let us first rewrite the arithmetic average $(x_{1} + x_{2} + x_{3}) / 3$ as a weighted average of two quantities as follows:

(x_{1} + x_{2} + x_{3}) / 3 = (2 x^{†} + x_{3}) / 3,

where $x^{†} = (x_{1} + x_{2}) / 2$ . By applying inequality (21) to this weighted average of $x^{†}$ and $x_{3}$ , we then obtain the inequality

\log [(2 x^{†} + x_{3}) / 3] \geq (2 \log x^{†} + \log x_{3}) / 3;

by applying inequality (20) to $x^{†}$ (i.e., to the midpoint between $x_{1}$ and $x_{2}$ ), we see that

(2 \log x^{†} + \log x_{3}) / 3 \geq {2 [(\log x_{1} + \log x_{2}) / 2] + \log x_{3}} / 3 .

(Note that the term on the left side of [22] is equal to the term on the left side of [23] and the term on the right side of [22] is equal to the term on the right side of [24].) Inequality (23) states that the logarithm of the weighted average of $x^{†}$ and $x_{3}$ is greater than or equal to the corresponding weighted average of $\log x^{†}$ and $\log x_{3}$ , and inequality (24) states that this weighted average of $\log x^{†}$ and $\log x_{3}$ is greater than or equal to the corresponding weighted average of (1) the midpoint between $\log x_{1}$ and $\log x_{2}$ and (2) $\log x_{3}$ . For specified values of $x_{1}$ , $x_{2}$ , and $x_{3}$ , a simple examination of the graph of the logarithmic function will make clear what is the magnitude of the difference between the term on the left side of inequality (23) (i.e., the logarithm of the weighted average of $x^{†}$ and $x_{3}$ , where $x^{†}$ is the midpoint between $x_{1}$ and $x_{2}$ ) and the term on the right side of inequality (23) (i.e., the height of the corresponding point on the line segment [or chord] between $[x^{†}, \log x^{†}]$ and $[x_{3}, \log x_{3}]$ ). And, similarly, a simple examination of the graph of the logarithmic function will make clear what is the magnitude of the difference between the term on the left side of inequality (24) (which was the same as the term on the right side of inequality [23]—that is, the height of the corresponding point on the line segment between $[x^{†}, \log x^{†}]$ and $[x_{3}, \log x_{3}]$ ) and the term on the right side of inequality (24)—that is, the height of the corresponding point on the line segment between $[x^{†}, \frac{1}{2} [\log x_{1} + \log x_{2}]]$ and $[x_{3}, \log x_{3}]$ . (Recall that $x^{†}$ is the midpoint between $x_{1}$ and $x_{2}$ and $[x^{†}, \frac{1}{2} [\log x_{1} + \log x_{2}]]$ is the midpoint on the line segment between $[x_{1}, \log x_{1}]$ and $[x_{2}, \log x_{2}]$ .) Thus, from the previous comments on inequalities (23) and (24), we see that inequality (22) can be viewed as a comparison of the logarithm of the weighted average of $x^{†}$ and $x_{3}$ (where $x^{†}$ is the midpoint between $x_{1}$ and $x_{2}$ ) and the height of the corresponding point on the line segment between $[x^{†}, \frac{1}{2} [\log x_{1} + \log x_{2}]]$ and $[x_{3}, \log x_{3}]$ .

We have considered inequality (18) in the special case in which $N = 2$ and $N = 3$ to gain more insight into the magnitude of the difference between the left side and right side of this inequality. The same approach can be applied in turn for $N = 4, 5, \dots$ .

The approach presented in this section is intended to shed some light on the difference between $\log A$ and $\log G$ in inequality (18), and earlier in this note we introduced explicit formulas—for example, see formulas (3), (10), (14) and (2), (8), (12), (15), as well as formulas in the Appendix—that shed light on the magnitude of the difference between $A$ and $G$ in the inequalities (4) and (19). We can rewrite the inequalities (4) and (19) as

A = G + V,

where $V \geq 0$ . Earlier in this note, we gave explicit formulas for the quantity $V$ —for example, see formulas (2), (8), (12), and (15) as well as the formulas in the Appendix. It may also be sometimes useful to view the quantity $V$ simply (tautologically) as the nonnegative quantity $A - G$ .

4. The Difference between the Slopes When a Continuous Dependent Variable Is Expressed in Raw Form Versus Logged Form

The sign-switching phenomenon noted in Petersen’s paper is presented there in the context in which comparison is made between results obtained when a continuous dependent variable is expressed in raw form and the corresponding results obtained when the dependent variable is expressed in logged form. It is possible that the slope obtained in the former case can be positive while the corresponding slope obtained in the latter case can be negative or that the slope in the former case can be negative while the corresponding slope in the latter case can be positive. In the special case when the independent variable can take on two possible scores—say, the scores 1 and 2—the observations on the continuous dependent variable corresponding to the independent variable at score 1 can be viewed as the observations in group 1 (or sample 1), and the observations on the dependent variable corresponding to the independent variable at score 2 can be viewed as the observations in group 2 (or sample 2). A negative slope obtained when the dependent variable is expressed in raw form and a positive slope obtained when the dependent variable is expressed in logged form would be described by the inequalities $A_{2} - A_{1} < 0$ and $G_{2} - G_{1} > 0$ ; similarly, a reversal of the sign of the slopes would be described by the inequalities $A_{2} - A_{1} > 0$ and $G_{2} - G_{1} < 0$ . With the formulas for $G$ and $\log G$ in (16) and (17), respectively, we recall that the arithmetic mean of the logarithm of the $x$ scores (i.e., the arithmetic mean of the logged form of the scores on the dependent variable) is equal to the logarithm of the geometric mean of the $x$ scores (i.e., the logarithm of the geometric mean of the raw form of the scores on the dependent variable).

Footnotes

Appendix

Author Biography

Leo A. Goodman is the Class of 1938 Professor Emeritus in the Department of Sociology and the Department of Statistics at the University of California, Berkeley. His research interests are in statistical methods of research, mathematical demography, mathematical sociology, and social mobility. Goodman is an elected member of the three main Learned Societies in the United States: The National Academy of Sciences, the American Academy of Arts and Sciences, and the American Philosophical Society. He is also the recipient of various other honors and awards from the American Statistical Association, the American Sociological Association, and the Institute of Mathematical Statistics, including the Samuel S. Wilks Memorial Medal presented by the American Statistical Association and the Career of Distinguished Scholarship Award presented by the American Sociological Association. He has also received the Samuel A. Stouffer Award from the American Sociological Association, the R. A. Fisher Lectureship from the Committee of Presidents of Statistical Societies, and the Henry L. Rietz Lectureship from the Institute of Mathematical Statistics. The University of Michigan conferred upon him the honorary degree Doctor of Science, and another honorary DS degree was also conferred upon him by Syracuse University. He was a faculty member in the Statistics Department and the Sociology Department at the University of Chicago until 1986. He served there as the Charles L. Hutchinson Distinguished Service Professor from 1970 to 1986. Since 1986 he has been a faculty member at the University of California, Berkeley.

References

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Varberg

Dale E.

1973. Complex Functions. New York: Academic Press.