Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean

Abstract

This paper presents three confidence intervals for the coefficient of variation in a normal distribution with a known population mean. One of the proposed confidence intervals is based on the normal approximation. The other proposed confidence intervals are the shortest-length confidence interval and the equal-tailed confidence interval. A Monte Carlo simulation study was conducted to compare the performance of the proposed confidence intervals with the existing confidence intervals. Simulation results have shown that all three proposed confidence intervals perform well in terms of coverage probability and expected length.

1. Introduction

The coefficient of variation of a distribution is a dimensionless number that quantifies the degree of variability relative to the mean [1]. It is a statistical measure for comparing the dispersion of several variables obtained by different units. The population coefficient of variation is defined as a ratio of the population standard deviation (𝜎) to the population mean (𝜇) given by 𝜅=𝜎/𝜇. The typical sample estimate of 𝜅 is given as 𝑆𝜅=𝑋, (1) where 𝑆 is the sample standard deviation, the square root of the unbiased estimator of population variance, and 𝑋 is the sample mean.

The coefficient of variation has been widely used in many areas such as science, medicine, engineering, economics, and others. For example, the coefficient of variation has also been employed by Ahn [2] to analyze the uncertainty of fault trees. Gong and Li [3] assessed the strength of ceramics by using the coefficient of variation. Faber and Korn [4] applied the coefficient of variation as a way of including a measure of variation in the mean synaptic response of the central nervous system. The coefficient of variation has also been used to assess the homogeneity of bone test samples to help determine the effect of external treatments on the properties of bones [5]. Billings et al. [6] used the coefficient of variation to study the impact of socioeconomic status on hospital use in New York City. In finance and actuarial science, the coefficient of variation can be used as a measure of relative risk and a test of the equality of the coefficients of variation for two stocks [7]. Furthermore, Pyne et al. [8] studied the variability of the competitive performance of Olympic swimmers by using the coefficient of variation.

Although the point estimator of the population coefficient of variation shown in (1) can be a useful statistical measure, its confidence interval is more useful than the point estimator. A confidence interval provides much more information about the population characteristic of interest than does a point estimate (e.g., Smithson [9], Thompson [10], and Steiger [11]). There are several approaches available for constructing the confidence interval for 𝜅. McKay [12] proposed a confidence interval for 𝜅 based on the chi-square distribution; this confidence interval works well when 𝜅<0.33 [13–17]. Later, Vangel [18] proposed a new confidence interval for 𝜅, which is called a modified McKay’s confidence interval. His confidence interval is based on an analysis of the distribution of a class of approximate pivotal quantities for the normal coefficient of variation. In addition, modified McKay’s confidence interval is closely related to McKay’s confidence interval but it is usually more accurate and nearly exact under normality. Panichkitkosolkul [19] modified McKay’s confidence interval by replacing the sample coefficient of variation with the maximum likelihood estimator for a normal distribution. Sharma and Krishna [20] introduced the asymptotic distribution and confidence interval of the reciprocal of the coefficient of variation which does not require any assumptions about the population distribution to be made. Miller [21] discussed the approximate distribution of 𝜅 and proposed the approximate confidence interval for 𝜅 in the case of a normal distribution. The performance of many confidence intervals for 𝜅 obtained by McKay’s, Miller’s, and Sharma-Krishna’s methods was compared under the same simulation conditions by Ng [22].

Mahmoudvand and Hassani [23] proposed an approximately unbiased estimator for 𝜅 in a normal distribution and also used this estimator for constructing two approximate confidence intervals for the coefficient of variation. The confidence intervals for 𝜅 in normal and lognormal were proposed by Koopmans et al. [24] and Verrill [25]. Buntao and Niwitpong [26] also introduced an interval estimating the difference of the coefficient of variation for lognormal and delta-lognormal distributions. Curto and Pinto [27] constructed the confidence interval for 𝜅 when random variables are not independently and identically distributed. Recent work of Gulhar et al. [28] has compared several confidence intervals for estimating the population coefficient of variation based on parametric, nonparametric, and modified methods.

However, the population mean may be known in several phenomena. The confidence intervals of the aforementioned authors have not been used for estimating the population coefficient of variation for the normal distribution with a known population mean. Therefore, our main aim in this paper is to propose three confidence intervals for 𝜅 in a normal distribution with a known population mean.

The organization of this paper is as follows. In Section 2, the theoretical background of the proposed confidence intervals is discussed. The investigations of the performance of the proposed confidence interval through a Monte Carlo simulation study are presented in Section 3. A comparison of the confidence intervals is also illustrated by using an empirical application in Section 4. Conclusions are provided in the final section.

2. Theoretical Results

In this section, the mean and variance of the estimator of the coefficient of variation in a normal distribution with a known population mean are considered. In addition, we will introduce an unbiased estimator for the coefficient of variation, obtain its variance, and finally construct three confidence intervals: normal approximation, shortest-length, and equal-tailed confidence intervals.

If the population mean is known to be 𝜇₀, then the population coefficient of variation is given by 𝜅₀=𝜎/𝜇₀. The sample estimate of 𝜅₀ is 𝜅₀=𝑆₀𝜇₀, (2) where 𝑆²₀=𝑛⁻¹∑^𝑛_𝑖=1(𝑋_𝑖−𝜇₀)². To find the expectation of (2), we have to prove the following lemma.

Lemma 1. Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be a random sample from normal distribution with known mean 𝜇₀ and variance 𝜎² and let 𝑆²₀=𝑛⁻¹∑^𝑛_𝑖=1(𝑋_𝑖−𝜇₀)². Then 𝐸𝑆₀=𝑐_𝑛+1𝑆𝜎,𝑣𝑎𝑟₀=1−𝑐²_𝑛+1𝜎², (3) where 𝑐_𝑛+1=2/𝑛(Γ((𝑛+1)/2)/Γ(𝑛/2)).

Proof of Lemma 1. By definition, 𝑆²₀=1𝑛^𝑛_𝑖=1𝑋_𝑖−𝜇₀²=𝜎²𝑛^𝑛_𝑖=1𝑍²_𝑖, (4) where 𝑍_𝑖=(𝑋_𝑖−𝜇₀/𝜎)∼𝑁(0,1).
Thus, 𝑛𝑆²₀𝜎²∼𝜒²_𝑛. (5)
Let 𝑆²=(𝑛−1)⁻¹∑^𝑛_𝑖=1(𝑋_𝑖−𝑋)² and 𝑆²_∗=𝑛⁻¹∑^𝑛+1_𝑖=1(𝑋_𝑖−𝑋)². From Theorem B of Rice [29, page 197], the distribution of (𝑛−1)𝑆²/𝜎² is central chi-square distribution with 𝑛−1 degrees of freedom. Similarly, the distribution of 𝑛𝑆²_∗/𝜎² is central chi-square distribution with 𝑛 degrees of freedom; that is, 𝑛𝑆²_∗𝜎²∼𝜒²_𝑛. (6)
One can see that [30, page 181] 𝐸(𝑆)=𝑐_𝑛𝜎, (7) where 𝑐_𝑛=2/(𝑛−1)(Γ(𝑛/2)/Γ((𝑛−1)/2)).
Similarly, 𝐸𝑆_∗=𝑐_𝑛+1𝜎, (8) where 𝑐_𝑛+1=2/𝑛(Γ((𝑛+1)/2)/Γ(𝑛/2)).
Equations (5) and (6) are equivalent. Thus, we obtain 𝐸(𝑆₀)=𝐸(𝑆_∗)=𝑐_𝑛+1𝜎. Next, we will find the variance of 𝑆₀: var𝑆₀=𝐸𝑆²₀−𝐸𝑆₀²=𝜎²−𝑐²_𝑛+1𝜎²=1−𝑐²_𝑛+1𝜎². (9)

By using Lemma 1, we can show that the mean and variance of 𝜅₀ are 𝐸𝜅₀=𝑐_𝑛+1𝜎𝜇₀=𝑐_𝑛+1𝜅₀,var𝜅₀=1−𝑐²_𝑛+1𝜇²₀𝜎²=1−𝑐²_𝑛+1𝜅²₀. (10)(11) Note that 𝑐_𝑛+1→1 as 𝑛→∞. Therefore, it follows that lim_𝑛→∞𝐸𝜅₀=𝜅₀. (12) It means that 𝜅₀ is asymptotically unbiased and asymptotically consistent for 𝜅₀. From (10), the unbiased estimator of 𝜅₀ is 𝜅₀=𝜅₀𝑐_𝑛+1. (13) Using Lemma 1, the mean and variance of 𝜅₀ are given by 𝐸𝜅₀𝜅=𝐸⁰𝑐_𝑛+1=𝜅₀,var𝜅₀𝜅=var⁰𝑐_𝑛+1=1𝑐²_𝑛+1𝑆var⁰𝜇₀=1𝑐²_𝑛+1𝜇²₀1−𝑐²_𝑛+1𝜎²=1−𝑐²_𝑛+1𝑐²_𝑛+1𝜅²₀. (14)(15) Thus, lim_𝑛→∞var𝜅₀=0. (16) Hence, 𝜅₀ is also asymptotically consistent for 𝜅₀. Next, we examine the accuracy of 𝜅₀ from another point view. Let us first consider the following theorem.

Theorem 2. Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be a random sample from a probability density function 𝑓(𝑥), which has unknown parameter 𝜃. If 𝜃 is an unbiased estimator of 𝜃, it can be shown under very general conditions that the variance of 𝜃 must satisfy the inequality 𝜃≥1𝑣𝑎𝑟𝑛𝐸−𝜕²/𝜕𝜃²(𝑥)=1ln𝑓(𝜃),𝑛𝐼 (17) where 𝐼(𝜃) is the Fisher information. This is known as the Cramér-Rao inequality. If 𝑣𝑎𝑟(𝜃)=1/(𝑛𝐼(𝜃)), the estimator 𝜃 is said to be efficient.

Proof of Theorem 2. See [31, pages 377–379].

By setting 𝜃=𝜅₀=𝜎/𝜇₀ in Theorem 2, it is easy to show that var𝜅̆₀≥𝜅²₀,2𝑛 (18) where 𝜅̆₀ is any unbiased estimator of 𝜅₀. This means that the variance for the efficient estimator of 𝜅₀ is 𝜅²₀/2𝑛.

From (15), we will show that (1−𝑐²_𝑛+1)/𝑐²_𝑛+1→1/(2𝑛−1). The asymptotic expansion of the gamma function ratio is [32] Γ𝑗+(1/2)Γ𝑗=𝑗11−+18𝑗128𝑗²+⋯. (19) Now, if 𝑗=𝑛/2 in (19), we have 𝑐_𝑛+1=2𝑛Γ((𝑛+1)/2)Γ()=𝑛/22𝑛𝑛211−+14𝑛32𝑛²1+⋯=1−14𝑛+o𝑛^3/2. (20) Thus, we obtain 𝑐²_𝑛+11=1−12𝑛+o𝑛²,1−𝑐²_𝑛+1𝑐²_𝑛+1⟶1.2𝑛−1 (21) Therefore, var(𝜅₀)→𝜅²₀/(2𝑛−1). This means that 𝜅₀ is asymptotically efficient (see (18)). In the following section, three confidence intervals for 𝜅₀ are proposed.

2.1. Normal Approximation Confidence Interval

Using the normal approximate, we have 𝜅𝑧=⁰−𝜅₀var𝜅₀=𝜅₀/𝑐_𝑛+1−𝜅₀1−𝑐²_𝑛+1𝜅²₀/𝑐²_𝑛+1=𝜅₀−𝑐_𝑛+1𝜅₀𝜅₀1−𝑐²_𝑛+1⟶𝑁(0,1). (22) Therefore, the 100(1−𝛼)% confidence interval for 𝜅₀ based on (22) is 𝜅₀𝑐_𝑛+1+𝑧_1−𝛼/21−𝑐²_𝑛+1≤𝜅₀≤𝜅₀𝑐_𝑛+1−𝑧_1−𝛼/21−𝑐²_𝑛+1, (23) where 𝑧_1−𝛼/2 is the 100(1−𝛼/2) percentile of the standard normal distribution.

2.2. Shortest-Length Confidence Interval

A pivotal quantity for 𝜎² is 𝑄=𝑛𝑆²₀𝜎²∼𝜒²_𝑛. (24) Converting the statement 𝑃𝑎≤𝑛𝑆²₀𝜎²≤𝑏=1−𝛼, (25) we can write 𝑃𝜅₀√𝑛√𝑏≤𝜅₀≤𝜅₀√𝑛√𝑎=1−𝛼. (26) Thus, the 100(1−𝛼)% confidence interval for 𝜅₀ based on the pivotal quantity 𝑄 is 𝜅₀√𝑛√𝑏≤𝜅₀≤𝜅₀√𝑛√𝑎, (27) where 𝑎,𝑏>0, 𝑎<𝑏, and the length of confidence interval for 𝜅₀ is defined as 𝐿=𝜅₀√𝑛1√𝑎−1√𝑏. (28) In order to find the shortest-length confidence interval for 𝜅₀, the following problem has to be solved: goal:min_𝑎,𝑏𝜅₀√𝑛1√𝑎−1√𝑏constraint:∫^𝑏_𝑎𝑓_𝑄𝑞𝑑𝑞=1−𝛼, (29) where 𝑓_𝑄 is the probability density function of central chi-square distribution with 𝑛 degrees of freedom. From Casella and Berger [33, pages 443-444], the 100(1−𝛼)% shortest-length confidence interval for 𝜅₀ based on the pivotal quantity 𝑄 is determined by the value of 𝑎 and 𝑏 satisfying 𝑎^3/2𝑓_𝑄(𝑎)=𝑏^3/2𝑓_𝑄(𝑏),^𝑏_𝑎𝑓_𝑄𝑞𝑑𝑞=1−𝛼. (30)

Table 1 is constructed for the numerical solutions of these equations by using the R statistical software [34–36].

Table 1 

The values of 𝑎 and 𝑏 for the shortest-length confidence interval for 𝜅₀.

2.3. Equal-Tailed Confidence Interval

The 100(1−𝛼)% equal-tailed confidence interval for 𝜅₀ based on the pivotal quantity 𝑄 is 𝜅₀√𝑛𝜒²_{𝑛,1−𝛼/2}≤𝜅₀≤𝜅₀√𝑛𝜒²_𝑛,𝛼/2, (31) where 𝜒²_𝑛,𝛼/2 and 𝜒²_{𝑛,1−𝛼/2} are the 100(𝛼/2) and 100(1−𝛼/2) percentiles of the central chi-square distribution with 𝑛 degrees of freedom, respectively.

3. Simulation Study

A Monte Carlo simulation was conducted using the R statistical software [34–36] version 3.0.1 to investigate the estimated coverage probabilities and expected lengths of three proposed confidence intervals and to compare them to the existing confidence intervals. The estimated coverage probability and the expected length (based on 𝑀 replicates) are given by #()1−𝛼=𝐿≤𝜅≤𝑈𝑀,∑Length=^𝑀𝑗=1𝑈_𝑗−𝐿_𝑗𝑀, (32) where #(𝐿≤𝜅≤𝑈) denotes the number of simulation runs for which the population coefficient of variation 𝜅 lies within the confidence interval. The data were generated from a normal distribution with a known population mean 𝜇₀=10 and 𝜅₀ = 0.05, 0.10, 0.20, 0.33, 0.50, and 0.67 and sample sizes (𝑛) of 5, 10, 15, 25, 50, and 100. The number of simulation runs (𝑀) is equal to 50,000 and the nominal confidence levels 1−𝛼 are fixed at 0.90 and 0.95. Three existing confidence intervals are considered, namely, Miller’s [7], McKay’s [12], and Vangel’s [18].

Miller: 𝜅₀∈𝜅₀−𝑧_1−𝛼/2𝜅²₀1𝑛−12+𝜅²₀,𝜅₀+𝑧_1−𝛼/2𝜅²₀1𝑛−12+𝜅²₀, (33)

McKay: 𝜅₀∈𝜅₀𝜒²_{𝑛,1−𝛼/2}𝑛−1𝜅²₀+𝜒²_{𝑛,1−𝛼/2}𝑛−1^−1/2,𝜅₀𝜒²_𝑛,𝛼/2𝑛−1𝜅²₀+𝜒²_𝑛,𝛼/2𝑛−1^−1/2, (34)

Vangel: 𝜅₀∈𝜅₀𝜒²_{𝑛,1−𝛼/2}+2𝑛−1𝜅²₀+𝜒²_{𝑛,1−𝛼/2}𝑛−1^−1/2,𝜅₀𝜒²_𝑛,𝛼/2+2𝑛−1𝜅²₀+𝜒²_𝑛,𝛼/2𝑛−1^−1/2. (35) The upper McKay’s limit will have to be set to ∞ under the following condition [25]: 𝜅₀≥√𝑛𝜒²_𝑛,𝛼/2(𝑛−1)𝑛−𝜒²_𝑛,𝛼/2, (36) and the upper Vangel’s limit will have to be set to ∞ under the following condition: 𝜅₀≥√𝑛𝜒²_𝑛,𝛼/2(𝑛−1)𝑛−𝜒²_𝑛,𝛼/2.−2 (37) As can be seen from Tables 2 and 3, the three proposed confidence intervals have estimated coverage probabilities close to the nominal confidence level in all cases. On the other hand, the Miller’s, McKay’s, and Vangel’s confidence intervals provide estimated coverage probabilities much different from the nominal confidence level, especially when the population coefficient of variation 𝜅₀ is large. In other words, the estimated coverage probabilities of existing confidence intervals tend to be too high. Additionally, the estimated coverage probabilities of existing confidence intervals increase as the values of 𝜅₀ get larger (i.e., for 95% McKay’s confidence interval, 𝑛=10, 0.9522 for 𝜅₀ = 0.05; 0.9539 for 𝜅₀ = 0.10; 0.9856 for 𝜅₀ = 0.67). However, Figure 1 shows that the estimated coverage probabilities of the three proposed confidence intervals do not increase or decrease according to the values of 𝜅₀.

Table 2 

The estimated coverage probabilities and expected lengths of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

Table 3 

The estimated coverage probabilities and expected lengths of 95% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

(a)

(b)

(c)

(d)

Figure 1 

The estimated coverage probabilities of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

As can be seen from Figure 2, McKay’s and Vangel’s confidence intervals have longer expected lengths than Miller’s and the proposed confidence intervals. While the expected lengths of the three proposed confidence intervals are shorter than the lengths of the existing ones in almost all cases. Additionally, when the sample sizes increase, the lengths become shorter (i.e., for 95% shortest-length confidence interval, 𝜅₀ = 0.20, 0.1553 for 𝑛=10; 0.0949 for 𝑛 = 25; 0.0665 for 𝑛 = 50).

(a)

(b)

(c)

(d)

Figure 2 

The expected lengths of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

4. An Empirical Application

To illustrate the application of the confidence intervals proposed in the previous section, we used the weights (in grams) of 61 one-month old infants listed as follows:4960513042605160405052404350436039304410461045504460294041604110441048005130367045504290495052103210403035804360436039204050463037564586533628284172425645944866478445205238432053303836591650104344349641484044519243684180410252104382507050443530 (38)The data are taken from the study by Ziegler et al. [37] (cited in Ledolter and Hogg [38], page 287). The histogram, density plot, Box-and-Whisker plot, and normal quantile-quantile plot are displayed in Figure 3. Algorithm 1 shows the result of the Shapiro-Wilk normality test.

Shapiro-Wilk normality test data: weight W = 0.978, P-value = 0.3383

Algorithm 1 

Shapiro-Wilk test for normality of the weights of 61 one-month old infants.

(a)

(b)

(c)

(d)

Figure 3 

(a) Histogram, (b) density plot, (c) Box-and-Whisker plot, and (d) normal quantile-quantile plot of the weights of 61 one-month old infants.

As they appear in Figure 3 and Algorithm 1, we find that the data are in excellent agreement with a normal distribution. From past research, we assume that the population mean of the weight of one-month old infants is about 4400 grams. An unbiased estimator of the coefficient of variation is 𝜅₀≃0.9091. The 95% of proposed and existing confidence intervals for the coefficient of variation are calculated and reported in Table 4. This result confirms that the three confidence intervals proposed in this paper are more efficient than the existing confidence intervals in terms of length of interval.

Table 4 

The 95% confidence intervals for the coefficient of variation of the weight of one-month old infants.

5. Conclusions

The coefficient of variation is the ratio of standard deviation to the mean and provides a widely used unit-free measure of dispersion. It can be useful for comparing the variability between groups of observations. Three confidence intervals for the coefficient of variation in a normal distribution with a known population mean have been developed. The proposed confidence intervals are compared with Miller’s, McKay’s, and Vangel’s confidence intervals through a Monte Carlo simulation study. Normal approximation, shortest-length, and equal-tailed confidence intervals are better than the existing confidence intervals in terms of the expected length and the closeness of the estimated coverage probability to the nominal confidence level.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to Professor Dr. Tonghui Wang, Professor Dr. John J. Borkowski, and anonymous referees for their valuable comments and suggestions, which have significantly enhanced the quality and presentation of this paper.