The plaintiffs are seven white male police officers employed by the Massachusetts Bay Transportation Authority (MBTA) who contend that they were unlawfully bypassed for promotions to the ranks of sergeant and lieutenant when the chief of the MBTA police department, with the approval of the personnel administrator (administrator) of the human resources division of the Executive Office for Administration and Finance (HRD),4 chose instead to promote six minority or female candidates to those positions. We consider in this case whether Personnel Administration Rules PAR. 10 (Rule 10), pertaining to special certifications in the civil service based on race, color, national origin, or sex, was validly enacted, procedurally satisfied, and constitutionally sound. For the reasons that follow, we conclude that Rule 10 is valid and constitutional, and that the actions of the HRD were lawful.
1. Rule 10. Rule 10 provides in pertinent part:
“(1) Prerequisites for special certifications based on race, color, national origin or sex shall be made whenever:
“(a) an appointing authority shall make requisition to *235fill one or more positions included in said appointing authority’s affirmative action plan on file with the administrator, and;
“(b) the administrator has made a written determination substantiating that previous practices of the department and/or of said appointing authority with respect to the filling of such position or positions have discriminated against members of a group, hereinafter referred to as a protected group, on the basis of race, color, sex, or national origin in contravention of any provision of the Constitution of the United States or the Constitution of the Commonwealth, Title VII of the federal Civil Rights Act of 1964 (42 U.S.C. [§ ] 2000e) or any other federal or state statute[,] the administrator may then certify, in addition to names certified in accordance with PAR. 09, the names of a like number of individuals who are members of the protected group and are on an eligible list for such position, in order of their standing.
“(2) At least ten days prior to certifying names under the provisions of PAR. 10(1), the administrator shall post a notice of his intention to do so in the offices of the Department of Personnel Administration and shall mail a copy of such notice to the appointing authority, with instructions to post copies of such notice at all locations where persons whose names may be certified under the provisions of section one may, if employed, be assigned.”
2. Factual and procedural background. We recite in some detail the lengthy procedural history of these consolidated cases because the evidence of discrimination, presented by the MBTA, unfolded over the course of several proceedings before the Civil Service Commission (commission). On September 9, 1996, the chief of the MBTA police department submitted three public safety civil service requisition forms to the HRD in accordance with Rule 10. One form requested special certification of a “females” list for the position of lieutenant from an existing departmental promotional eligible list; the other two forms requested special certification of a “females” list and a “minori*236ties”5 list for the position of sergeant from an existing departmental promotional eligible list.6 Attached to each requisition form was a statistical breakdown of the racial and gender composition of the MBTA’s police force as of May, 1996. The statistics showed that out of a total of 211 police officers in the MBTA police department, fifty-two officers (or 24.6%) were minorities, and fifteen officers (or 7.1%) were women. With respect to the 181 patrol officers, fifty officers (or 27.6%) were minorities, and twelve officers (or 6.6%) were women. With respect to the thirty police officers occupying the positions of sergeant, lieutenant, captain, major, and chief, two officers (or 6.6%) were minorities, and three officers (or 10%) were women.
The HRD approved a “females” list for one lieutenant position, a “females” list for two sergeant positions, and a “minorities” list for three sergeant positions. The chief of the MBTA police department posted notices of his intention to certify names under the provisions of Rule 10 for these positions, and eligible candidates from both the regular lists7 and the special certification lists were considered. Effective October 5, 1996, the MBTA promoted a total of nine police officers, with the approval of the HRD. Two lieutenant positions were filled by one white male (William Fleming from the regular list) and one white female (Nancy O’Loughlin from the special certification *237list).8 Seven sergeant positions were filled by two white males (Salvatore Venturelli and Michael Moms from the regular list), two white females (Anne McCall and Irene Reardon from the special certification list), two black males (Herman Wheeler and Michael Lopes from the special certification list), and one black female (Gloria Andrews from the special certification list).9 Rule 10 did not establish any quotas with respect to promotions, and the MBTA’s affirmative action program was scheduled to expire on January 1, 2000.
The plaintiffs filed an appeal with the commission, pursuant to G. L. c. 31, § 2 (b), alleging that they were unlawfully bypassed for promotion in favor of minority and female candidates. By decision dated November 5, 1997, the commission denied the plaintiffs’ appeal, concluding that (1) Rule 10 was a valid exercise of the HRD’s rule making authority under G. L. c. 31, § 3; (2) the plaintiffs’ constitutional claims could not be decided by the commission; and (3) both the MBTA and the HRD attempted in good faith to comply with the procedural requirements of Rule 10. As such, the commission declined to substitute its own judgment for that of the MBTA and the HRD.
The plaintiffs sought judicial review of the commission’s determination pursuant to G. L. c. 30A, § 14, and G. L. c. 31, § 44.10 By decision dated October 20, 1998, a judge in the Superior Court (first Superior Court judge) denied the plaintiffs’ *238motion for judgment on the pleadings with respect to the minority promotions, affirming the commission’s determination in that regard. The judge concluded that the statistical data presented by the MBTA, together with its well-documented judicial history of racial discrimination, constituted a “strong basis in evidence” to support the MBTA’s use of special certification lists for minorities. The judge further concluded that Rule 10 was narrowly tailored to serve a compelling State interest. In addition, the judge agreed with the commission that the promulgation of Rule 10 was a valid exercise of the HRD’s rule mating authority under G. L. c. 31, § 3, and that the MBTA and the HRD satisfactorily complied with its provisions.
The judge allowed the plaintiffs’ motion for judgment on the pleadings with respect to the gender-based promotions, reversing the commission’s determination in that regard and remanding the matter to the commission for further findings whether there was a “strong basis in evidence” to justify the use of a gender-based classification in determining eligibility for promotions. The judge opined that the mere absence of women in the positions of sergeant and lieutenant did not warrant a finding that women had been subjected to past gender discrimination by the MBTA, and she concluded that the statistical data presented by the MBTA did not constitute a “strong basis in evidence” to support its use of special certification lists for women. The plaintiffs filed a timely notice of appeal with respect to the judge’s decision affirming the minority promotions; the MBTA filed a timely notice of appeal with respect to the judge’s decision regarding the gender-based promotions.
On remand to the commission, the MBTA presented the testimony of Dr. Leonard A. Cupingood, an expert in the field of statistics, who performed, as part of his work, utilization analyses comparing the gender composition of an employer’s work force with the gender composition of similar occupations in the relevant labor market, using the census as his primary data base.11 Dr. Cupingood compared, inter alla, the number of women employed by the MBTA police department with the *239number of women in the field of “protective services’’ who were performing police-related work in the Boston metropolitan area,12 taking into consideration the age, education, and income constraints that would be associated with an MBTA position. He concluded that there was a statistically significant underutilization of women in the MBTA police department in 1996. Nonetheless, by decision dated December 6, 2000, the commission found that the evidence presented by the MBTA was insufficient to show past discrimination justifying the use of gender-based classifications in determining eligibility for promotions. The MBTA filed a complaint for judicial review of the commission’s determination pursuant to G. L. c. 30A, § 14, and G. L. c. 31, § 44. See note 3, supra.
Before this appeal was considered, the MBTA filed a motion under G. L. c. 30A, § 14 (6), to present additional evidence to the commission pertaining to the gender of actual applicants for various officer positions in the MBTA police department. A judge in the Superior Court allowed the MBTA’s motion, concluding that the evidence was material, that the MBTA had shown good cause for not presenting the evidence earlier “inasmuch as the [HRD], which had exclusive access to the information, acknowledged at the [prior] hearing that the information now being offered to the [commission] by the [MBTA] was timely requested but not provided,” and that the plaintiffs had not shown that they would be prejudiced by the admission of such evidence.13 After considering the new evidence, the commission concluded, by decision dated January 22, 2002, that the additional testimony provided by Dr. Cupingood still did not demonstrate prior discriminatory practices by the MBTA police department that would justify the use of gender-based classifications in promotion decisions. Therefore, the commission affirmed its decision of December 6, 2000.
The MBTA then filed a motion for judgment on the pleadings with respect to its complaint for judicial review of the commis*240sion’s determinations relating to the gender-based promotions. By decision dated November 21, 2003, a judge in the Superior Court (second Superior Court judge) allowed the MBTA’s motion, reversing and vacating the determinations of the commission dated December 6, 2000, and January 22, 2002, and affirming the determination of the commission dated November 5, 1997, which upheld the MBTA’s gender-based promotions. The judge concluded that at the hearings before the commission on remand, the MBTA presented uncontroverted expert testimony that the statistical disparity between women in the labor force for the positions at issue and women in the MBTA police department did, in fact, exceed the benchmark of three standard deviations established by the first Superior Court judge as the measure for evaluating whether there was a “strong basis in evidence” as to past discrimination.14 The judge further opined that the MBTA was not required to provide direct evidence of past gender discrimination, but instead could present evidence of “gross statistical disparities” between the proportion of women hired and the proportion of women willing and able to do the work. Because the MBTA presented unrebutted statistical evidence to meet the benchmark of three standard deviations, the judge concluded that the commission was obligated to uphold the MBTA’s promotions of female police officers to the positions of sergeant and lieutenant.
The plaintiffs appealed from the judgments affirming the *241minority and gender-based promotions, and the Appeals Court consolidated the parties’ actions. We then transferred the cases to this court on our own motion.15 The plaintiffs now contend that (1) the actions of the MBTA in requesting, and of the HRD in approving, special certifications based on race and gender, pursuant to Rule 10, for promotions to the positions of sergeant and lieutenant in the MBTA police department violated the plaintiffs’ rights to equal protection under the Fourteenth Amendment to the United States Constitution and arts. 1 and 12 of the Massachusetts Declaration of Rights; (2) there was no statutory authority for the HRD’s promulgation of Rule 10; and (3) the HRD and the MBTA failed to comply with the procedural requirements of Rule 10. We affirm.
3. Standard of review. General Laws c. 31, § 2 (b), requires the commission to determine, on the basis of the evidence before it, whether the appointing authority sustained its burden of proving, by a preponderance of the evidence, that there was reasonable justification for the action taken by the appointing authority. See Massachusetts Ass’n of Minority Law Enforcement Officers v. Abban, 434 Mass. 256, 260 (2001); Cambridge v. Civil Serv. Comm’n, 43 Mass. App. Ct. 300, 303 (1997). Reasonable justification in this context means “done upon adequate reasons sufficiently supported by credible evidence, when weighed by an unprejudiced mind, guided by common sense and by correct rules of law.” Selectmen of Wakefield v. Judge of First Dist. Court of E. Middlesex, 262 Mass. 477, 482 (1928). See Massachusetts Ass’n of Minority Law Enforcement Officers v. Abban, supra. “In reviewing [the commission’s] action under G. L. c. 30A, § 14 (7), it was not open to the Superior Court judge to substitute [her] judgment for that of the commission.” Thomas v. Civil Serv. Comm’n, 48 Mass. App. Ct. 446, 451 (2000). The judge’s task was limited to determining whether the commission’s decision was supported by substantial evidence. See McIsaac v. Civil Serv. Comm’n, 38 Mass. App. Ct. 473, 476 (1995). Further, a judge is required to “give due weight to the experience, technical competence, and specialized knowledge of the agency, as well as to the *242discretionary authority conferred upon it.” Iodice v. Architectural Access Bd., 424 Mass. 370, 375-376 (1997), quoting G. L. c. 30A, § 14 (7). “This standard of review is highly deferential to the agency on questions of fact and reasonable inferences drawn therefrom.” Flint v. Commissioner of Pub. Welfare, 412 Mass. 416, 420 (1992).
Pursuant to G. L. c. 31, § 44, “we review the commission’s decision to determine if it violates any of the standards set forth in G. L. c. 30A, § 14 (7), and cases construing those standards.” Plymouth v. Civil Serv. Comm’n, 426 Mass. 1, 5 (1997). “[W]e are required to overturn commission decisions that are inconsistent with governing law.” Id. See Boston Police Superior Officers Fed’n v. Labor Relations Comm’n, 410 Mass. 890, 892 (1991) (considerable deference generally accorded agency decision, unless agency commits error of law). The party appealing from an administrative decision has the burden of proving its invalidity. See Coggin v. Massachusetts Parole Bd., 42 Mass. App. Ct. 584, 587 (1997).
4. Constitutional claims. We generally analyze claims of constitutional violations last because they need not always be considered. See Manor v. Superintendent, Mass. Correctional Inst., Cedar Junction, 416 Mass. 820, 824 (1994). Here, however, because of the importance of the constitutional issues, which are the focus of the present appeal, we discuss them first. The plaintiffs contend that the actions of the MBTA, with the approval of the HRD, in promoting minority and female police officers through the use of special certifications under Rule 10 violated the plaintiffs’ rights to equal protection under the Fourteenth Amendment and under the cognate provisions of the Massachusetts Declaration of Rights. The thrust of the plaintiffs’ argument is that the MBTA failed to provide a “strong basis in evidence” that it had engaged in prior discriminatory practices with respect to the promotion of officers, the effects of which continue to affect its police department, such that the MBTA was justified in bypassing the plaintiffs in favor of women and minority candidates for the positions of sergeant and lieutenant. The plaintiffs assert that, at the time of the present dispute, the MBTA was in full compliance with the minority hiring practices for civil service jobs set forth in Castro v. Beecher, 334 F. *243Supp. 930, 944 (D. Mass. 1971), afPd in part and rev’d in part, 459 F.2d 725, 735-737 (1st Cir. 1972), as evidenced by the fact that 24.6% of all police officers employed by the MBTA police department were minorities. Further, the plaintiffs take issue with the testimony of Dr. Cupingood, who opined that the MBTA had “underutilized” women in the police department. The plaintiffs claim that any such purported underutilization did not equate with past discrimination by the MBTA, necessitating the application of Rule 10 to promotion decisions. Accordingly, the plaintiffs argue that the promotions made from special certifications based on race and gender must be deemed unconstitutional and invalid. We disagree.
The Fourteenth Amendment to the United States Constitution provides, in relevant part, that no State shall “deny to any person within its jurisdiction the equal protection of the laws.” The standard for equal protection analysis under our Declaration of Rights is the same as under the Federal Constitution. See Dickerson v. Attorney Gen., 396 Mass. 740, 743 (1986); Commonwealth v. Franklin Fruit Co., 388 Mass. 228, 235 (1983). The equal protection clause of the Fourteenth Amendment does not require that every citizen be treated identically but, rather, that an adequate explanation be given for treating citizens differently. See Cotter v. Boston, 323 F.3d 160, 168 (1st Cir.), cert. denied, 540 U.S. 825 (2003).
As to the minority promotions, classifications based on race are inherently suspect and, therefore, are subject to strict judicial scrutiny. See Adarand Constructors, Inc. v. Pena, 515 U.S. 200, 227 (1995); Wygant v. Jackson Bd. of Educ., 476 U.S. 267, 273 (1986). See also Doe v. Commissioner of Transitional Assistance, 437 Mass. 521, 533 (2002). A classification based on race will be upheld where its proponents demonstrate that such classification serves a compelling State interest and is narrowly tailored to achieve that interest. See Richmond v. J.A. Croson Co., 488 U.S. 469, 493, 505 (1989) (city failed to demonstrate compelling interest in apportioning public contracts on basis of race); Wygant v. Jackson Bd. ofEduc., supra at 274. The burden of production is on the governmental actor to show that a racial classification is justified. See Cotter v. Boston, supra at 168 n.6. See also Adarand Constructors, Inc. v. Pena, supra at 224. *244However, the ultimate burden of persuasion rests on those challenging an affirmative action plan to demonstrate that it is unconstitutional. See Wygant v. Jackson Bd. of Educ., supra at 277-278.
The remedying of past racial discrimination is a compelling State interest as long as there is a “ ‘strong basis in evidence’ for the conclusion that the [affirmative action plan] at issue serves a remedial purpose with respect to past discrimination.” Stuart v. Roache, 951 F.2d 446, 450 (1st Cir. 1991), cert, denied, 504 U.S. 913 (1992), quoting Richmond v. J.A. Croson Co., supra at 500. See Wygant v. Jackson Bd. of Educ., supra at 277; Boston Police Superior Officers Fed’n v. Boston, 147 F.3d 13, 20 (1st Cir. 1998). “The ‘strong basis in evidence’ may consist of ‘either a contemporaneous or antecedent finding of past discrimination by a court or other competent body, or evidence approaching a prima facie case of a constitutional or statutory violation.’ ” Cotter v. Boston, supra at 169, quoting Boston Police Superior Officers Fed’n v. Boston, supra. The evidence of discrimination must be specific to the agency seeking to use a racial preference; discrimination in society as a whole will not support an affirmative action plan. See Wygant v. Jackson Bd. of Educ., supra at 274, 276. “[T]he relevant analysis in cases involving proof of discrimination by statistical disparity focuses on those disparities that demonstrate such prior governmental discrimination.” Id. at 274, citing Hazelwood Sch. Dist. v. United States, 433 U.S. 299 (1977). “In the absence of particularized findings, a court could uphold remedies that are ageless in their reach into the past, and timeless in their ability to affect the future.” Wygant v. Jackson Bd. of Educ., supra at 276. The question whether current remedial action is warranted by past discrimination is, accordingly, fact driven. See Boston Police Superior Officers Fed’n v. Boston, supra at 21. We turn, therefore, to consider whether there was a strong basis in evidence to show that the MBTA’s past employment practices discriminated against minorities, necessitating the implementation of Rule 10.
Racial discrimination in hiring practices pertaining to Massachusetts police officers has been established in prior judicial determinations. In Castro v. Beecher, 459 F.2d 725, 728, 735-*245736 (1st Cir. 1972), the court found that civil service examinations used before 1970 had a racially discriminatory impact on minorities who applied for jobs as police officers in various Massachusetts municipalities and agencies, including the MBTA. Nearly twenty years later, in Stuart v. Roache, supra at 452, the court recognized that discriminatory hiring practices, such as those found in Castro v. Beecher, supra, have a significant impact beyond the entry level, affecting the number of minorities who ultimately reach the supervisory level. “[Rjemedial action takes time, and discrimination may linger for many years in an organization that had excluded blacks from its ranks.” Stuart v. Roache, supra. See United States v. Paradise, 480 U.S. 149, 168-169 (1987) (plurality opinion) (where “[djiscrimination at the entry level necessarily precluded blacks from competing for promotions, and resulted in a departmental hierarchy dominated exclusively by nonminorities[,j . . . [the department cannot] segregate the results achieved by its hiring practices and those achieved by its promotional practices”); Boston Police Superior Officers Fed’n v. Boston, supra at 20-21. Cf. Clifton v. Massachusetts Bay Transp. Auth., 445 Mass. 611, 624 (2005) (MBTA management “must put an end to any legacy of discrimination that still pervades that authority”).
In May, 1996, minorities comprised 27.6% of the patrol officers employed by the MBTA police department (fifty officers out of 181).16 In comparison, minorities constituted only 7% of all sergeants in the MBTA police department (one sergeant out of fourteen), and there were no minority lieutenants. Of the sixteen highest ranking officers employed by the MBTA police department, comprised of nine lieutenants, one captain, five majors, and one chief, only one such officer (a major) was *246black, or 6.2% of this group. While the MBTA has made significant progress in remedying discriminatory hiring practices, as evidenced by the number of minority patrol officers, the effects of such discrimination linger at the supervisory level.17 See Cotter v. Boston, supra at 170-171 (“Past discrimination in the hiring of minorities has limited the opportunity for minorities to move up through the ranks, and recent statistics show that these effects remain”). If the MBTA had based its promotion decisions on strict rank order, as determined by scores on a competitive examination, the statistical disparity between minority and nonminority officers would have been even more pronounced. No black officers would have been promoted to fill any of the seven sergeant positions. Consequently, minorities would have comprised only 4.8% of all sergeants in the MBTA police department (one sergeant out of twenty-one). Contrary to the plaintiffs’ assertion, we do not think that the MBTA’s evidence of past racial discrimination is too temporally remote or statistically insignificant to justify a conclusion that such discrimination has had a readily apparent effect on the promotion of minority officers. Rule 10 serves a compelling State interest in remedying the ongoing effects of past racial discrimination. The plaintiffs have failed to make a sufficient demonstration to the contrary.
As to the promotions of women, classifications based on gender, unlike race, are not inherently suspect under the United States Constitution and, therefore, are subject only to an intermediate level of scrutiny. See United States v. Virginia, 518 U.S. 515, 532-533 & n.6 (1996). See also Clark v. Jeter, 486 U.S. 456, 461 (1988). To withstand such judicial scrutiny, the State must show that the challenged classification is “substantially related to an important governmental objective.” Id. See *247United States v. Virginia, supra at 533. In contrast, under the Massachusetts Equal Rights Amendment,18 “classifications on the basis of sex are subject to a degree of constitutional scrutiny ‘at least as strict as the scrutiny required by the Fourteenth Amendment for racial classifications.’ ” Attorney Gen. v. Massachusetts Interscholastic Athletic Ass’n, 378 Mass. 342, 354 (1979), quoting Commonwealth v. King, 374 Mass. 5, 21 (1977). Such classifications “will be upheld only if a compelling interest justifies the classification and if the impact of the classification is limited as narrowly as possible consistent with its proper purpose.” Lowell v. Kowalski, 380 Mass. 663, 666 (1980).
As alluded to in our discussion of racial discrimination, “absent explanation, it is ordinarily to be expected that nondiscriminatory hiring practices will in time result in a work force more or less representative of the . . . composition of the population in the community from which employees are hired.” International Bhd. of Teamsters v. United States, 431 U.S. 324, 339 n.20 (1977). A statistical imbalance, however, may be a telltale sign of discrimination. Id. “Where gross statistical disparities can be shown, they alone may in a proper case constitute prima facie proof of a pattern or practice of discrimination.” Hazelwood Sch. Dist. v. United States, 433 U.S. 299, 307-308 (1977) (analyzing statistical proof that school district engaged in pattern or practice of teacher employment discrimination in violation of Title VII of Civil Rights Act of 1964). See International Bhd. of Teamsters v. United States, supra at 339. Cf. Smith College v. Massachusetts Comm’n Against Discrimination, 376 Mass. 221, 228 & n.9 (1978) (discrimination may be inferred from disparate treatment of two groups as evidenced by statistical disparities). The United States Supreme Court has cautioned, however, that “statistics are not irrefutable; they come in infinite variety and, like any other kind of evidence, they may be rebutted.” International Bhd. of Teamsters v. United States, supra at 340. To the extent that a significant statistical disparity exists between the composition of a public employer’s workforce and the relevant qualified applicant pool, *248such employer has the requisite basis for believing that remedial action is necessary. See Cotter v. Boston, 323 F.3d 160, 170 (1st Cir. 2003). See also Wygant v. Jackson Bd. of Educ., 416 U.S. 267, 292 (1986) (O’Connor, J., concurring in part)19; Stuart v. Roache, 951 F.2d 446, 451 (1st Cir. 1991).
When, as here, special qualifications are necessary to fill particular jobs,20 “the relevant statistical pool for purposes of demonstrating discriminatory exclusion must be the number of [women] qualified to undertake the particular task.” Richmond v. J.A. Croson Co., 488 U.S. 469, 501-502 (1989). More specifically, the proper comparison is between the gender composition of the MBTA police department and the gender composition of the qualified police officer population in the relevant labor market. See Hazelwood Sch. Dist. v. United States, supra at 308. When evaluating discriminatory exclusion with respect to certain entry level positions or those requiring minimal training, a court need only view statistical comparisons between the composition of an employer’s workforce and the composition of *249the general population as probative of discriminatory practices. See Richmond v. J.A. Croson Co., supra at 501; International Bhd. of Teamsters v. United States, supra at 337-338.
As an initial matter, data from the 1990 census showed that women constituted 13.9% of the potential applicants in the labor force available for “protective service” positions. See note 12, supra. However, in 1996, women comprised only 6.6% of the patrol officers employed by the MBTA police department (twelve officers of 181). Women constituted 7% of all sergeants in the MBTA police department (one sergeant of fourteen), and 11% of all lieutenants (one lieutenant of nine). Of the sixteen highest ranking officers employed by the MBTA police department, comprised of nine lieutenants, one captain, five majors, and one chief, only two such officers (one lieutenant and one captain) were female, or 12.5% of this group. If the MBTA had based its promotion decisions on strict rank order, as determined by scores on a competitive examination, no female officers would have been promoted to fill any of the seven sergeant positions. Consequently, women would have constituted only 4.8% of all sergeants in the MBTA police department (one sergeant of twenty-one).
Following a remand to the commission, the MBTA had been allowed to present additional evidence as to the statistical disparity between the number of women who were qualified to be police officers in Massachusetts and the number of women who were employed by the MBTA police department. See note 13, supra. In his supplemental report setting forth a “Statistical Analysis of Female Representation in the MBTA Police Force,” Dr. Cupingood presented data for the years 1986, 1988, 1990, and 1994, as to the numbers of male and female applicants for police officer positions in Massachusetts, the numbers of those applicants who took the required examination, and the numbers of applicants who passed it. He opined that, based on his analysis of this data, women were, from a statistical perspective, significantly underrepresented in the MBTA police department. Women comprised 15.4% of the total applicants, 15.0% of the applicants who took the examination, and 14.8% of those who passed it. Dr. Cupingood applied the same statistical methodology as used in an earlier report, which was the *250same methodology relied on by the first Superior Court judge in her October 20, 1998, decision. Using the 14.8% benchmark of female availability (women who had passed the examination), Dr. Cupingood opined that there was an underrepresentation of 16.2 females in the MBTA police department in May, 1996.21 Dr. Cupingood stated that a shortfall of 16.2 women was significant and corresponded to a statistical disparity of 3.14 standard deviations, meaning that there was approximately a one in 600 probability that such occurrence had happened by chance. This statistical disparity of 3.14 standard deviations exceeded the benchmark of three standard deviations adopted by the first Superior Court judge as the measure for a “strong basis in evidence” of past discrimination.
Taking a broader view of hiring practices by the MBTA police department, Dr. Cupingood opined that, using the 14.8% benchmark of female availability, there was a statistically significant shortfall of 18.2 women hired from 1986 to 1996.22 A shortfall of 18.2 females corresponded to a statistical disparity of 3.78 standard deviations, meaning that there was approximately a one in 6,000 probability that such occurrence had happened by chance. Again, this statistical disparity of 3.78 standard deviations exceeded the three standard deviations benchmark adopted by the first Superior Court judge. The plaintiffs did not challenge Dr. Cupingood’s statistical methodology or present any witnesses to discredit it.
As to promotional decisions made by the MBTA police department, Dr. Cupingood opined that the data with respect to the positions of sergeant and lieutenant was simply too small to be amenable to statistical analysis. This deficiency, however, was not fatal to the MBTA’s case for showing that a gender classification was justified. Dr. Cupingood still opined that the *251small number of women occupying the positions of sergeant and lieutenant was consistent with a pattern of underutilization of women in the MBTA police department. Moreover, as we discussed in our analysis of racial discrimination, to the extent that women were not being hired in statistically significant numbers by the MBTA police department in the first instance, there was no way that they could be promoted in statistically significant numbers. See United States v. Paradise, 480 U.S. 149, 168 (1987); Boston Police Superior Officers Fed’n v. Boston, 147 F.3d 13, 20-21 (1st Cir. 1998); Stuart v. Roache, 951 F.2d 446, 452 (1st Cir. 1991).
We conclude that the MBTA’s evidence of gross statistical disparities between the number of women who were qualified to be police officers in Massachusetts and the number of women who were hired for positions in the MBTA police department constituted prima facie evidence of a past practice of discrimination. The chief of the MBTA police department had the requisite firm basis for believing that remedial action was necessary. As such, Rule 10 served a compelling State interest in addressing the ongoing effects of past gender discrimination. The plaintiffs have failed to meet their ultimate burden of demonstrating that the implementation of Rule 10, permitting special certifications based on gender, was not justified.
Once the proponents of an affirmative action plan show that it serves a compelling State interest, a court then considers whether such remedial action is narrowly tailored to address the particular harm at issue. See Wygant v. Jackson Bd. of Educ., supra at 274; Stuart v. Roache, supra at 453. See also Lowell v. Kowalski, supra at 666. Narrow tailoring ensures that the “means chosen ‘fit’ [the government’s] compelling goal so closely that there is little or no possibility that the motive for the classification was illegitimate . . . prejudice or stereotype.” Richmond v. J.A. Croson Co., supra at 493. We analyze several factors in determining whether a remedial measure is narrowly tailored, including “the extent to which (i) the beneficiaries of the order are specially advantaged; (ii) the legitimate expectancies of others are frustrated or encumbered; (iii) the order interferes with other valid state or local policies; and (iv) the order contains (or fails to contain) built-in mechanisms which *252will, if time and events warrant, shrink its scope and limit its duration.” Boston Police Superior Officers Fed’n v. Boston, supra at 23, quoting Mackin v. Boston, 969 F.2d 1273, 1278 (1st Cir. 1992), cert, denied, 506 U.S. 1078 (1993).
The plaintiffs contend that Rule 10 is not narrowly tailored to redress purported discrimination against minorities and women in the MBTA police department because the evidence did not demonstrate that those groups, in fact, had suffered discriminatory treatment in the past. We disagree and conclude that Rule 10 was narrowly tailored to satisfy the MBTA’s compelling interest in eliminating racial and gender discrimination in the hiring and promotion practices of its police department. When evaluating an affirmative action plan, the fact that only qualified minority and female candidates, meaning those who have passed the required examinations, are “specially advantaged” is an “important indicium of narrow tailoring.” Mackin v. Boston, supra. See Stuart v. Roache, supra at 454. It suggests that promotions are not being made illegitimately or gratuitously to achieve racial or gender balance. See United States v. Paradise, supra at 177-178 (relief appropriate where qualified candidates promoted and State agency not obligated to make unnecessary or gratuitous promotions). Here, the police officers on the eligible lists for minority and female candidates had passed the required examinations and, thus, were qualified for promotion. See G. L. c. 31, § 65. See also Boston Police Superior Officers Fed’n v. Boston, supra at 24. The plaintiffs have not asserted that the officers who were promoted to the ranks of sergeant and lieutenant were unqualified to perform the responsibilities of those jobs. See id. See also Stuart v. Roache, supra.
All of the MBTA police officers who hoped to be promoted were competing for a limited number of positions. The nature of competition is such that success is not automatic, but typically subject to a combination of variables. See Boston Police Superior Officers Fed’n v. Boston, supra (expectation of promotion not firmly rooted where more candidates than available vacancies). See also Cotter v. Boston, 323 F.3d 160, 171 (1st Cir. 2003); Mackin v. Boston, supra at 1278. The civil service system does not guarantee an applicant a promotion “at a certain *253date or upon a specific event.” Bielawski v. Personnel Adm’r of the Div. of Personnel Admin., 422 Mass. 459, 466 (1996). See Callanan v. Personnel Adm’r for the Commonwealth, 400 Mass. 597, 601 (1987) (“individuals do not have a vested right in their particular positions on the eligibility list once it is established”); Bums v. Sullivan, 619 F.2d 99, 104 (1st Cir.), cert, denied, 449 U.S. 893 (1980) (expectation of promotion not property interest entitled to constitutional protection where subjective factors may be considered in addition to written examination score). The civil service system confers only limited rights on those individuals, like the plaintiffs, who pass the relevant examinations, but it gives broad discretion to the HRD to establish eligibility lists for promotion. See Callanan v. Personnel Adm’r for the Commonwealth, supra. Thus, while the plaintiffs have claimed that they were entitled to be promoted, and undoubtedly hoped that they would be, there was simply no guarantee that they would be successful in that endeavor.
We add that no valid policies have been disturbed by promoting qualified minorities and women to higher ranks within the MBTA police department. “While Massachusetts law requires an explanation for promotions made outside of strict rank order, there is no prohibition on such out-of-rank decisions.” Cotter v. Boston, supra at 171-172. See G. L. c. 31, § 27. Finally, no quotas with respect to promotions were established under Rule 10, and the MBTA’s affirmative action program expired on January 1, 2000. Therefore, Rule 10 was limited in scope and duration. See Mackin v. Boston, supra (duration of affirmative action plan is significant factor in determining whether remedial action is overly broad). Rule 10 was a narrowly tailored means of addressing racial and gender discrimination in the hiring and promotion practices of the MBTA police department. The plaintiffs have failed to satisfy their ultimate burden of demonstrating that Rule 10 violated their rights to equal protection under the United States Constitution and the Massachusetts Declaration of Rights.
5. Validity of Rule 10. The plaintiffs contend that there is no statutory authority for the existence of Rule 10. The thrust of their argument is that nothing in G. L. c. 31 suggests a legislative intent to confer on the HRD the authority to create special *254classifications for promotions within the civil service. We disagree and conclude that, contrary to the plaintiffs’ assertions, the HRD had the statutory authority, pursuant to G. L. c. 31, § 3, to certify separate lists of candidates for promotion, based on race and gender, as part of the MBTA’s affirmative action plan.
The power of an administrative agency to promulgate rules or regulations is conferred by the Legislature. See Borden, Inc. v. Commissioner of Pub. Health, 388 Mass. 707, 721, cert, denied sub nom. Formaldehyde Inst., Inc. v. Frechette, 464 U.S. 936 (1983). With respect to the present action, G. L. c. 31, § 3, states that the “administrator [of the HRD] shall make and amend rules which shall regulate the recruitment, selection, training and employment of persons for civil service positions.” These rules shall include provisions for “[promotional appointments, on the basis of merit as determined by examination, performance evaluation, seniority of service or any combination of factors which fairly test the applicant’s ability to perform the duties of the position as determined by the administrator” (emphasis added). G. L. c. 31, § 3 (e).
“An administrative agency . . . has considerable leeway in interpreting a statute it is charged with enforcing, and regulations adopted by the agency stand on the same footing as statutes, with reasonable presumptions to be made in favor of their validity.” Student No. 9 v. Board of Educ., 440 Mass. 752, 762-763 (2004). See Foss v. Commonwealth, 437 Mass. 584, 587 (2002) (court will give “due weight and deference to an agency’s interpretation of statutes within its charge”). Such deference “is necessary to maintain the separation between the powers of the Legislature and administrative agencies and the powers of the judiciary.” Borden, Inc. v. Commissioner of Pub. Health, supra at 723. When considering a challenge to a rule or regulation promulgated by an administrative agency, a court generally needs to find only “some rational relation between the regulation and the empowering statute” to uphold its validity. White Dove, Inc. v. Director of the Div. of Marine Fisheries, 380 Mass. 471, 477 (1980). See Borden, Inc. v. Commissioner of Pub. Health, supra. The burden is on the party challenging the rule or regulation to show that there is no *255conceivable basis on which it can be upheld. See Arthur D. Little, Inc. v. Commissioner of Health & Hosps. of Cambridge, 395 Mass. 535, 553 (1985); Purity Supreme, Inc. v. Attorney Gen., 380 Mass. 762, 776 (1980). “A court will not declare a regulation void unless its provisions cannot, in any appropriate way, be interpreted in harmony with the legislative mandate.” Student No. 9 v. Board of Educ., supra at 763. See Massachusetts Fed’n of Teachers, AFT, AFL-CIO v. Board of Educ., 436 Mass. 763, 771 (2002).
Rule 10 states that, once specified conditions are met, the HRD may certify the names of a designated number of individuals, who are members of a protected group and are on an eligible list for one or more available positions, in order of their standing. It does not dispense with the statutory requirement that an applicant take a promotional examination in the first instance, or with consideration of the applicant’s performance on such examination. See G. L. c. 31, § 65 (promotional appointments in MBTA police department “shall be made only after competitive examination”). It merely allows a candidate’s membership in a protected group to be considered along with the regular criteria for promotion in circumstances where it has been determined that prior practices of the appointing authority have discriminated against that protected group. Nothing in G. L. c. 31, § 3 (<?), mandates that promotions be made in strict rank order based only on examination results. Rather, the statute allows consideration of “any combination of factors which fairly test the applicant’s ability to perform the duties of the position as determined by the administrator.”23 Id. See G. L. c. 31, § 27 (allowing promotion of qualified person other than one at top of eligibility list where appointing authority files written statement of reasons for such appointment). Cf. G. L. c. 31, § 21 (allowing HRD to limit eligibility for any examination for original appointment to either men or women “if the duties and responsibilities of such position require special physi*256cal or medical standards or require custody or care of a person of a particular sex”). Moreover, this court acknowledged the commission’s approval of Rule 10 in Massachusetts Ass’n of Minority Law Enforcement Officers v. Abban, 434 Mass. 256, 264 (2001) (“civil service law allows consideration of race in promotion decisions as part of an approved affirmative action plan”).
Here, both the commission and the first Superior Court judge concluded that Rule 10 was a valid exercise of the HRD’s rule making authority under G. L. c. 31, § 3, and we agree. On the record before us, we have no difficulty discerning a rational relation between Rule 10 and the empowering statute where the evidence demonstrates that prior practices of the MBTA police department resulted in discriminatory treatment of minorities and women. “A regulation is essentially an expression of public policy.” Borden, Inc. v. Commissioner of Pub. Health, supra at 721. Rule 10 was specifically designed to ensure that civil service hiring and promotion procedures did not run afoul of constitutional and statutory prohibitions against unlawful racial and gender discrimination. The plaintiffs have not come close to showing that the promulgation of Rule 10 was beyond the statutory authority conferred on the HRD by G. L. c. 31, § 3.
6. Compliance with Rule 10. The plaintiffs next contend that the MBTA and the HRD failed to satisfy the procedural requirements of Rule 10. They argue that when the chief of the MBTA police department requested special certifications based on race and gender, he submitted to the HRD only statistical data of the racial and gender composition of the MBTA’s police force as of May, 1996. The plaintiffs point out that, pursuant to Rule 10 (1) (b), special certifications in the civil service shall be made when, inter alla, the HRD “has made a written determination substantiating that previous practices of the . . . appointing authority with respect to the filling of [particular] positions have discriminated against members of a group, hereinafter referred to as a protected group, on the basis of race, color, sex, or national origin.” The plaintiffs assert that the mere requests by the MBTA police department for special certifications, together with the HRD’s pro forma approval of such requests, did not constitute a written determination that the *257MBTA police department had engaged in prior discriminatory practices against minorities and women with respect to promotional appointments. Therefore, the plaintiffs continue, the approval of such special certifications was unlawful, and the promotions based on eligible lists of minority and female candidates were invalid. We disagree.
The commission found, and the first Superior Court judge agreed, that the HRD’s written approval of the requests for special certifications and the posting of notices of intention to certify eligible lists of minority and female candidates constituted a good faith attempt by the HRD and the MBTA to comply with Rule 10. Further, the commission concluded that there was “no suggestion or evidence in the record to show that the processing of these selective certifications was in any manner different from accepted procedure.” A reviewing court “shall give due weight to the experience, technical competence, and specialized knowledge of the agency, as well as to the discretionary authority conferred upon it.” G. L. c. 30A, § 14 (7). See Flynn v. Civil Serv. Comm’n, 15 Mass. App. Ct. 206, 210 (1983) (decision by commission that civil service rule not violated entitled to weight). See also Amherst-Pelham Regional Sch. Comm. v. Department of Educ., 376 Mass. 480, 491 (1978).
We recognize that the HRD did not issue a written determination specifically enunciating that past hiring and promotion practices of the MBTA police department had resulted in discriminatory treatment of minorities and women. Nonetheless, we conclude that the statistical evidence presented by the MBTA, which was unrebutted and persuasive, and the HRD’s written approval of the requests for special certifications were sufficient to satisfy the technical requirements of Rule 10. The plaintiffs have not demonstrated that the commission’s decision was unlawful or that their substantial rights have been prejudiced in this regard. See G. L. c. 30A, § 14 (7). Moreover, there is no evidence that meticulous compliance with the technical provisions of Rule 10 would have resulted in a different outcome for the plaintiffs.
7. Conclusion. The judgment of the first Superior Court judge, *258dated October 20, 1998, is affirmed. The judgment of the second Superior Court judge, dated November 21, 2003, is also affirmed.
So ordered.
Prior to July 2, 1998, the human resources division of the Executive Office for Administration and Finance was known as the Department of Personnel Administration. See St. 1998, c. 161, § 234.
The MBTA considers “minority” officers to be those who are black and Hispanic.
General Laws c. 31, § 1, defines “ [certification” as “the designation to an appointing authority by the administrator of sufficient names from an eligible list or register for consideration of the applicants’ qualifications for appointment pursuant to the personnel administration rules.” An “[appointing authority” is “any person, board or commission with power to appoint or employ personnel in civil service positions.” Id. Here, the appointing authority was the chief of police of the MBTA police department. The “[ajdministratar” is “the personnel administrator of the human resources division within the executive office for administration and finance.” Id. An “[eligible list” is defined as “a list established by the administrator, pursuant to the civil service law and rules, of persons who have passed an examination ... or any other list established pursuant to the civil service rules from which certifications are made to appointing authorities to fill positions in the official service.” Id. The “[ojfficial service” is “the composite of all civil service positions not in the labor service.” Id.
The “regular” list was comprised of individuals who were not members of a protected group.
William Fleming received an examination score of 82.00, and Nancy O’Loughlin received a score of 78.00. William Brackett, the plaintiff who was not promoted to the position of lieutenant, received an examination score of 82.00.
The individuals who were promoted to the position of sergeant received the following examination scores: Salvatore Venturelli received a score of 86.00; Michael Morris received a score of 86.00; Anne McCall received a score of 79.00; Irene Reardon received a score of 79.00; Herman Wheeler received a score of 79.00; Michael Lopes received a score of 77.00; and Gloria Andrews received a score of 76.00. The plaintiffs who were not promoted to the position of sergeant received the following examination scores: Robert Vitale received a score of 84.00; Steven Douglas received a score of 82.00; Robert Fitzsimmons received a score of 80.00; Joseph Lyons received a score of 80.00; Joseph O’Connor received a score of 80.00; and Peter Roy received a score of 80.00.
General Laws c. 31, § 44, provides, in pertinent part: “Any party aggrieved by a final order or decision of the [civil service] commission following a hearing . . . may institute proceedings for judicial review in the superior court within thirty days after receipt of such order or decision. Any proceed*238ings in the superior court shall, insofar as applicable, be governed by the provisions of [G. L. c. 30A, § 14] . . . .”
The plaintiffs did not present any additional evidence at this hearing *239before the commission.
“Protective service” positions encompass several occupational categories, including police officers, fire fighters, and correctional institution officers.
In their brief to this court, the plaintiffs do not challenge the admission of this additional evidence.
According to the testimony of Dr. Cupingood, “statisticians measure how far away an actual outcome is from what’s expected in statistical units called standard deviations. A standard deviation of two approximately corresponds to a probability of occurrence of five percent.” “Two standard deviations is normally enough to show that it is extremely unlikely (that is, there is less than a 5% probability) that the disparity is due to chance, giving rise to a reasonable inference that the hiring was not race-neutral; the more standard deviations away, the less likely the factor in question played no role in the decisionmaking process.” Adams v. Ameritech Servs., Inc., 231 F.3d 414, 424 (7th Cir. 2000). See Hazelwood Sch. Dist. v. United States, 433 U.S. 299, 308 n.14 (1977). Dr. Cupingood opined that three standard deviations, the benchmark being used in this case, “is particularly rare, and might occur only three times in a thousand.” The plaintiffs have not challenged the adoption of this three standard deviations benchmark by the first Superior Court judge, and as such, it is the law of the case. See Peterson v. Hopson, 306 Mass. 597, 599 (1940) (“Where there has been no change of circumstances, a court or judge is not bound to reconsider a case, an issue, or a question of fact or law, once decided”).
We acknowledge the amicus brief filed by the Massachusetts Association of Minority Law Enforcement Officers.
When the chief of the MBTA police department requested special certifications based on race and gender, he supported such requests with statistical data setting forth the demographics of the MBTA police department in May, 1996, and September, 1996. In proceedings before the Superior Court, the respective judges found that the statistical data from May, 1996, varied only slightly from the statistical data from September, 1996. We rely here on the statistics from May, 1996, because they appear to be more detailed and comprehensive, and because any differences are not significant enough to be material.
We note that on February 6, 1997, the MBTA entered into an equal employment opportunity agreement and compliance program with the Attorney General’s office and numerous labor organizations, pursuant to which the MBTA agreed to “take effective action to prevent and eliminate discrimination and harassment on the basis of race, color, national origin, ancestry, sex, sexual orientation, religious creed, age and disability with regard to employment or the terms and conditions of employment including, but not limited to, promotions, terminations, transfers, job assignments and discipline” (emphasis added).
Article 1 of the Massachusetts Declaration of Rights, as amended by art. 106 of the Amendments, provides in part: “Equality under the law shall not be denied or abridged because of sex, race, color, creed or national origin.”
In her concurring opinion in Wygant v. Jackson Bd. of Educ., 476 U.S. 267, 286 (1986), Justice O’Connor pointed out that “[t]he Court is in agreement that. . . remedying past or present racial discrimination by a state actor is a sufficiently weighty state interest to warrant the remedial use of a carefully constructed affirmative action program. This remedial purpose need not be accompanied by contemporaneous findings of actual discrimination to be accepted as legitimate as long as the public actor has a firm basis for believing that remedial action is required.”
General Laws c. 31, § 6, provides that each “original appointment in the official service shall be made after certification from an eligible list established as the result of a competitive examination for which civil service employees and non-civil service employees were eligible to apply.” Further, pursuant to G. L. c. 31, § 8, “[a]n appointing authority, with the approval of the administrator, may promote an employee in the official service from one title to the next higher title in the same series in the same departmental unit, subject to the following requirements: (a) such employee shall have been employed in such unit in such lower title as a permanent employee for three years or longer immediately preceding the request for approval of such promotion; (b) such employee is the first, second, or third ranking employee in length of service in such departmental unit who is willing to accept the promotion; and (c) such employee shall demonstrate, by the passing of an examination prescribed by the administrator, that he possesses the qualifications and abilities necessary to perform the duties of the higher title.” See G. L. c. 31, § 64 (specific qualifications for employment as MBTA police officer); G. L. c. 31, § 65 (specific qualifications for promotion in MBTA police department).
This underrepresentation of 16.2 females was calculated as follows. As of May, 1996, the number of police officers in all ranks of the MBTA police department totaled 211 officers. Based on a female availability of 14.8% it would be expected that 31.2 women would he employed by the MBTA police department. However, only fifteen women actually were so employed.
This underrepresentation of 18.2 females was calculated as follows. From 1986 to 1996, the MBTA police department hired 184 individuals for the position of patrol officer. Based on a female availability of 14.8% it would be expected that 27.2 women would be hired by the MBTA police department. However, only nine women were actually hired.
Prior to 1982, the only criteria that were considered by the HRD in making promotional appointments were the examination and seniority of service. See St. 1978, c. 393, § 11. In 1981, the Legislature amended G. L. c. 31, § 3 (e), to expand the criteria for promotional appointments, suggesting an intent to look beyond statistically quantifiable measures of an applicant’s merit. See St. 1981, c. 767, § 12.