These are consolidated actions for declaratory and injunctive relief, both brought by taxpayers of the city of Woburn (city) against the city’s board of assessors (board) and tax collector. The Chomerics case was brought on March 23,1973, and the Globe Ticket case was brought on March 14,1974. Both actions seek relief from allegedly disproportionate and discriminatory assessment of the taxpayers’ real estate.
In June, 1973, a judge of the Superior Court ordered that a third case (Clifford J. Martin & others us. Leo C. Keating & others, Superior Court, Middlesex County, Eq. 35141 [1973]), also attacking the assessment practices by the board, be consolidated with the Chomerics case. In September, 1973, a final decree (which apparently has not been appealed and which is not before us) was entered in the Martin case. That decree declared that the board had a constitutional and statutory duty to assess taxable real property in Woburn at its full and fair cash value and ordered the board to "file in the Superior Court within thirty days ... a comprehensive plan for the revaluation at full and fair cash value ... of all taxable property in Woburn in an orderly manner and with all deliberate speed, such revaluation to take effect as of July 1, 1975.”2 On the same day that this decree was entered, the parties to the Chomerics case entered into a stipulation which permitted the board and the tax collector to proceed with the then temporarily enjoined collection of the 1973 taxes and which provided that "[p]ending final determination of [the plaintiffs’] assessed valuations for 1973 and subsequent years, each of the named [plaintiffs] will pay an amount equal to 50% of their 1972 tax bills, and [the defendants] will not take any action to collect any unpaid balance claimed by them to be due from the named [plaintiffs].”
*396A demurrer in the Chomerics case was overruled on July 9,1973, and a motion to dismiss in the Globe Ticket case was denied on July 2,1975: Evidence was presented in the Superior Court, and the parties rested with respect to the issues whether the board had engaged in an illegal and discriminatory policy of tax assessment to the plaintiffs’ detriment and, if so, what percentages should be applied to the full and fair cash values of the plaintiffs’ properties to determine the taxes due for the years 1972 and 1973.3 The judge filed findings of fact and rulings of law (Mass.R. Civ.P. 52[a], 365 Mass. 816 [1974]) on those issues. In accordance with those findings and rulings a “Partial Final Judgment” (Mass.R.Civ.P. 54[b], 365 Mass. 821 [1974]) was entered in each case which declared that since at least 1972 the defendants’ assessment practices had violated the Constitution and the laws of the Commonwealth, resulting in excessive assessed valuations of the plaintiffs’ properties, and that the plaintiffs’ properties should be assessed at no more than 25% and 23% of their full and fair cash value for the years 1972 and 1973 respectively. In addition, the partial final judgments authorized the defendants, in accordance with these declarations, to issue amended tax bills for 1972 and the tax years thereafter, ordered the defendants to refund any tax payments the plaintiffs had made in excess of the amended tax bills, and ordered the plaintiffs to pay any deficits by which the tax bills exceeded the amounts already paid by the taxpayers. All parties filed notices of appeal from the partial final judgments; however, since none of the defendants has filed a brief, we deem their appeals to have been waived. Mass.R.A.P. 16 (a) (4), as amended, effective February 24, 1975, 367 Mass. 921 (1975).
*397Accordingly, the only issue before us is that raised by the plaintiffs.4 We summarize some of the facts and proceedings to indicate the scope of the plaintiffs’ contention. The plaintiffs are all owners of commercial and industrial properties in the city. Beginning in 1972 the board, employing a professional appraiser,, regularly and periodically appraised all commercial and industrial properties and assessed them at approximately 40% of their full and fair cash value. At the same time residential property, whether built upon or vacant, was revalued only when sold in an arm’s length sale. The sale price was taken by the assessors as determinative of the full and fair cash value, and the property was thereafter assessed at 40% of that value. Until residential properties were sold their valuations remained unchanged. The judge found that this difference in method of valuation created two classes of property and discriminated against business and commercial property. He determined that the median level at which the class of residential property was assessed was *39825% of the full and fair cash value in 1972 and 23% of the full and fair cash value in 1973.5 Applying the rule that a taxpayer who has been a victim of discrimination is entitled to a reduction in his tax assessment "which will make ... [his] assessment proportional to other assessments”6 (Shoppers’ World, Inc. v. Assessors of Framingham, 348 Mass. 366, 377-378 [1965]; First Natl. Stores, Inc. v. Assessors of Somerville, 358 Mass. 554, 559 [1971]; Coomey v. Assessors of Sandwich, 367 Mass. 836, 838 [1975]; Assessors of Kingston v. Sgarzi, 367 Mass. 840, 843 [1975]), the judge, as indicated above, ordered that the 1972 and 1973 assessed valuations of the plaintiffs’ properties be reduced to 25% and 23% respectively of the properties’ full and fair cash values.
The plaintiffs have no quarrel with this analysis so far as it goes. They argue, however, that vacant residential property in the city comprises a third class of property for assessment purposes with a lower level of assessment than that of built upon residential property and that note 10 of the Shoppers’ World case (348 Mass. at 377-378) requires reduction to the level of assessment for that class.7 They derive the level of assessment for each of the *399years 1972 and 1973 by taking the vacant residential properties sold during each of those years and comparing the sale prices of those properties with their assessed valuations during the year of sale. They view the series of ratios of assessed valuation to sale price, thus derived, as a representative sample of all the assessment ratios for vacant residential property in the city and the medians of the series for each of the years 1972 and 1973 — 9% % for 1972 and 10%% for 1973 — as representative of a common level of assessment ratios of all vacant residential property during those years.* ****8
*400The plaintiffs contend that the judge was compelled as a matter of law to accept each of these medians as representative of the level of assessment ratios of vacant residential properties and to use these medians as a basis for reducing the plaintiffs’ taxes for the years before a citywide revaluation.
The issue is thus quite narrow. The plaintiffs’ contention cannot succeed unless the medians are representative of the level of assessment. But neither we nor the trial judge has, so far as the record appendix indicates, been given any statistical analysis of these series of ratios of assessed valuation to sale price which might demonstrate their representative character or the representative character of the medians. We have been told nothing of the number of parcels of vacant residential property in the city and so cannot tell whether (and certainly cannot say as a matter of law that) the number of sales was adequate to yield a representative sample. See Sudbury v. Commissioner or Corps. and Taxn., 366 Mass. 558, 560-561 (1974). Nor have we been given any expert analysis of the pattern of the ratios of assessed valuation to sale price contained in the two series which might shed light on whether those ratios cluster about the medians to a sufficient extent to warrant a finding that vacant residential properties constitute a most favored class of the kind referred to in footnote 10 of the Shoppers’ World case.9 See Beardsley v. Assessors of Foxborough, 369 Mass. *401855, 859 &n.6 (1976). A mere inspection of the two series set out in the plaintiffs’ brief, apart from any statistical analysis, indicates widely disparate assessment ratios with no discernible cluster around the medians. See Appeals of Kents 2124 Atlantic Ave. Inc., 34 N.J. 21, 27 (1961). Contrast Bettigole v. Assessors of Springfield, 343 Mass. 223, 225-226 (1961), referred to in note 10 of the Shoppers’ World case, 348 Mass. at 377, and Beardsley v. Assessors of Foxborough, 369 Mass. at 857, in both of which the most favored class had an ascertained common level of assessment.
Nor is this a case where the plaintiffs remain without any remedy if these series are not utilized. Cf. Appeals of Kents 2124 Atlantic Ave. Inc., 34 N.J. at 30. The reduction of the plaintiffs’ assessed valuations to the 25% and 23% medians of the assessment ratios of residential property sold in 1972 and 1973 (see note 5, supra) — also the medians, as computed in the plaintiffs’ brief, of the assessment ratios for all property sold in those years — would seem to be an equitable remedy, since adjustment on this basis ”assure[s] that each taxpayer [will] bear ... his proportionate share of the tax burden” (Coomey v. Assessors of Sandwich, 367 Mass. at 837) more nearly than if his taxes *402were reduced to the 9%% and 10%% medians. If the latter reductions were made, the ratios of the 1972 and 1973 assessed valuations of the plaintiffs’ properties to their full and fair cash value would be less than half the medians of the assessment ratios of all properties sold in each of those years. The ratios of the 1972 assessed valuations of the plaintiffs’ properties to their full, fair cash value would be less than those of 94.3% of all the properties sold in 1972; the ratios of the 1973 assessed valuations of the plaintiffs’ properties to their full, fair cash values would be less than those of 90.7% of all the properties sold in 1973. (See note 8, part B, supra.) It is evident that such a reduction would only aggravate the existing inequality in the city’s 1972 and 1973 assessments.
We need not decide whether note 10 of the Shoppers’ World case, supra, would require such a result if we were bound to accept the 9% % and 10% % medians as representative of a common level of assessment arising from action taken by the assessors intended to circumvent the constitutional and statutory requirements or from the natural and probable consequence of action by the assessors (Leto v. Assessors of Wilmington, 348 Mass. 144, 148 n.5 [1964]; Shoppers’ World, Inc. v. Assessors of Framingham, 348 Mass. at 377) and if, as here, the result would collide with the principle that each piece of property should so far as possible bear an equal share of the tax burden and the city’s "taxpayers ... should not be compelled to suffer in effect a forfeiture to ... [the plaintiffs] of tax revenues to which the city is entitled.”10 Assessors *403of Lynn v. Shop-Lease Co., 364 Mass. 569, 572 (1974). Here, as we have seen, we are not required to accept the 9%% and 10%% medians as the indices of assessment ratios of a class of vacant land.11
Judgments affirmed.
According to the plaintiffs’ brief, the July 1,1975, deadline for the revaluation of all property in Woburn has, on the board’s motion, been extended "so as to delay the revaluation until issuance of the tax bills for the 1976-1977 tax year.”
The parties have stipulated to the full and fair cash value of certain of the plaintiffs’ properties but the determination of the remainder has been referred to a master. They have also stipulated that the percentage to be used in determining the plaintiffs’ subsequent assessed valuations at issue shall not exceed the percentage used to calculate their 1973 assessed valuations.
The various stipulations, which we have examined, and the failure of the defendants to appeal the overruling of their demurrer and the denial of their motion to dismiss (see Leto v. Assessors of Wilmington, 348 Mass. 144, 148-149 [1964]) have eliminated any question whether the claim for reduction of1972 taxes by the plaintiffs in the Chomerics case brought in March, 1973, and the claim for reduction of1973 taxes by the plaintiffs in the Globe Ticket case brought in March, 1974, were moot or otherwise barred (see Carr v. Assessors of Springfield, 339 Mass. 89, 92 [1959], Second Church in Dorchester v. Boston, 343 Mass. 477, 478-479 [1962], and Gallo v. Division of Water Pollution, 374 Mass. 278, 287-288 [1978]), and whether claims for reductions in taxes for other years should more appropriately have been brought before the Appellate Tax Board pursuant to G. L. c. 59, §§ 59,64, and 65. See Shoppers’ World, Inc. v. Assessors of Framingham, 348 Mass. 366, 377-378 (1965); Sears, Roebuck & Co. v. Somerville, 363 Mass. 756 (1973). Cf. Bettigole v. Assessors of Springfield, 343 Mass. 223, 237 (1961); Coan v. Assessors of Beverly, 349 Mass. 575, 577 (1965). Cf. also Sydney v. Commissioner of Corps. and Taxn., 371 Mass. 289, 293-294 (1976) (citing Squantum Gardens, Inc. v. Assessors of Quincy, 335 Mass. 440, 443 [1957]), and S.J. Groves & Sons v. State Tax Commn., 372 Mass. 140, 142 (1977), holding that in the tax field a judge may exercise discretion as to whether to entertain a declaratory action in spite of the plaintiff’s failure to exhaust his administrative remedies.
The judge’s finding appears to be with reference to all residential property both built upon and vacant. The tables in the appendix to the plaintiffs’ brief indicate the same medians for built upon residential property. Any discrepancy thus appears to be immaterial. We note that the median ratios for all property sold during 1972 and 1973 are also 25% and 23% respectively.
This rule rests on the principle, established by Sioux City Bridge Co. v. Dakota County, Nebraska, 260 U.S. 441, 446 (1923), that "where it is impossible to secure both the standard of the true value, and the uniformity and equality required by law, the latter requirement is to be preferred as the just and ultimate purpose of the law.”
Note 10 of the Shoppers’ World case states: "If, as in the Bettigole case, 343 Mass. 223, 227 [1961], it should be shown that several different percentages of full, fair cash value were employed in valuing different classes of property, the principle discussed in the Sioux City Bridge Co. case [see n. 6] would logically require reduction of the assessment of a taxpayer against whom there had been discrimination so that such taxpayer’s assessment would be proportional to the as*399sessments of the class of property valued at the lowest percentage of fair cash value. It is not necessary for us now to consider whether patterns of illegal and discriminatory assessment may exist which are so complicated as to require the use of remedies which would treat the whole tax levy as invalid.”
A. The series of ratios of assessed valuation to sale price for 1972 and 1973 which the plaintiffs ask us to accept are set out in an appendix to their brief. The 1972 series contains 10 ratios; the 1973 series contains 28 ratios. The two series may be summarized as follows:
Ratio of assessed valuation Percent of vacant resident-in year of sale to sale ial properties sold in price 1972 and 1973 having the indicated ratios
1972 sales 1973 sales
1% - 5% 20% 7.1%
6% -10% 40% 42.9%
11% -15% 0 17.9%
16%-20% 20% 0
21% - 25% 0 10.7%
26% - 30% 10% 0
31% - 35% 10% 3.6%
36%-40% 0 7.1%
over 40% 0 10.7%
B. For comparison purposes the corresponding series of ratios of assessed valuation to sale price for all properties sold in 1972 and 1973 may be summarized as follows:
Ratio of assessed valuation Percent of all properties sold in year of sale to sale in 1972 and 1973 having price the indicated ratios
*4001972 sales 1973 sales
l%-5% 4.3% 5%
6% -10% 2.4% 4.3%
11% -15% 2.2% 5.8%
16%-20% 17.5% 19%
21% - 25% 26.8% 27.1%
26% - 30% 22.7% 12.8%
31%-35% 9.1% 11.3%
36% - 40% 3.6% 5.5%
over 40% 11.5% 9.3%
The 1972 series contains 418 ratios; the 1973 series contains 399 ratios.
The general statement by a witness who qualified as a statistician that "the median is normally a better definition of central tendency *401[than the mean]” is too general to be of assistance. Further, some doubt is cast on the significance of the medians of our series when we apply the method which has been used by the Bureau of the Census to measure uniformity of assessment. For this purpose a "coefficient of dispersion” is calculated in the manner explained in Oldman and Schoettle, State and Local Taxes and Finance c. 3 at 266 (1974), Bird, The General Property Tax: Findings of the 1957 Census of Governments 53 et seq. (1960), and 75 Harv. L. Rev. 1377, fn. 23 (1962). Using this method, the 1972 series has a coefficient of dispersion of 78% and the 1973 series 92.5%. These results would seem to indicate an abnormally high variation, for it has been said that "[w]hile some have suggested a lower figure, most experts see a coefficient of 20% as a reasonable maximum.” 75 Harv. L. Rev. 1377 n.23. And "an index as high as forty five [indicates a lack of uniformity which] should be judged cause for the gravest concern.” Bird, The General Property Tax 54.
See Siegal v. Newark, 38 N.J. 57, 60-64 (1962), and Bade v. Drachman, 4 Ariz. App. 55, 63-65 (1966), holding that the taxpayers against whom there had been discrimination were entitled to have the assessed valuations of their properties reduced to a percentage of their full and fair cash value equal to the average of the percentages of full and fair cash value at which all properties were ¿ssessed rather than to the assessment ratio applicable to the most favored class. See also Rick Appeal, 402 Pa. 209, 210-211 (1961); Deitch v. Board of Property Assessment, 417 Pa. 213, 218-221 (1965); Kays, Inc. v. Board of Tax Review of New Haven, 170 Conn. 477 (1976).
In this connection the plaintiffs point to the depositions of the assessors as evidence that vacant land was treated as a special class in 1970 or 1971. We cannot say that the trial judge was clearly wrong in refusing to accept this testimony as determinative of the establishment of such a class. At best the depositions indicate that something was done about "homes” and presumably nothing was done about vacant land, business or commercial property or any other kind of property for that matter. Further, just what was done is not clear. One of the assessors did indeed testify that "the homes in the whole city were reassessed by the previous board” in 1970 and 1971 but he did not know in what way. Another testified that in 1970 and 1971 "I know that the whole city was supposed to have been reassessed, that is the homes.” In later testimony he seems to be referring to the action of the previous board as "a survey of the city.” He also did not know just what was done. In any event the judge did not find that whatever was done was reflected in any particular uniformity of increased assessment ratios.