Where the words come from
Progressive, proportional and regressive taxes, the average (effective) tax rate, and the marginal rate as the rate on the band the last pound falls in — Principles of Economics 2e, OpenStax, licensed CC BY 4.0. https://openstax.org/books/principles-economics-2e/pages/30-2-taxation and https://openstax.org/books/principles-economics-2e/pages/15-5-government-policies-to-reduce-income-inequality. Changes: the three definitions from the 2e Key Terms for chapter 30, the proportional one reworded from "a flat percentage of income earned, regardless of level of income" and each given its average-rate consequence; the US payroll and Social Security examples dropped; the card examples are UK or ours; "dollar" to "pound".
UK income tax bands and rates for 2026-27 — Income Tax rates and Personal Allowances, GOV.UK (HM Revenue and Customs), Open Government Licence v3.0. https://www.gov.uk/income-tax-rates. Changes: the rates table for England, Wales and Northern Ireland put into words; the tapering of the allowance above £100,000 and the Scottish bands left out; the £60,000 and £30,000 calculations are ours.
The test for a progressive and a regressive tax by average tax rate — Effects of taxes and benefits on UK household income: financial year ending 2017, Office for National Statistics, Open Government Licence v3.0. https://www.ons.gov.uk/peoplepopulationandcommunity/personalandhouseholdfinances/incomeandwealth/bulletins/theeffectsoftaxesandbenefitsonhouseholdincome/financialyearending2017 and https://www.ons.gov.uk/peoplepopulationandcommunity/personalandhouseholdfinances/incomeandwealth/bulletins/theeffectsoftaxesandbenefitsonhouseholdincome/latest. Changes: the bulletin's "What does it mean for a benefit or tax to be progressive?" box condensed; the 4.4 percentage point effect of direct taxes on the Gini coefficient is from the FYE 2024 bulletin (25 September 2025).
Indirect taxes as a share of disposable income and of spending by income fifth, 2017-18 — Effects of taxes and benefits on UK household income: financial year ending 2018, Office for National Statistics, Open Government Licence v3.0. https://www.ons.gov.uk/peoplepopulationandcommunity/personalandhouseholdfinances/incomeandwealth/bulletins/theeffectsoftaxesandbenefitsonhouseholdincome/financialyearending2018. Changes: "one-fifth of people" given as a fifth of households; £12,300 against £4,600 given as about 2.7 times; the reason, that low-income households spend a larger share of income, condensed from the bulletin's discussion of income and expenditure measures.
Indirect taxes as a share of disposable income for the poorest and richest fifths, 2019-20 — Effects of taxes and benefits on UK household income: financial year ending 2020, Office for National Statistics, 28 May 2021, Open Government Licence v3.0. https://www.ons.gov.uk/peoplepopulationandcommunity/personalandhouseholdfinances/incomeandwealth/bulletins/theeffectsoftaxesandbenefitsonhouseholdincome/financialyearending2020. Changes: "the poorest fifth of people" given as households; figures quoted as published.
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