incomeInequalityR Documentation

Income Inequality in the US


Data on quantiles of the distributions of family incomes in the United States. This combines three data sources:

(1) US Census Table F-1 for the central quantiles

(2) Piketty and Saez for the 95th and higher quantiles

(3) Gross Domestic Product and implicit price deflators from Measuring Worth. (NOTE: The Measuring Worth Web site,, often gives security warnings. The desired data still seems to be available and not corrupted, however.)




A data.frame containing:


numeric year 1947:2012


number of families in the US

quintile1, quintile2, median, quintile3, quintile4, p95

quintile1, quintile2, quintile3, quintile4, and p95 are the indicated quantiles of the distribution of family income from US Census Table F-1. The media is computed as the geometric mean of quintile2 and quintile3. This is accurate to the extent that the lognormal distribution adequately approximates the central 20 percent of the income distribution, which it should for most practical purposes.

P90, P95, P99, P99.5, P99.9, P99.99

The indicated quantiles of family income per Piketty and Saez

realGDP.M, GDP.Deflator, PopulationK, realGDPperCap

real GDP in millions, GDP implicit price deflators, US population in thousands, and real GDP per capita, according to (NOTE: The web address for this,, seems to be functional but may not be maintained to current internet security standards. It is therefore given here as text rather than a hot link.)


ratio of the estimates of the 95th percentile of distributions of family income from the Piketty and Saez analysis of data from the Internal Revenue Service (IRS) and from the US Census Bureau.

The IRS has ranged between 72 and 98 percent of the Census Bureau figures for the 95th percentile of the distribution, with this ratio averaging around 75 percent since the late 1980s. However, this systematic bias is modest relative to the differences between the different quantiles of interest in this combined dataset.


average number of persons per family using the number of families from US Census Table F-1 and the population from MeasuringWorth. (Note: The web site for Measuring Worth,, often gives security warnings. It still seems to work. It seems that the web site is not maintained to current internet security standards.)


personsPerFamily * realGDPperCap


ratio of realGDPperFamily to the median. This is a measure of skewness and income inequality.


For details on how this data.frame was created, see "F1.PikettySaez.R" in system.file('scripts', package='fda'). This provides links for files to download and R commands to read those files and convert them into an updated version of incomeInequality. This is a reasonable thing to do if it is more than 2 years since max(incomeInequality$year). All data are in constant 2012 dollars.


Spencer Graves


United States Census Bureau, Table F-1. Income Limits for Each Fifth and Top 5 Percent of Families, All Races,, accessed 2016-12-09.

Thomas Piketty and Emmanuel Saez (2003) "Income Inequality in the United States, 1913-1998", Quarterly Journal of Economics, 118(1) 1-39,, update accessed February 28, 2014.

Louis Johnston and Samuel H. Williamson (2011) "What Was the U.S. GDP Then?" MeasuringWorth. (Note: Their web address,, often gives security warnings. The desired data still seems to be available there. However, it seems that the site is not maintained to current internet security standards. The data used in the current USGDPpresidents data set was extracted February 28, 2014.)


## Rato of IRS to census estimates for the 95th percentile
plot(P95IRSvsCensus~Year, incomeInequality, type='b')
# starts ~0.74, trends rapidly up to ~0.97,
# then drifts back to ~0.75
# check
# The Census data runs to 2011;  Pikety and Saez runs to 2010.
quantile(incomeInequality$P95IRSvsCensus, na.rm=TRUE)
# 0.72 ... 0.98

## Persons per Family

plot(personsPerFamily~Year, incomeInequality, type='b')
# ranges from 3.72 to 4.01 with median 3.84
#  -- almost 4

## GDP per family
plot(realGDPperFamily~Year, incomeInequality, type='b', log='y')

## Plot the mean then the first quintile, then the median,
##            99th, 99.9th and 99.99th percentiles
plotCols <- c(21, 3, 5, 11, 13:14)
kcols <- length(plotCols)
plotColors <- c(1:6, 8:13)[1:kcols] # omit 7=yellow
plotLty <- 1:kcols

matplot(incomeInequality$Year, incomeInequality[plotCols]/1000,
        log='y', type='l', col=plotColors, lty=plotLty)

#*** Growth broadly shared 1947 - 1970, then began diverging
#*** The divergence has been most pronounced among the top 1%
#*** and especially the top 0.01%

## Growth rate by quantile 1947-1970 and 1970 - present
keyYears <- c(1947, 1970, 2010)
(iYears <- which(is.element(incomeInequality$Year, keyYears)))

(dYears <- diff(keyYears))
kk <- length(keyYears)
(lblYrs <- paste(keyYears[-kk], keyYears[-1], sep='-'))

(growth <- sapply(incomeInequality[iYears,], function(x, labels=lblYrs){
    dxi <- exp(diff(log(x)))
    names(dxi) <- labels
} ))

# as percent
(gr <- round(100*(growth-1), 1))

# The average annual income (realGDPperFamily) doubled between
# 1970 and 2010 (increased by 101 percent), while the median household
# income increased only 23 percent.

## Income lost by each quantile 1970-2010
## relative to the broadly shared growth 1947-1970
(lostGrowth <- (growth[, 'realGDPperFamily']-growth[, plotCols]))
# 1947-1970:  The median gained 20% relative to the mean,
#           while the top 1% lost ground
# 1970-2010:  The median lost 79%, the 99th percentile lost 29%,
#           while the top 0.1% gained

(lostIncome <- (lostGrowth[2, ] *
                incomeInequality[iYears[2], plotCols]))
# The median family lost $39,000 per year in income
# relative to what they would have with the same economic growth
# broadly shared as during 1947-1970.
# That's slightly over $36,500 per year = $100 per day

(grYr <- growth^(1/dYears))
(grYr. <- round(100*(grYr-1), 1))

## Regression line:  linear spline

(varyg <- c(3:14, 21))
Varyg <- names(incomeInequality)[varyg]
str(F01ps <- reshape(incomeInequality[c(1, varyg)], idvar='Year',
                     times=Varyg, timevar='pctile',
                     varying=list(Varyg), direction='long'))
names(F01ps)[2:3] <- c('variable', 'value')
F01ps$variable <- factor(F01ps$variable)

# linear spline basis function with knot at 1970
F01ps$t1970p <- pmax(0, F01ps$Year-1970)

table(nas <-$value))
# 6 NAs, one each of the Piketty-Saez variables in 2011
F01i <- F01ps[!nas, ]

# formula:
# log(value/1000) ~ b*Year + (for each variable:
#     different intercept + (different slope after 1970))

Fit <- lm(log(value/1000)~Year+variable*t1970p, F01i)
# all highly significant
# The residuals may show problems with the model,
# but we will ignore those for now.

# Model predictions
str(Pred <- predict(Fit))

## Combined plot
#  Plot to a file?  Wikimedia Commons prefers svg format.
## Not run: 
#  If you want software to convert svg to another format 
#  such as png, consider GIMP (

#  Base plot

# Leave extra space on the right to label 
# with growth since 1970
  op <- par(mar=c(5, 4, 4, 5)+0.1)

      log='y', type='l', col=plotColors, lty=plotLty,
      xlab='', ylab='', las=1, axes=FALSE, lwd=3)
  axis(1, at=seq(1950, 2010, 10),
     labels=c(1950, NA, 1970, NA, 1990, NA, 2010), 
  yat <- c(10, 50, 100, 500, 1000, 5000, 10000)
  axis(2, yat, labels=c('$10K', '$50K', '$100K', '$500K',
             '$1M', '$5M', '$10M'), las=1, cex.axis=1.2)

#  Label the lines
  pctls <- paste(c(20, 40, 50, 60, 80, 90, 95, 99, 
      99.5, 99.9, 99.99),
              '%', sep='')
  lineLbl0 <- c('Year', 'families K', pctls,
     'realGDP.M', 'GDP deflator', 'pop-K', 'realGDPperFamily',
     '95 pct(IRS / Census)', 'size of household',
     'average family income', 'mean/median')
  (lineLbls <- lineLbl0[plotCols])
  sel75 <- (incomeInequality$Year==1975)

  laby <- incomeInequality[sel75, plotCols]/1000

  text(1973.5, c(1.2, 1.2, 1.3, 1.5, 1.9)*laby[-1], 
    lineLbls[-1], cex=1.2)
  text(1973.5, 1.2*laby[1], lineLbls[1], cex=1.2, srt=10)

## Add lines + points for the knots in 1970
  End <- numeric(kcols)
  F01names <- names(incomeInequality)
  for(i in seq(length=kcols)){
    seli <- (as.character(F01i$variable) == 
#  with(F01i[seli, ], lines(Year, exp(Pred[seli]), 
#  col=plotColors[i]))
    yri <- F01i$Year[seli]
    predi <- exp(Pred[seli])
    lines(yri, predi, col=plotColors[i])
    End[i] <- predi[length(predi)]
    sel70i <- (yri==1970)
    points(yri[sel70i], predi[sel70i], 

##  label growth rates
  table(sel70. <- (incomeInequality$Year>1969))
  (lastYrs <- incomeInequality[sel70., 'Year'])
  (lastYr. <- max(lastYrs)+4)
#text(lastYr., End, gR., xpd=NA)
  text(lastYr., End, paste(gr[2, plotCols], '%', sep=''), 
  text(lastYr.+7, End, paste(grYr.[2, plotCols], '%', 
    sep=''), xpd=NA)

##  Label the presidents
  abline(v=c(1953, 1961, 1969, 1977, 1981, 1989, 1993, 
    2001, 2009))
  (m99.95 <- with(incomeInequality, sqrt(P99.9*P99.99))/1000)

  text(1949, 5000, 'Truman')
  text(1956.8, 5000, 'Eisenhower', srt=90)
  text(1963, 5000, 'Kennedy', srt=90)
  text(1966.8, 5000, 'Johnson', srt=90)
  text(1971, 5*m99.95[24], 'Nixon', srt=90)
  text(1975, 5*m99.95[28], 'Ford', srt=90)
  text(1978.5, 5*m99.95[32], 'Carter', srt=90)
  text(1985.1, m99.95[38], 'Reagan' )
  text(1991, 0.94*m99.95[44], 'GHW Bush', srt=90)
  text(1997, m99.95[50], 'Clinton')
  text(2005, 1.1*m99.95[58], 'GW Bush', srt=90)
  text(2010, 1.2*m99.95[62], 'Obama', srt=90)
##  Done
  par(op) # reset margins # for plot to a file
## End(Not run)