# Dictionary, Census of Population, 2016

Appendix 4.0 Derived Statistics

**Release date:**September 13, 2017

Median income and average income are two statistics commonly computed on income variables to measure central tendency.

These statistics can be derived for total income, after-tax income, employment income, wages and salaries, or any other particular source of income on a variety of universes: individuals, families, persons not in families and households.

Typically, for total income and after-tax income, median and average incomes of individuals are calculated for those with income (positive or negative); median and average incomes of households are calculated for all units, whether or not they had income; median and average incomes of families are calculated for all units, whether or not they had income; median and average incomes of persons not in families are calculated for all units, whether or not they had income.

However, for the income components, median and average incomes are sometimes calculated for units with income (positive or negative) only for all universes.

In 2016, as part of the measures to ensure non-disclosure of individual characteristics, the average income statistic is only available from the sampled population, i.e., information from the long-form census questionnaire. The median income statistic is the measure of central tendency and is available for 100% of the population (short-form census questionnaire).

## Median income

The median income of a specified group is the amount that divides the income distribution of that group into two halves, i.e., the incomes of half of the units in that group are below the median, while those of the other half are above the median.

When median income is computed from the census short-form questionnaire, no weighting is required because each unit represents itself. When median income is computed from the census long-form questionnaire, certain units would represent multiple units (known as weight) due to sampling.

For an income size distribution, the median is usually estimated as follows:

$M\text{\hspace{1em}}\text{\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{1em}}{L}_{m}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{m}\text{\hspace{0.17em}}\left(d/{f}_{m}\right)$, where

$M\text{\hspace{1em}}\text{\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{1em}}$Median value

${L}_{m}\text{\hspace{1em}}=\text{\hspace{1em}}$Lower boundary of the income group in which

$\text{\hspace{1em}}\text{\hspace{0.17em}\hspace{0.17em}}\text{\hspace{1em}}\frac{N}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{{\displaystyle \sum W}}_{i}}{2}$ falls, where

$\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{1em}}N\text{\hspace{0.17em}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{0.17em}}$Number of (weighted) units in the specified group for which the distribution is being shown

$\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{1em}}{W}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}$ Weight of each unit in the specified group, for the short-form census questionnaire, the weight is equal to 1 for each unit.

${c}_{m}\text{\hspace{1em}}=\text{\hspace{1em}}$Size (range) of the median income group

$d\text{\hspace{1em}}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{\hspace{1em}}$Number of units in the specified group necessary from the median income group to reach the middle

i.e., $\frac{N}{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\displaystyle \sum _{i}^{m-1}\text{}}{f}_{i}$

${f}_{m}\text{\hspace{1em}}=\text{\hspace{1em}}$Frequency or (weighted) total number of units in the median income group

In a similar fashion, decile income values (the 9 dollar amounts which divide the income recipients in 10 equal groups), quintiles (5 equal groups) and quartiles (4 equal groups) can also be derived for the population with income.

## Average income

Average income of a specified group is calculated by dividing the aggregate income of that group by the number of units in that group.

This statistic is calculated for any specified group as follows:

$\overline{Y}\text{\hspace{1em}}\text{\hspace{0.17em}}=\text{\hspace{1em}}\frac{{\displaystyle \sum ({Y}_{i}{W}_{i})}}{{\displaystyle \sum {W}_{i}}}$, where

$\overline{Y}\text{\hspace{1em}}\text{\hspace{0.17em}}=\text{\hspace{1em}}$ Average income of the group specified

${Y}_{i}\text{\hspace{1em}}\text{\hspace{0.17em}}=\text{\hspace{1em}}$ Actual income of each unit in the group specified

${W}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{1em}}$ Weight of each unit in the group specified