# Analytical concepts

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## Current dollars versus constant dollars

"Current dollars" are what we usually mean when we refer to a currency in the current time period. The term "constant dollars" refers to dollars of several years expressed in terms of their value ("purchasing power") in a single year, called the base year. This type of adjustment is done to eliminate the impact of widespread price changes.

Current dollars are converted to constant dollars using an index of price movements. The most widely used index for household or family incomes, provided that no specific uses of the income are identified, is the all-items annual Consumer Price Index (CPI), which reflects average spending patterns by consumers in Canada.

The all-items annual Consumer Price Index can be found in CANSIM Table 326-0021. To convert current dollars of any year to constant dollars, divide them by the index of that year and multiply them by the index of the base year you choose (remember that the numerator contains the index value of the year you want to move to). For example, using this index, \$10,000 in 1997 would be 12,887 in 2010 constant dollars (\$10,000 × 116.5/ 90.4 = \$12,887).

## Earner/Income recipient

An earner is a person who received income from employment (wages and salaries) and/or self-employment during the reference year. The term income recipient is generally used for someone who received a positive (or negative) amount of income of any given type.

## Mean income (average income)

The mean or average income is computed as the total or "aggregate" income divided by the number of units in the population. It offers a convenient way of tracking aggregate income while adjusting for changes in the size of the population.

There are two drawbacks to using average income for analysis. First, since everyone's income is counted, the mean is sensitive to extreme values: unusually high income values will have a large impact on the estimate of the mean income, while unusually low ones, i.e. highly negative values, will drive it down. (See also Recipients versus non-recipients and Negative values.) Secondly, it does not give any insight into the allocation of income across members of the population. To examine allocation of income, measures such as Percentiles or Gini coefficients may be used.

## Recipients versus non-recipients (zero values)

For every table showing average incomes, it must be kept in mind whether non-recipients of that type of income are included or excluded from the population. In the case of total family income, the difference from including or excluding units with zero income is small since there are very few such families. However, if one is interested in the average amount of individual self-employment earnings, the value will be quite different if one includes those persons who were not self-employed.

## Negative values

Negative income amounts can arise in two ways: net losses from self-employment (expenses exceed receipts), or net investment losses (losses exceed gains). As with zero values, negative values can have a large impact on results. In general, the published income tables treat negative values no differently than positive values, but there are a few exceptions: for the calculation of both Gini coefficients and the low income gap, negative values are converted to zeroes; and in the derivation of the major income earner of a family or household, the absolute value is used instead (see Major income earner).

## Percentiles

Income percentiles, like quintiles and deciles, are a convenient way of categorizing units of a given population from lowest income to highest income for the purposes of drawing conclusions about the relative situation of people at either end or in the middle of the scale. Rather than using fixed income ranges, as in a typical distribution of income, it is the fraction of each population group that is fixed.

First, all the units of the population, whether individuals or families, are ranked from lowest to highest by the value of their income of a specified type, such as after-tax income. Then the ranked population is divided into five groups of equal numbers of units, called quintiles. Analogously, dividing the population ranked by income into ten groups, each comprising the same number of units, produces deciles.

Most analyses should be carried out on the people of different percentiles within one population distribution. Care should be taken in making comparisons between percentiles that resulted from different distributions, because any difference in either the population or the income concept used to rank units could have a large effect. It is probable that both the income ranges represented by each percentile and the people making up each percentile will be different.

## Median income

The median income is the value for which half of the units in the population have lower incomes and half has higher incomes. To derive the median value of income, units are ranked from lowest to highest according to their income and then separated into two equal-sized groups. The value that separates these groups is the median income (50th percentile).

Because the median corresponds exactly to the midpoint of the income distribution, it is not, contrary to the mean, affected by extreme income values. This is a useful feature of the median, as it allows one to abstract from unusually high values held by relatively few people.

Since income distributions are typically skewed to the left - that is, concentrated at the low end of the income scale - median income is usually lower than mean income.

## Implicit rate of government transfers or taxes

The implicit rate of government transfers or taxes is a way of showing the relative importance of transfers received or taxes paid for different families or individuals. This concept is similar, but not identical, to the effective rate of taxation. For a given individual or family, the effective rate is the amount of transfers/taxes expressed as a percentage of their market income, total income, or after-tax income. The implicit rate for a given population is the average (or aggregate) amount of transfers/taxes expressed as a percentage of their average (or aggregate) income.

## Family or household size adjustment (equivalence scale)

When comparing grouping unit (family or household) incomes to study such things as income adequacy or socio-economic status, one often wants to take the unit size and composition into account. The income amount itself is not sufficient to understand a unit's financial well-being without knowing how many people are sharing it. In general, two approaches have been used to help with the analysis of grouping unit income. One is to produce data by detailed unit types, so that within a given type, differences in unit size are not significant. In fact, many income measures have been reported by detailed unit types in the published tables. The other way to take into account unit size and composition is to adjust the income amount by an adjustment factor.

The simplest method is to use per capita income, that is, to divide the grouping unit income by the number of members it includes. A limitation of per capita income, however, is that it tends to underestimate economic well-being for larger units as compared to smaller units. This is due to the fact that it assumes equal living costs for each member of the unit, but some costs, primarily those related to shelter, decrease proportionately with unit size (they may also be lower for children than for adults). For example, the shelter costs for an adult married couple with no children are arguably not much more than those for an adult living alone.

To take such economies of scale into account, it is common to use an "equivalence scale" to adjust grouping unit incomes. Instead of implicitly assuming equal costs for additional unit members as the per capita approach does, the equivalence scale is a set of decreasing factors assigned to the first member, the second member, and so on. The adjusted income amount for the unit is obtained by dividing the unit's income by the sum of the factors assigned to each member. The concept can be applied to the grouping unit after-tax income, grouping unit market income as well as any other grouping unit income sources, even grouping unit tax paid.

In order to ensure international consistency and to facilitate the calculation of adjusted family or household income, a new scale was introduced with the release of the 2008 data. Adjusted income is now obtained by dividing family or household income by the square root of the number of members in the family or household. The estimates for years prior to 2008 have been revised accordingly. The square root adjustment is very close to the previous scale, particularly for families with six members or less.

## Gini coefficient

The Gini coefficient measures the degree of inequality in the income distribution. Gini coefficients are published for market income, total income and after-tax income, and are used to compare the uniformity of income allocation between different income concepts, across different populations or within the same population over time.

Values of the Gini coefficient can range from 0 to 1. A value of zero indicates income is equally divided among the population with all units receiving exactly the same amount of income. At the opposite extreme, a Gini coefficient of 1 denotes a perfectly unequal distribution where one unit possesses all of the income in the economy. A decrease in the value of the Gini coefficient can, by and large, be interpreted as reflecting a decrease in inequality, and vice versa.

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