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# Appendix Q Derived Statistics

Modified on May 1, 2008

There are three statistics not resident on the database that can readily be computed on income variables: (1) average, (2) median, and (3) standard error.

These statistics can be derived for total income, after-tax income, earnings, wages and salaries, or any other particular source of income on a variety of universes. The statistic calculation is explained for two groups of universes: individuals and all other universes which include families, persons not in families and households.

## A. Income of individuals

Average and median incomes and standard errors for average income of individuals will be calculated for those individuals who are at least 15 years of age and who have an income or after-tax income (positive or negative). For all other universes (for example, census families or private households), these statistics will be calculated over all units, whether or not they reported any income.

### 1. Average income of individuals

Average income of individuals refers to the dollar amount obtained by adding up the total income of all individuals 15 years of age and over who reported income for 2005 and dividing this sum by the number of individuals with income.

Average income is calculated from unrounded data by dividing the weighted aggregate income of a specified group of individuals (for example, males 45 to 54 years of age) by the number of individuals with income in that group.

Average income is calculated for any group as follows:

=    , where
=   Average income of the individuals 15 years of age and over with income in the group
=   Actual income of each individual 15 years of age and over with income in the group
=   Weight of each individual 15 years of age and over with income in the group

### 2. Median income of individuals

The median income of a specified group of income recipients is that dollar amount which divides their income size distribution ranked by size of income, into two halves, i.e., the incomes of the first half of individuals are below the median, while those of the second half are above the median. Median income is calculated from the unrounded number of individuals (for example, males 45 to 54 years of age) with income in that group.

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

=    + , Where

=    Median value

=   Lower boundary of the income group in which
=    falls, where

=
Number of (weighted) individuals 15 years of age and over with income in the category for which the distribution is being shown

=
Weight of each individual 15 years of age and over with income in the category

=   Size (range) of the median income group

=
Number of individuals 15 years of age and over necessary from the median income group to reach the middle
i.e.,

=
Frequency or total (weighted) of individuals 15 years of age and over in the median income group

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

### 3. Standard error of average income

The standard error of average income refers to a dollar value which indicates the precision of the estimate of average income.

If interpreted as shown below, it serves as a rough indicator of the precision of the corresponding estimate of average income. For about 68% of the samples which could be selected from the sample frame, the difference between the sample estimate of average income and the corresponding figure based on complete enumeration would be less than one standard error. For about 95% of the possible samples, the difference would be less than two standard errors and, in about 99% of the samples, the difference would be less than approximately two and one half standard errors.

This statistic only attempts to measure for average income the imprecision due to sampling error.

## B. Income of families, persons not in families, and households

Average and median income of families (both census and economic families), persons 15 years of age and over not in families, households and the standard error for average income are normally calculated for all units in the specified group, whether or not they reported income.

For the universe of individuals, these statistics are normally calculated for those individuals who are at least 15 years of age and have reported an income (positive or negative).

### 1. Average income of families (census/economic), persons not in families, and households

Dollar amount obtained by adding up the total income of all family members (census/economic), persons 15 years of age and over not in families, or households and dividing this sum by the number of families, persons 15 years of age and over not in families, or households.

Average income of families (census/economic) or persons 15 years of age and over not in families or households refers to the weighted mean total income of families (census/economic), persons 15 years of age and over not in families, or households in 2005.

Average income is calculated from unrounded data by dividing the aggregate income of a specified group of families (for example, husband-wife families with working wives), persons 15 years of age and over not in families, or households (for example, family households) by the number of families, persons 15 years of age and over not in families, or households in that group, whether or not they reported income

This statistic is calculated for any group as follows:

=    , where
=   Average income of the group
=

Actual income of each family, person 15 years of age and over not in a family, or household in the group

=
Weight of each family, person 15 years of age and over not in family, or household in the group

### 2. Median income of families (census/economic), persons not in families, and households

The median income of a specified group of families (census/economic), persons 15 years of age and over not in families, or households is that amount which divides their income size distribution ranked by size of income, into two halves. That is, the incomes of the first half of the families, persons 15 years of age and over not in families, or households are below the median, while those of the second half are above the median. Median incomes of families (census/economic), persons 15 years of age and over not in families, or households are normally calculated for all units in the specified group, whether or not they reported income.

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

=    + , Where

=    Median value

=   Lower boundary of the income group in which
=    falls, where

=
Number of families (census/economic), persons 15 years of age and over not in families, or households in the category for whom the distribution is being shown

=
Weight of each family (census/economic), person 15 years of age and over not in a family, or household in the category

=   Size (range) of the median income group

=
Number of families (census/economic), persons 15 years of age and over not in families, or households necessary from the median income group to reach the middle
i.e.,

=
Frequency or total (weighted) families, persons 15 years of age and over not in families, or households in the median income

In a similar fashion, decile income values (nine dollar amounts which divide the units in 10 equal groups), quintiles (five equal groups) and quartiles (four equal groups) can also be derived for the groups of interest.

### 3. Standard error of average income

The standard error of average income refers to a dollar value which indicates the precision of the estimate of average income.

If interpreted as shown below, it serves as a rough indicator of the precision of the corresponding estimate of average income. For about 68% of the samples which could be selected from the sample frame, the difference between the sample estimate of average income and the corresponding figure based on complete enumeration would be less than one standard error. For about 95% of the possible samples, the difference would be less than two standard errors and, in about 99% of the samples, the difference would be less than approximately two and one half standard errors.

This statistic only attempts to measure for average income the imprecision due to sampling error.