30 Consumer Expenditure Survey Anthology, 2003

Standard Errors in the

Consumer Expenditure

Survey

JEFFREY L. BLAHA

Jeffrey L. Blaha is a mathematical statistician in the Survey Methods Division, Consumer Expenditure Survey, Bureau of Labor

Statistics.

D

ata for the Consumer Expenditure (CE) Survey are collected

from a sample of all the consumer units (CUs) in the United States.

Estimates of the average (mean) annual

expenditure per CU in the CE tables for

the year 2000 were based on a sample

of about 30,000 CUs, out of a total of

about 109 million in the Nation. These

mean estimates differ from the true

population values because a subset,

rather than the whole population, is

observed. Sampling error is the difference between the survey estimate and

the true population value. The most

common measure of the magnitude of

the sampling error is the standard error of the estimate. The standard error

provides data users with information

about the variability associated with

the estimate.

Prior to the publication of the 2000

data, the CE program made available

separate tables of standard errors for

the Interview and Diary components

of the CE Survey. Starting with the 2000

data, the CE program began using

tables with integrated data from both

surveys. Integrated data provide a complete accounting of consumer expenditures and income, which neither survey component alone is designed to

do. The tables, which correspond to

standard integrated tables of CU expenditures published in the CE reports and

on the CE Web site, are provided by

standard demographic characteristics

(except for region).1 This article gives

a summary description of the halfsample replication method used to calculate the standard error statistics and

demonstrates the proper interpretation

of these statistics.

Methodology

Standard textbook formulas for calculating standard errors assume simple

random sampling and do not apply to

the CE Survey, because it does not use

a simple random sample. Instead, the

Survey uses stratified random sampling, with systematic sampling within

the strata. Hence, a different method

for calculating standard errors is

needed. Replication methods make up

a class of techniques that provides a

way to produce unbiased and designconsistent estimates of standard error

for complex survey designs when the

usual assumptions are not satisfied.

The fundamental idea behind replication methods is to select

subsamples repeatedly from the full

sample, calculate the statistic of interest for each subsample, and use

the variability among the subsamples

to estimate the standard error of the

full-sample statistic.

1 The replication methodology used to

calculate the standard errors is designed to

work at the national level and is not applicable to regional estimates.
 Consumer Expenditure Survey Anthology, 2003 31

the square root of the variance:

The coefficient of variation (CV)is

the standard error expressed as a percentage of the sample mean estimate

and thus is independent of the scale of

measurement. The CV is calculated as

These formulas apply to aggregated

categories as well as individual component items. In producing a table that

uses integrated data from both surveys,

the aggregated categories may be composed of items from one survey or the

other, or they can be based on integrated data from both surveys.

Interpretation of the statistics

The primary purpose of calculating

standard errors for the mean estimates

is to provide data users with a measure

of the variability associated with the

estimates. This variability measures

how close different estimates would be

to each other if it were possible to repeat the survey over and over, using

different samples of CUs. While it is

not feasible to repeat the survey over

and over, statistical theory allows the

standard error to be estimated anyway.

A small standard error indicates that

multiple independent samples would

produce values that are consistently

very close to each other, whereas a large

standard error indicates that multiple

independent samples would produce

values that are consistently not very

close to each other.

The balanced repeated replication

method is used to estimate the standard error in the CE Survey. In this

method, sampled geographic locations

are divided into 40 groups (called

strata). The CUs within each stratum

are randomly divided into two half

samples. Half of the CUs are assigned

to one half sample, and the other half

are assigned to the other half sample.

Because there are 40 strata and 2 groups

of CUs in each stratum, we can compute 240 (approximately 18 trillion) different estimates of expenditure in which

we use exactly half of the collected

data. With this information, we can estimate the standard error of CE estimates by examining how the different

estimates compare with the full-sample

estimate. In the balanced repeated replication method, we use a 44 × 44

Hadamard matrix to choose the 44

“best” combinations of groups out of

the 18 trillion possible combinations.

A variance estimate for each category of item is obtained by first computing the mean estimate of the item

for each replicate, then summing the

squared deviations of the replicate

mean estimates from the full-sample

mean estimate, and then dividing by

the number of replicates. Thus,

where xi r is a calendar-period estimate

of the mean expenditure for item i, using the rth replicate data and xi

 is the

calendar-period estimate of the mean

expenditure for item i, using the fullsample data.

The standard error is calculated as

( ) ( ) ,

44

1 2

44

1

i r i

r

i V x = ∑ x − x

−

SE( ) ( ).

i i

x = V x

( )

( )

100.

SE CV = ×

i

i

i

x

x

x

Table 1 is an extract from one of the

standard published CE demographic

tables. The table shows the mean estimate, standard error, and coefficient of

variation for a list of expenditure items

and categories, using integrated data

from both the Interview and Diary Surveys in 2000. For example, the table

shows that the average annual expenditure by all CUs on personal care products and services for 2000 was $563.62,

with a standard error of $7.94. Because

it was impossible to ask all CUs in the

country how much they spent on personal care products and services, the

$563.62 mean figure is an estimate, and

we have a margin of error, usually defined as ± 2 standard errors. In this example, the average annual expenditure

on personal care products and services

has a margin of error of ± $15.88. Thus,

we can say that the average CU probably spent between $547.74 and $579.50

($563.62 ± $15.88) annually on personal

care products and services.

Because the CV is the standard error as a proportion of the mean estimate, it provides an indication of the

spread of the data around the mean.

The smaller the CV, the smaller is the

spread of the data around the mean.

The CV also makes possible comparisons of the spread of data around the

mean of different items. For example, in

the 2000 integrated survey, the CV for

education is 4.55 percent and the CV

for personal care products and services

is 1.41 percent. Comparing the CVs for

the two items, we can say that the

spread of the data around the mean for

education expenditure is larger than the

spread of the data around the mean for

personal care products and services.
32 Consumer Expenditure Survey Anthology, 2003

Table 1. Quintiles of income before taxes, annual means, standard errors, and coefficients of variation,

Consumer Expenditure, 2000

Mean ................................................... $44,649 $44,649 $7,683 $19,071 $32,910 $53,295 $110,118 (

1

)

SE ....................................................... 517.9 517.9 137.6 87.1 84.6 193.3 1613.4 (

1

)

CV(percent) .......................................... 1.2 1.2 1.8 0.5 0.3 0.4 1.5 (

1

)

Average annual expenditures

Mean ................................................... $38,041.03 $40,234.86 $17,939.45 $26,547.37 $34,713.42 $46,791.00 $75,093.08 $32,059.31

SE ....................................................... 336.7 356.8 399.9 622.2 412.3 626.7 850.8 713.0

CV(percent) .......................................... 0.9 0.9 2.2 2.3 1.2 1.3 1.1 2.2

Food

Mean ................................................... $5,157.88 5,434.76 2,673.31 4,178.21 5,183.19 6,451.56 8,679.37 4,516.56

SE ....................................................... 65.8 77.4 97.0 106.8 143.6 127.7 184.0 93.3

CV(percent) .......................................... 1.3 1.4 3.6 2.6 2.8 2.0 2.1 2.1

Alcoholic beverages

Mean ................................................... 371.81 422.87 206.4 247.93 366.12 512.9 780.2 253.67

SE ....................................................... 15.6 21.2 32.7 24.7 22.3 53.7 75.4 17.9

CV(percent) .......................................... 4.2 5.02 15.9 10.0 6.1 10.5 9.7 7.1

Housing

Mean ................................................... 12,318.51 12,527.38 6,508.78 8,482.33 10,857.48 14,151.75 22,610.61 11,788.92

SE ....................................................... 148.8 172.1 202.2 125.9 144.9 267.4 326.7 267.7

CV(percent) .......................................... 1.2 1.4 3.1 1.5 1.3 1.9 1.5 2.3

Apparel and services

Mean. .................................................. 1,852.53 2,000.22 843.47 1,298.60 1,612.83 2,261.46 3,980.12 1,500.88

SE ....................................................... 38.9 54.3 61.3 72.3 86.6 79.1 198.4 62.1

CV(percent) .......................................... 2.1 2.7 7.3 5.6 5.4 3.5 5.0 4.1

Transportation

Mean ................................................... 7,417.36 7,567.51 3,211.97 5,042.68 7,028.41 9,223.30 13,315.32 6,985.40

SE ....................................................... 101.2 110.7 133.4 222.5 234.9 211.0 322.1 211.2

CV(percent) .......................................... 1.4 1.5 4.2 4.4 3.3 2.3 2.4 3.0

Health

Mean ................................................... 2,065.67 2,120.04 1,469.87 1,987.62 1,964.07 2,312.36 2,864.12 1,919.38

SE ....................................................... 30.1 31.6 71.7 63.7 50.6 61.7 64.0 60.3

CV(percent) .......................................... 1.5 1.5 4.9 3.2 2.6 2.7 2.2 3.1

Entertainment

Mean ................................................... 1,863.50 1,957.63 836.92 1,146.62 1,609.18 2,324.39 3,866.21 1,602.97

SE ....................................................... 35.6 39.1 66.4 50.7 64.0 74.7 117.7 70.3

CV(percent) .......................................... 1.9 2.0 8.0 4.4 4.0 3.2 3.0 4.4

Personal care products and services

Mean ................................................... 563.62 595.33 318.28 441.98 533.59 698.91 982.91 490.56

SE ....................................................... 8.0 9.8 13.1 14.8 22.7 23.0 23.0 14.3

CV(percent) .......................................... 1.4 1.7 4.1 3.4 4.3 3.3 2.4 3.0

Reading

Mean ................................................... 146.47 156.11 73.04 105.17 135.61 174.92 291.47 118.38

SE ....................................................... 2.2 2.3 3.2 4.3 3.9 5.9 7.0 5.0

CV(percent) .......................................... 1.5 1.5 4.4 4.1 2.9 3.4 2.4 4.3

Education

Mean ................................................... 631.93 635.52 430.25 290.47 393.09 600.05 1,461.94 625.79

SE ....................................................... 28.8 35.7 59.5 41.8 48.9 53.5 107.4 56.3

CV(percent) .......................................... 4.6 5.6 13.8 14.4 12.4 8.9 7.4 9.0

Tobacco products and smoking supplies

Mean ................................................... 318.62 333.3 257.24 316.91 366.31 390.04 335.94 275.75

SE ....................................................... 8.1 11.1 13.0 18.3 19.6 16.9 17.4 10.4

CV(percent) .......................................... 2.5 3.3 5.1 5.8 5.3 4.3 5.2 3.8

Miscellaneous

Mean ................................................... 775.78 831.81 364.53 594.94 832.71 1,047.19 1,318.23 619.2

SE ....................................................... 19.6 22.8 50.1 53.4 55.0 67.1 55.8 49.6

CV(percent) .......................................... 2.5 2.7 13.7 9.0 6.6 6.4 4.2 8.0

Cash contributions

Mean ................................................... 1,192.44 1,344.06 332.27 1,162.95 953.01 1,217.29 3,050.11 749.99

SE ....................................................... 96.8 116.2 39.0 452.9 125.6 117.0 285.9 128.1

CV(percent) .......................................... 8.1 8.6 11.7 38.9 13.2 9.6 9.4 17.1

Personal insurance and pensions

Mean ................................................... 3,364.92 4,308.33 413.14 1,250.97 2,877.80 5,424.88 11,556.55 611.86

SE ....................................................... 54.7 58.2 28.0 34.0 76.5 110.3 229.4 30.5

CV(percent) .......................................... 1.6 1.4 6.8 2.7 2.7 2.0 2.0 5.0

Item All

consumer

units

Complete

income

reporters

Lowest 20

percent

Secondlowest 20

percent

Thirdlowest 20

percent

Fourthlowest 20

percent

Incomplete

income

reporters

Highest 20

percent

Complete reporting of income

1

 Components of income and taxes are derived from complete income reporters only; see glossary.