September, 2017 1 

Standard Errors in the 2016

Consumer Expenditure Survey 

David Swanson 

Consumer Expenditure Surveys Program Report Series

“Standard errors” are a measure of the uncertainty in a survey’s estimates caused by the use of data from a 

representative sample of households instead of the complete universe of households when making the survey’s 

estimates. The United States has approximately 129 million households, and the Consumer Expenditure Survey 

(CE) interviews approximately 17,500 households per year.1 The fact that the CE does not interview every 

household means its estimates have a margin-of-error or standard error associated with them. This article explains 

the use of standard errors of the CE’s expenditure and income estimates. 

Standard errors are the most common measure of sampling variability of a survey’s estimates. Defined as the 

square root of the survey estimates’ variances, they measure how much the CE’s estimates would vary if the 

survey could be repeated over-and-over using a different sample of households every time. Of course it is not 

feasible to repeat the survey over-and-over, but statistical theory allows standard errors to be estimated anyway. 

Standard errors serve two important purposes. First, they provide a general measure of the accuracy of the CE’s 

estimates. And second, they are used to determine whether differences between various expenditure estimates are 

statistically significant. The statistical significance of the estimates is usually described with a 95% confidence 

interval, which is informally called the estimate’s margin-of-error. 

Presentation of the estimates 

The CE’s expenditure tables for 2016 show four numbers for every item category: the mean, the share, the 

standard error, and the coefficient of variation (see example in Table 1). 

The “mean” is an estimate of the average annual expenditure per household on a particular item category. As 

mentioned above, it comes from interviewing 17,500 households per year. In 2016 the average annual expenditure 

per household on all items (“average annual expenditures”) was estimated to be $57,311.14, and on food it was 

estimated to be $7,203.16. These are sometimes called the survey’s “point estimates.”

The “share” is an item category’s expenditure amount expressed as a percent of total expenditures. For example, 

in 2016 the mean expenditure per household for “food” was $7,203.16, and for all items it was $57,311.14, so the 

share of total expenditures for “food” was 12.6 percent ($7,203.16 divided by $57,311.14 equals 0.126).

 

1

 The CE collects data from approximately 6,500 households per quarter in the Interview survey and 6,000 households per year in the Diary 

survey. Each household in the Interview survey is interviewed four times, and each household in the Diary survey fills out two weekly 

diaries, making the total amount of data collected 26,000 quarterly interviews and 12,000 weekly diaries per year. Because of the rotating 

nature of the Interview survey’s sample design, with old addresses constantly dropping out of the sample and new addresses replacing 

them, there are about 11,500 unique households in the Interview survey each year. Adding this to the 6,000 households in the Diary survey 

gives 17,500 unique households per year in both surveys. 
September, 2017 2 

The “standard error” (SE) is a measure of how much the mean expenditure would vary if the survey could be 

repeated over-and-over using a different sample of households every time. In 2016 the standard error for all items 

was $593.55, and for food it was $94.56. That means if the CE survey could have been repeated over-and-over 

with different samples of households, the mean expenditure for all items would have varied by approximately 

plus-or-minus $593.55, and the mean expenditure for food would have varied by approximately plus-or-minus 

$94.56. 

Table 1. Expenditure Estimates

Item

All 

consumer 

units2

Average annual expenditures

Mean $57,311.14

Share 100.0

SE 593.55

CV (%) 1.04

Food

Mean $7,203.16

Share 12.6

SE 94.56

CV (%) 1.31

In mathematical terms, the variability of a survey’s estimates is usually described by a 95 percent confidence 

interval, which is equal to the survey’s point estimate plus-or-minus 1.96 times its standard error. Thus the usual 

way of describing the variability of these point estimates is $57,311.14 plus-or-minus $1,163.36 (= $57,311.14 

1.96593.55) and $7,203.16 plus-or-minus $185.34 (= $7,203.16  1.9694.56).

The last number is the “coefficient of variation” (CV). It is the standard error divided by the mean expenditure, 

and it is expressed as a percent. It gives the relative amount of variability instead of the absolute amount of 

variability in the expenditure estimates. It is useful when comparing expenditure categories whose mean and 

standard error differ in magnitude. For example, in 2016 the average annual expenditure per household on all 

items was $57,311.14 and its standard error was $593.55, so its coefficient of variation was 1.04 percent ($593.55 

divided by $57,311.14 equals 0.0104). Likewise, the average annual expenditure per household on food was 

$7,203.16 and its standard error was $94.56, so its coefficient of variation was 1.31 percent ($94.56 divided by 

$7,203.16 equals 0.0131). These two item categories have approximately the same coefficient of variation, which 

means they have approximately the same amount of variability relative to the size of their point estimates. 

Analysis of the standard errors 

An examination of the data reveals three significant observations. First, standard errors tend to be smaller for the 

nationwide estimates than for the individual regions of the country. This is primarily due to their different sample 

sizes. In general standard errors decrease as the sample size “n” increases, and the CE interviews more households 

in the whole country than in each individual region of the country. 

Second, standard errors increase as the number of individual item categories in the broader item category 

increases. For example, the standard error of the “all items” category is larger than the standard error for “food,” 

which in turn is larger than the standard error for “fruits and vegetables.” This is primarily due to the heterogeneity 

of items in broader item categories. 

 

2 The unit from which the CE collects expenditure data is called a “consumer unit.” It is a group of people who pool their incomes to make 

joint expenditure decisions. Consumer units include families; groups of unrelated people who live together and pool their incomes to make 

joint expenditure decisions; and single persons who live alone or with other individuals but who are financially independent of the other 

individuals. There can be multiple consumer units in a household, but approximately 99 percent of all households have one consumer unit 

so the terms are often used interchangeably. The term “consumer unit” is used in the titles of CE’s tables, but the term “household” will 

be used in this paper for simplicity. 
September, 2017 3 

And third, standard errors decrease as the frequency of purchases increases. For example, the standard error for 

“telephone services” is smaller than the standard error for the purchase of new cars and trucks even though both 

item categories have approximately the same mean expenditure. This is due to the tendency of households to 

spend roughly the same amount of money every month on frequently purchased items like telephone services, 

while for infrequently purchased items like new cars and trucks there are a lot of $0 expenditures in the CE 

database interspersed with a small number of large expenditures, causing the variance of infrequently purchased 

items to be large. 

All three of these observations can be explained by a single statistical formula. Let xhi be the expenditure of an 

individual household on an individual item category (e.g., “beef” or “bakery products”), let “n” be the number of 

households in the sample, and let 

I

x

 be the average expenditure per household on an aggregate item category 

(e.g., “food”). Then a simple calculation from elementary statistics shows 





 

Ii

I I hi xV

n

SE xVx )(

1

)()( .

The term on the right-hand side has three components corresponding to the three observations made here. The 

first observation noted that the variance decreases as the sample size “n” increases. This is clear from the formula 

by observing that “n” is in the denominator. The second observation noted that the variance increases as the 

number of individual item categories in the broader item category increases. This is clear from the formula by the 

number of terms in the sum (iI) increasing as the number of individual item categories increases. The third 

observation noted that the variance decreases as the frequency of purchases increases. This can be seen in the 

formula by the variance of individual item categories (V(xhi)) being smaller for frequently purchased items.

Standard errors change from year to year as the range of expenditures reported by the households in the survey 

changes. Although it is not often discussed, every statistic has a standard error, and since the standard error is a 

statistic it has a standard error as well. In recent years (2012-2016) the standard error for the nationwide all items 

category ranged from $519.52 to $603.93, which gives a rough idea of the true standard error – $560 plus-orminus $50. That means the standard error of the standard error is approximately $50. From this it can be seen 

that the estimated mean is generally more stable than the estimated standard error in relative terms, which is a 

well-known fact from statistical theory. 

Nonsampling error 

Sampling error is not the only source of uncertainty in a survey’s estimates. There is also “nonsampling error,”

which is everything else. It includes incorrect information provided by the survey’s respondents, households 

missing from the survey’s master address file, data processing errors, and so forth. Nonsampling errors occur 

regardless of whether data are collected from a sample of households or the complete universe of households. 

Standard errors do not measure these other sources of error. 

Variance estimation methodology 

Standard textbook formulas for computing variances usually assume the survey’s data come from a “simple 

random sample” of households. Those formulas do not apply to the CE because, like most real-world surveys, 

the CE does not collect data from a simple random sample of households. Instead the CE draws a stratified random 

sample of geographic areas around the United States, and then draws a systematic sample of households within 

the selected areas. As a result, a different method of variance estimation is needed. 

There is a class of variance estimation techniques that use “replicates” to produce unbiased variance estimates for 

surveys with any sample design, simple or complex. In those techniques a number of subsamples are drawn from 

the survey’s full sample of data, and the mean expenditure is computed for both the full sample and each 
September, 2017 4 

subsample. Then the variance of the full sample expenditure estimate is computed by looking at the variability of 

all the different subsample expenditure estimates. 

The specific technique used by the CE is called “balanced repeated replication” (BRR). In this technique the CE 

divides the geographic areas in which it collects data into 43 subgroups called “strata.” Then the households 

within each stratum are randomly divided into two “half samples,” with half of the households assigned to one 

half sample, and the other half assigned to the other half sample. There are 43 strata with 2 half samples per 

stratum, making 243

 (approximately 9 trillion) different estimates of the mean expenditure that can be computed 

from exactly half of the data. 

Fortunately it is not necessary to compute all 9 trillion estimates of the mean expenditure to get a reasonable 

variance estimate. By carefully picking 44 estimates in a balanced way (hence the name of the technique), the 

same variance estimate can be obtained with considerably less work. Then after generating 44 balanced estimates, 

the variance of the full sample estimate is computed as the average squared difference between the half sample 

and full sample estimates: 



 

44

1

2

( )

44

1

)()(

HS

HS SampleFull SE xVx xx .

This gives an unbiased estimate of the CE’s variance.

To be more precise, the CE uses a 4444 Hadamard matrix to pick its 44 balanced estimates. A Hadamard matrix 

is an nn matrix whose entries are 1 and where the product of the matrix and its transpose is equal to n times the 

identity matrix. If H = [hij] is such a matrix, 

21

,

ii xx

are the mean expenditures from half samples 1 and 2 in the ith stratum, and wi is the weight of the i-th stratum (wi0 and w1+...+w43=1), then 

















 







43

1

1 2

2

1

2

1

i

i

ij

i

ij

j i x

h

x

h

wx

is one of the 44 balanced estimates of the mean expenditure, and 



 

44

1

2

)(

44

1

)(

ˆ

j

j

xV xx .

is the unbiased estimate of the variance used by the CE.3

More Information 

For more information, contact the Consumer Expenditure Surveys Program by telephone at (202) 691-6000, by 

email at cexinfo@bls.gov, or online at http://www.bls.gov/cex/. 

 

3

 The CE uses a 4444 Hadamard matrix, but only 43 of its rows. One of its rows cannot be used because it consists of all 

+1’s, which means it does not satisfy the full orthogonal balancing property needed for the technique.