Standard Errors in the 2011 Consumer Expenditure Survey 

by David Swanson 

This article presents standard errors of the expenditure and income estimates in the 2011 Consumer Expenditure 

(CE) Survey. “Standard errors” are a measure of the uncertainty in a survey’s estimates caused by the use of a 

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

estimates. The United States has approximately 120 million households, and the CE interviews approximately 

21,000 of them per year.1

 The fact that the CE does not interview every household means its estimates have a 

standard error or margin-of-error associated with them. 

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

root of the estimates’ variances, and 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 measure of the amount of accuracy in 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 standard error tables for 2011 show three numbers for every item category: the mean, the standard error, 

and the coefficient of variation. The tables look like this: 

Item 

All 

consumer 

units 

Average annual expenditures 

Mean $49,704.88 

SE 436.12 

CV (%) 0.88 

Food 

Mean $6,457.65 

SE 65.65 

CV (%) 1.02 

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

mentioned above, it comes from interviewing 21,000 households per year. In 2011 the average annual expenditure 

per household on all items (“average annual expenditures”) was estimated to be $49,704.88, and on food it was 

estimated to be $6,457.65. These are sometimes called “point estimates.” 

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 2011 the standard error for all items 

was $436.12, and for food it was $65.65. 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-orminus $436.12, and the mean expenditure for food would have varied by approximately plus-or-minus $65.65. 

In technical 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 

 

1

 The CE collects data from approximately 7,000 households per quarter in the Interview survey and 7,000 

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 28,000 

quarterly interviews and 14,000 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 14,000 unique households in the Interview survey each year. 
describing the variability of these point estimates is $49,704.88 plus-or-minus $854.80 (= $49,704.88 

1.96436.12) and $6,457.65 plus-or-minus $128.67 (= $6,457.65  1.9665.65). 

The third 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 2011 the average annual expenditure per household on all items 

was $49,704.88 and its standard error was $436.12, so its coefficient of variation was 0.88 percent ($436.12 divided 

by $49,704.88 equals 0.0088). Likewise, the average annual expenditure per household on food was $6,457.65 and 

its standard error was $65.65, so its coefficient of variation was 1.02 percent ($65.65 divided by $6,457.65 equals 

0.0102). 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 estimates 

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 larger sample sizes lead to smaller standard errors, and the CE interviews more households in the 

whole country than in each individual region of the country. 

Second, standard errors tend to be larger for broader item categories. 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 broader range of items purchased in higher-level item categories. 

And third, standard errors tend to be smaller for frequently purchased items than for infrequently purchased items. 

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. 

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 xVxSE )( 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 aggregate 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. This is due to the tendency of 

households to spend roughly the same amount of money 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. 

The standard errors can 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 (2009-2011) the standard error for the nationwide all 

items category ranged from $436.12 to $637.58, which gives a rough idea of the standard error of the standard error 

– $550 plus-or-minus $100. From this it can be seen that the estimated mean is generally more stable than the 

estimated standard error in relative terms. 

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 them. 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 subsample. 

Then the variance of the full sample expenditure estimate is computed from 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. Thus 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. The 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 xVxSE 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 i-th 

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.2

More Information 

 

2

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

of all +1’s, which means it does not satisfy the full orthogonal balancing property that is needed. 
For more information, contact the Consumer Expenditure Survey Division by telephone at (202) 691-6000, by email 

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