U.S. Bureau of Labor Statistics 

Current Population Survey (CPS) 

Technical Documentation 

April 2024

www.bls.gov/cps/documentation.htm#reliability

Calculating Approximate Standard 

Errors and Confidence Intervals for 

Current Population Survey Estimates

Each month, at the time of the Employment Situation news release, BLS publishes a statistical 

significance test online for selected labor force indicators. For those who wish to generate their 

own standard errors and confidence intervals, this document provides information about how to 

do this for estimates from the Current Population Survey (CPS).

1 See the accompanying 

Standard error illustrations XLSX file for calculations behind the illustrations in this document. 

Contents

What is a confidence interval?...................................................................................................................... 2

Calculating approximate standard errors and confidence intervals............................................................. 3

Parameter and factor tables PF-1 through PF-16 ..................................................................................... 3

Formulas to calculate standard errors...................................................................................................... 4

1. Standard errors of estimated levels.................................................................................................. 4

2. Standard errors of estimated rates, ratios, and percentages........................................................... 8

3. Standard errors of estimated mean duration of unemployment................................................... 11

4. Standard errors of estimated median duration of unemployment................................................13

Borrowing αα-ββ parameters from another series........................................................................................ 16

Guidelines for selecting an appropriate series to lend αα and ββ parameters: ........................................16

Formulas to calculate standard errors when borrowing αα and ββ parameters and factors....................17

A. Borrowing parameters from one level for use by another level .................................................... 17

B. Borrowing parameters from one rate for use by another rate ...................................................... 18

C. Borrowing parameters from a level for use by a rate..................................................................... 19
2

What is a confidence interval?

When a sample, rather than the entire population, is surveyed, estimates differ from the true 

population values that they represent. The component of this difference that occurs because 

samples differ by chance is known as sampling error, and its variability is measured by the 

standard error of the estimate. A sample estimate and its estimated standard error can be used to 

construct confidence intervals. A confidence interval is a range of values centered around the 

sample estimate that is likely to include the true population value with a degree of confidence. 

When sample estimates are unbiased, the statistical properties of confidence interval “coverage” 

are known. 

The following are examples of confidence intervals:

• A 90% confidence interval is the range from 1.645 standard errors below the estimate to 

1.645 standard errors above the estimate. The true population value is unknown, but there 

is an approximate 90% probability that the interval includes or “covers” the true 

population value. 

• A 95% confidence interval is the range from 1.96 standard errors below the estimate to 

1.96 standard errors above the estimate. The true population value is unknown, but there 

is an approximate 95% probability that the interval includes or “covers” the true 

population value. 

Confidence interval statements like these are approximately true for the CPS, and this is 

especially so for rates and changes over time. For a more complete explanation of confidence 

interval coverage refer to a standard survey methodology text, such as chapter 1.7 in the 3rd

edition of Sampling Techniques by William G Cochran (Wiley, 1977). 

It is important to know if the difference between two estimates is statistically significant. The 

difference may be a comparison of the same characteristic in different time periods (like the 

employment-population ratio in May compared with June) or a comparison between different 

groups in the same time period (like the March unemployment rate for men compared with that 

for women). Given an unbiased estimate of the difference and an unbiased estimate of standard 

error for that difference, confidence intervals can be constructed and statements such as the 

following can be made:

• If zero lies outside of the 90% confidence interval, from 1.645 standard errors below the 

estimated difference to 1.645 standard errors above the estimated difference, the 

difference is statistically significant at a 90% confidence level. If zero lies within the 

interval, the difference is not statistically significant at a 90% confidence level.

• If zero lies outside of the 95% confidence interval from 1.96 standard errors below the 

estimated difference to 1.96 standard errors above the estimated difference, the difference 

is statistically significant at a 95% confidence level. If zero lies within the interval, the 

difference is not significantly different at a 95% confidence level.
3

Calculating approximate standard errors and confidence intervals

Parameter and factor tables PF-1 through PF-16

Approximate standard errors and confidence intervals for CPS estimates can be calculated using 

the following instructions and parameter and factor tables (PF-1 through PF-16) that are 

available online. (These tables mirror tables A-1 through A-16 of the monthly Employment 

Situation news release.) The parameters and factors are used in formulas that are commonly 

called generalized variance functions. The parameter and factor tables allow users to calculate 

approximate standard errors for a wide range of estimated levels, rates, and percentages, and also 

changes over time. 

These tables do not include parameters and factors for all possible estimates from the CPS; they 

only include parameters and factors for series that are published in the Employment Situation 

news release. The next section of this document provides instructions on how to borrow 

parameters and factors from the estimates covered in these tables to calculate an approximate 

standard error for estimates that are not covered. 

Information presented here may be of use to researchers working with official BLS estimates as 

well as those computing estimates using the public use CPS microdata files, available from the 

U.S. Census Bureau's website or from Census’s Microdata Access Tool (MDAT). Note that 

estimates generated using the public use microdata files, except for a few topside estimates, will 

not match official BLS estimates.2

CPS parameters and factors tables give α and β parameters that can be used with formulas to 

calculate approximate monthly standard errors for a wide range of estimated levels, proportions, 

and rates. Factors are provided to convert monthly standard error measures into approximate 

standard errors of estimates for other periods and approximate standard errors for changes over 

time. Specifically, the tables show factors (f) that are used in calculating standard errors for:

• Consecutive month-to-month changes

• Changes in monthly estimates 1 year apart

• Quarterly averages

• Changes in consecutive quarterly averages

• Yearly averages

• Changes in consecutive yearly averages

The standard errors for estimated changes in level estimates from one time period to the next (for 

example, one month to the next or one year to the next) depend more on the monthly levels than 

on the size of the changes. Likewise, the standard errors for changes in rates (or percentages), 

means, and medians depend more on the monthly figures than on the size of the changes. 

Accordingly, the factors presented in tables PF-1 through PF-16 are applied to the monthly 

standard error approximations for levels, percentages, or rates; the magnitudes of the changes do 

not come into play. Factors are not given for estimated changes between nonconsecutive months 

(except for changes of monthly estimates 1 year apart); however, the standard errors for changes 
4

between nonconsecutive months are generally assumed to be larger than those for consecutive 

months.

IMPORTANT: CPS data are published as levels in thousands, so 4 million unemployed 

people would be displayed as 4,000 in a table. Similarly, an entry of 2,460 in a CPS table 

represents 2,460,000 people. When calculating standard errors and confidence intervals, 

you must use the actual level at full precision, not the data as published in thousands. 

Formulas to calculate standard errors

1. Standard errors of estimated levels

Standard errors for estimated monthly levels

The approximate standard error se(x) of x, an estimated monthly level, can be obtained using the 

formula below, where α and β are the parameters from parameter and factor tables PF-1 through 

PF-16 associated with a particular characteristic, and N is the total civilian noninstitutional 

population 16 years and over.

ssss(xx; NN) = �(α + βNN) �xx − xx2

NN �

Note that x is the monthly level (not in thousands).

Illustration 1.1 – calculating a standard error for a single-month level

In a given month, there are an estimated 4 million unemployed men, and the civilian 

noninstitutional population is 250 million. Obtain the appropriate α and β parameters from table 

PF-1 (in row 22 for unemployed men, 16 years and over). Use the formula above for se(x; N) to 

compute an approximate standard error on the estimate of x = 4,000,000 where N = 250,000,000.

α = -6127.75

β = 0.00003934

ssss(4,000,000; 250,000,000)

= ��−6127.75 + (0.00003934 ∗ 250,000,000)� ∗ (4,000,000 − � 4,000,0002

250,000,000�)

= 120,796

ssss(4,000,000; 250,000,000) ≈ 121,000
5

Standard errors for levels: Estimated quarterly averages, annual averages, and changes over time

The approximate standard error se(x) of x, an estimated level that is a quarterly average, annual 

average, or change over time, can be obtained using the formula below, where α and β are the 

parameters and f is the factor associated with a particular characteristic (from tables PF-1 

through PF-16), and N is the total civilian noninstitutional population 16 years and over.

The f in the formula is frequently called an adjustment factor because it appears to adjust a 

monthly standard error se(x). However, the x and N in the formula are not monthly levels, but 

averages of monthly levels (see examples listed below).

ssss(xx;N; ff) = ff ∗ ssss(xx; NN) = ff ∗ �(αα + ββ )(xx − xx2

NN )

Step 1. If quarterly or annual averages are not available, you will need to calculate them 

from the monthly estimates. Average levels appropriately to obtain x and N. Levels for 3 

months are averaged to calculate quarterly averages, and levels for 12 months are 

averaged to calculate annual averages. For changes in consecutive levels, average over 

the 2 months, 2 quarters, or 2 years involved. For changes in monthly estimates 1 year 

apart, average the 2 months involved.

Step 2. Calculate an approximate standard error se(x), treating the average x from step 1 

as if it were an estimated level for a single month. For a given characteristic, obtain 

parameters α and β from the applicable PF table. 

Step 3. Determine the standard error se (x; f) on the average level or on the change in 

level. Multiply the result from step 2 by the appropriate factor f. The α and β parameters 

used in step 2 and the factor f used in this step come from the same line in the 

corresponding PF table.

Illustration 1.2 – calculating a standard error for a consecutive month change in 

level

Continuing from illustration 1.1 above, suppose that in the next month the estimated number of 

unemployed men increases by 150,000, from 4,000,000 to 4,150,000. Also suppose the civilian 

noninstitutional population 16 years and over increases by 200,000, from 250,000,000 to 

250,200,000.

Step 1. Average the two monthly levels for the estimate and the total population to obtain

x = 4,075,000 and N = 250,100,000.
6

Step 2. Apply the α and β parameters from table PF-1 (in row 22 for unemployed men, 

16 years and over) to the average x and N, treating them like an estimate and a population 

value for a single month.

α = -6127.75

β = 0.00003934

ssss(4,075,000; 250,100,000)

= �� −6127.75 + (0.00003934 ∗ 250,100,000)� ∗ (4,075,000 − 4,075,0002

250,100,000)

= 121,970

ssss(4,075,000; 250,100,000) ≈ 122,000

Step 3. Obtain f = 1.12 from the same row of table PF-1 in the column “Consecutive 

month-to-month change,” and multiply the factor by the result from step 2 to calculate 

the standard error for the change between the 2 months.

ssss(150,000) = ff ∗ ssss(4,075,000; 250,100,000) = 1.12 ∗ 121,970 = 136,606

ssss(150,000) ≈ 137,000

For an approximate 90-percent confidence interval, compute 1.645 * 136,606 = 224,717 ≈

225,000. Next, subtract the number from and add the number to 150,000 to obtain an interval 

of -75,000 to 375,000. This is an approximate 90-percent confidence interval for the true change, 

and since this interval includes zero, one cannot assert at this level of confidence that any real 

change has occurred in the unemployment level over the month. The result also can be expressed 

by saying that the apparent change of 150,000 is not statistically significant at a 90-percent 

confidence level.

Illustration 1.3 – calculating a standard error for a quarterly average level

Suppose that an approximate standard error is desired for a quarterly average of the Black or 

African American employment level. Suppose that the estimated employment levels for the 3 

months making up the quarter are 14,900,000; 15,000,000; and 15,100,000, and the 

corresponding levels of the civilian noninstitutional population 16 years and over are 

249,800,000; 250,000,000; and 250,200,000.

Step 1. The averages of the 3 monthly levels are x = 15,000,000 and N = 250,000,000.
7

Step 2. Apply the α and β parameters from table PF-2 (in row 40 for employed Black or 

African American people) to the average x and N, treating them like an estimate for a 

single month.

α = -1478.12

β = 0.00001191

ssss(15,000,000; 250,000,000)

= ��−1478.12 + (0.00001191 ∗ 250,000,000)� ∗ (15,000,000 − 15,000,0002

250,000,000)

= 145,400

ssss(15,000,000; 250,000,000) ≈ 145,000

Step 3. Obtain f = 0.85 from the same row of table PF-2 in the column “Quarterly 

averages,” and multiply the factor by the result from step 2 to calculate the standard error 

of the quarterly average.

ssss(15,000,000) = 0.85 ∗ 145,400 = 123,590

ssss(15,000,000) ≈ 123,600

Illustration 1.4 – calculating a standard error for a consecutive quarter change in 

level

Continuing the example from illustration 1.3, suppose that, in the next quarter, the estimated 

average employment level for Blacks is 15,400,000, based on monthly levels of 15,300,000;

15,400,000; and 15,500,000. This is an estimated increase of 400,000 over the previous quarter.

Also suppose the average population level in the next quarter is 250,600,000, based on monthly 

levels of 250,400,000; 250,600,000; and 250,800,000.

Step 1. The average of the two quarterly levels (15,000,000 and 15,400,000) is x = 

15,200,000. The average of the two quarterly population levels (250,000,000 and 

250,600,000) is N = 250,300,000.

Step 2. Apply the α and β parameters from table PF-2 (in row 40 for employed Black or 

African American people) to the average x and N, treating them like an estimate for a 

single month.
8

α = -1478.12

β = 0.00001191

ssss(15,200,000; 250,300,000)

= ��−1478.12 + (0.00001191 ∗ 250,300,000)� ∗ (15,200,000 − 15,200,0002

250,300,000)

= 146,484

ssss(15,200,000; 250,300,000) ≈ 146,000

Step 3. Obtain f = 0.81 from the same row of table PF-2 (row 40 for employed Black or 

African American people) in the column “Change in consecutive quarterly averages,” and 

multiply the factor by the result from step 2 to calculate the standard error of the change 

in quarterly averages.

ssss(400,000) = 0.81 ∗ ssss(15,200,000) = 0.81 ∗ 140,195 = 118,652

ssss(400,000) ≈ 119,000

For an approximate 90-percent confidence interval, compute 1.645 * 118,652 = 195,183 ≈

195,000. Subtract the number from and add the number to 400,000 to obtain an interval of 

205,000 to 595,000. The interval excludes zero. Another way of stating this is to observe that the 

estimated change of 400,000 clearly exceeds 1.6 standard errors, or 195,000. One can conclude 

from these data that the change in quarterly averages is statistically significant at a 90-percent 

confidence level.

2. Standard errors of estimated rates, ratios, and percentages

Standard errors for estimated monthly rates, ratios, and percentages

As shown in the formula below, the approximate standard error se(p;y) of an estimated rate or 

percentage p depends, in part, upon the number of people y in its base or denominator (which is

the number of people represented by a particular estimate). The α and β parameters are obtained 

from the parameter and factor (PF) tables. 

For a rate, ratio, or percentage covered in the PF tables, use the α and β parameters for that 

estimate.

For a percentage that is not itself covered in the PF tables but that is calculated from two series 

that are INCLUDED in the PF tables, use the α and β parameters from the PF table relevant to the 

numerator of the rate or percentage. For example, when calculating a standard error for the 

proportion of the of the labor force made up by women (two series that are INCLUDED in the PF 

tables), use the α and β parameters associated with the number of women in the labor force (the 

numerator) rather than the total number of people in the labor force (the denominator). 
9

For a rate or percentage calculated from two series that are NOT INCLUDED in the PF tables, see 

the instructions on how to borrow parameters and factors below. 

ssss(pp; yy) = �(αα + ββ )

yy pp(100 − pp)

Note that y (not in thousands) is the base of percent p, and se(p; y) is in percent.

Illustration 2.1 – calculating a standard error for a single-month percent

For a given month, suppose y = 156,000,000 employed people. Of this total, 27,000,000, or p = 

17.3 percent, are classified as part-time workers. Obtain the α and β parameters from table PF-9

(in row 49 for part-time workers) that are relevant to the numerator of the percentage. Apply the 

formula to obtain:

αα = −4841.52

ββ = 0.00003413

ssss(17.3; 156,000,000)

= �(−4841.52 + (0.00003413 ∗ 156,000,000))

156,000,000 ∗ 17.3 ∗ (100 − 17.3)

= 0.067 ≈ 0.1 percent

For an approximate 90-percent confidence interval, compute 1.645 * 0.067 = 0.109 percent ≈ 0.1

percent. Subtract this from and add this to the estimate of p = 17.3 percent to obtain an 

approximate confidence interval of 17.2 percent to 17.4 percent.

Standard errors for percentages: Estimated for quarterly averages, annual averages, and changes over 

time 

Factors from tables PF-1 through PF-16 can be used to compute approximate standard errors on 

rates, ratios, and percentages for quarterly averages, annual averages, and for changes over time. 

As with levels, there are three steps in the procedure for applying the formula.

ssss(pp; yy; ff) = ff ∗ ssss(pp; yy) = ff ∗ �(αα + ββ )

yy pp(100 − pp)

Note that p and y are averages for the designated period, 

and se(p; y; f) is a percent.
10

Step 1. If you need a quarterly or annual average that is not available, then you will need 

to calculate one by averaging the monthly level estimates for the numerator and 

denominator and dividing the average for the numerator by the average for the 

denominator to obtain p. The average for the denominator will be y. For changes in 

consecutive averages, average over the 2 months, 2 quarters, or 2 years involved. For 

changes in monthly estimates 1 year apart, average the 2 months involved.

Step 2. Calculate an approximate standard error se(p; y), treating the averages p and y 

from step 1 as if they were estimates for a single month. Obtain the α and β parameters

from the PF table that correspond to the numerator of the rate or percentage. 

Step 3. Determine the standard error se(p; y; f) for the average percentage or the change 

in percentage. Multiply the result from step 2 by the appropriate factor f. The α and β

parameters used in step 2 and the factor f used in this step will come from the same row

in the appropriate PF table.

Illustration 2.2 – calculating a standard error for a consecutive month change in 

percentage

Continuing the example from illustration 2.1, suppose that, in the next month, there are 

156,600,000 employed people, and that 28,000,000, or 17.9 percent, are part-time workers.

Step 1. The month-to-month change is 0.6 percentage point—that is, the share of the

employed who worked part time changed from 17.3 percent to 17.9 percent over the 

month. The average of the two monthly percentages of 17.3 percent and 17.9 percent is 

needed (p = 17.6 percent), as is the average of the two bases of 156,000,000 and 

156,600,000 (y = 156,300,000).

Step 2. Apply the α and β parameters from table PF-9 (in row 49 for part-time workers)

to the averaged p and y, treating the averages like estimates for a single month.

αα − 4841.52

ββ = 0.00003413

ssss(17.6; 156,300,000)

= �(−4841.52 + (0.00003413 ∗ 156,300,000))

156,300,000 ∗ (17.6) ∗ (100 − 17.6) = 0.068

≈ 0.1 percent
11

Step 3. Obtain f = 1.00 from the same row of table PF-9 in the column “Consecutive 

month-to-month change,” and multiply the factor by the result from step 2.

ssss(0.6 percent) = 1.00 ∗ 0.068 = 0.068 percent ≈ 0.1 percent

For an approximate 90-percent confidence interval, compute 1.645 * 0.068 = 0.111 ≈ 0.1

percent. Subtract this from and add this to the 0.6-percentage point estimate of change to obtain 

an interval of 0.5 percent to 0.7 percent. Because this interval does not include zero, it can be 

concluded that the over-the-month change is statistically significant at a 90-percent confidence 

level.

3. Standard errors of estimated mean duration of unemployment

Standard errors for estimated single-month means

The approximate standard error ssss(xx̅; yy) of xx̅, the monthly estimate of mean duration of 

unemployment, can be obtained using the formula below, where αα and ββ are the parameters from 

parameter and factors table PF-12. The formula depends, in part, upon the total number of 

unemployed people yy in its denominator.

ssss(xx̅; yy) = αα + ββxx̅

�yy

Note that yy is a monthly level (not in thousands).

Illustration 3.1 – calculating a standard error for a single-month mean

In a given month, suppose there are an estimated 7 million unemployed people, and the 

estimated mean duration of unemployment is 22.7 weeks. To estimate the standard error for the 

mean duration of unemployment, start by finding the appropriate αα and ββ parameters from table 

PF-12 (in row 12 for the average duration of unemployment in weeks). Use the formula ssss(xx̅; yy)

to compute an approximate standard error for the estimate of xx̅= 22.7 where yy = 7,000,000.

αα = 521.77

ββ = 55.85768590

ssss(22.7; 7,000,000) = 521.77 + (55.85768590 ∗ 22.7)

√7,000,000 = 0.6765

ssss(22.7; 7,000,000) ≈ 0.68 weeks
12

Standard errors for means: Estimated for quarterly averages, annual averages, and changes over time

The approximate standard error ssss(xx̅; yy; ff) of xx̅, the average of monthly estimate of mean 

duration of unemployment over the designated period, can be obtained using the formula below. 

Table PF-12 provides factors that can be used to compute approximate standard errors of mean 

duration of unemployment for quarterly averages, annual averages, and changes over time. The 

formula depends, in part, upon the total number of unemployed people yy in its denominator. As 

with levels and rates, there are 3 steps in the procedure for using the formula.

ssss(xx̅; yy; ff) = ff ∗ ssss(xx̅; yy) = ff ∗ �

αα + ββxx̅

�yy �

Note that xx̅is the mean duration of unemployment over the designated period, yy is the 

average number of unemployed people over the designated period, and ssss(xx̅; yy; ff) is in 

weeks. If quarterly or annual averages are not available, you will need to calculate them 

by averaging the monthly estimates.

Step 1. Average monthly estimates appropriately to obtain xx̅and yy. Means (xx̅) and levels 

(yy) for 3 months are averaged for quarterly averages, and those for 12 months are 

averaged for yearly averages. For changes in consecutive mean estimates, average over 

the 2 months, 2 quarters, or 2 years involved. For changes in monthly estimates 1 year 

apart, average the 2 months involved.

Step 2. Calculate an approximate standard error ssss(xx̅; yy), treating the averages xx̅and yy

from step 1 as if they were estimated means and levels, respectively, for a single month. 

Obtain the appropriate αα and ββ parameters from table PF-12.

Step 3. Determine the standard error ssss(xx̅; yy; ff) on the average (average of multiple 

monthly mean estimates) or on the change in mean estimates. Multiply the result from 

step 2 by the appropriate factor ff. The αα and ββ parameters used in step 2 and the factor ff

used in this step come from the same row of table PF-12.

Illustration 3.2 – calculating a standard error for a consecutive month change in 

mean duration of unemployment

Continuing the example from illustration 3.1, suppose that in the next month the estimated 

number of unemployed people decreased by 200,000, from 7,000,000 to 6,800,000. Also,

suppose the estimate of the mean duration of unemployment increased by 2.0 weeks, from 22.7 

to 24.7 weeks.

Step 1. The averages of the two monthly means (xx̅) and levels (yy) are xx̅= 23.7 and 

yy = 6,900,000.
13

Step 2. Apply the αα and ββ parameters from table PF-12 (Unemployed people by duration 

of unemployment) to the averages xx̅and yy, treating them like estimates for a single 

month.

αα = 521.77

ββ = 55.85768590

ssss(23.7; 6,900,000) = 521.77 + (55.85768590 ∗ 23.7)

√6,900,000 = 0.7026

ssss(23.7; 6,900,000) ≈ 0.70 weeks

Step 3. Obtain ff = 1.11 from the same row of table PF-12 (in row 12 for the average 

duration of unemployment in weeks) in the column “Consecutive month-to-month 

change,” and multiply the factor by the result from step 2 to calculate the approximate 

standard error for the change between the two months.

ssss(2.0) = ff ∗ ssss(23.7; 6,900,000) = 1.11 ∗ 0.7026 = 0.7799

ssss(2.0) ≈ 0.78 weeks

For an approximate 90-percent confidence interval, compute 1.645 ∗ 0.7799 = 1.2829 ≈ 1.3

weeks. Next, subtract the number from and add the number to 2.0 weeks to obtain an interval of 

0.7 to 3.3 weeks. Because this interval does not include zero, it can be concluded that the change 

in the mean duration of unemployment is statistically significant at a 90-percent confidence 

level. 

4. Standard errors of estimated median duration of unemployment

Standard errors for estimated single-month medians

The approximate standard error ssss(mm; yy) of mm, the monthly estimate of median duration of 

unemployment, can be obtained using the formula below, where αα and ββ are the parameters from 

parameter and factors table PF-12. The formula depends upon both the estimates of mean weeks 

unemployed xx̅and the total number of unemployed people yy but does not depend directly on the 

estimated median mm.

ssss(mm; xx̅; yy) = αα + ββ(xx̅yy)

�yy

Note that yy is a monthly level (not in thousands).
14

Illustration 4.1 – calculating a standard error for a single month median

Suppose that in a given month there are an estimated 7 million unemployed people, the mean 

duration of unemployment is 22.7 weeks, and the median duration of unemployment is 9.4 

weeks. Obtain the appropriate αα and ββ parameters from table PF-12 (in row 13 for the median 

duration of unemployment in weeks). Use the formula ssss(mm; xx̅; yy) to compute an approximate 

standard error on the estimate of mm = 9.4, where xx̅= 22.7 and yy = 7,000,000.

αα = 274.00

ββ = 0.00000405

ssss(9.4; 22.7; 7,000,000) = 274.00 + 0.00000405 ∗ (22.7 ∗ 7,000,000)

√7,000,000 = 0.3468

ssss(9.4; 22.7; 7,000,000) ≈ 0.35 weeks

Standard errors for medians: Estimated quarterly, annually, and for changes over time

The approximate standard error ssss(mm; xx̅; yy; ff) of mm, the quarterly or annual median duration of 

unemployment, or change over time in this measure, can be obtained using the formula below. 

Table PF-12 provides factors that can be used in the computation. The formula depends upon 

estimates of both the mean weeks unemployed xx̅and the total number of unemployed people yy,

but does not depend directly on the estimated median mm. As with levels, rates, and means, there 

are 3 steps in the procedure for using the formula.

ssss(mm; xx̅; yy; ff) = ff ∗ ssss(mm; xx̅; yy) = ff ∗ �

αα + ββ(xx̅yy)

�yy �

Note that xx̅is the mean duration of unemployment over the designated period, yy is the 

number of unemployed people, and ssss(mm; xx̅; yy; ff) is in weeks.

Step 1. Average monthly estimates appropriately to obtain xx̅and yy. Means (xx̅) and levels 

(yy) for 3 months are averaged for quarterly averages, and those for 12 months are 

averaged for yearly averages. To calculate a standard error for changes in consecutive 

median estimates, average the medians (mm), means (xx̅) and levels (yy) over the 2 months, 

2 quarters, or 2 years involved. For changes in monthly estimates 1 year apart, average 

the 2 months involved.

Step 2. Calculate an approximate standard error ssss(mm; xx̅; yy), treating the averages from 

step 1 as if they were estimates for a single month. Obtain the appropriate αα and ββ

parameters from table PF-12.
15

Step 3. Determine the standard error ssss(mm; xx̅; yy; ff) on the average median or on the 

change in median estimates. Multiply the result from step 2 by the appropriate factor ff. 

The αα and ββ parameters used in step 2 and the factor ff used in this step come from the 

same row of table PF-12.

Illustration 4.2 – calculating a standard error for a quarterly median duration of 

unemployment

Suppose the median duration of unemployment estimates for a quarter is 10.5 weeks. Also 

suppose the mean duration of unemployment estimates for 3 consecutive months are 22.7, 24.7, 

and 24.3 weeks, and the estimates of total unemployed people for 3 consecutive months are 

7,000,000; 6,800,000; and 6,000,000. 

Step 1. The averages of the means (xx̅) and levels (yy) are xx̅= 23.9 and yy = 6,600,000.

Step 2. Apply the αα and ββ parameters from table PF-12 (in row 13 for the median 

duration of unemployment in weeks to the averages xx̅and yy, treating them like estimates 

for a single month.

αα = 274.00

ββ = 0.00000405

ssss(10.5; 23.9; 6,600,000) = 274.00 + 0.00000405 ∗ (23.9 ∗ 6,600,000)

√6,600,000 = 0.3553

ssss(10.5; 23.9; 6,600,000) ≈ 0.36 weeks

Step 3. Obtain ff = 0.69 from the same row of table PF-12 in the column “Quarterly 

averages,” and multiply the factor by the result from step 2 to calculate the approximate 

standard error for the quarterly average of median duration of unemployment.

ssss(10.5) = ff ∗ ssss(10.5; 23.9; 6,600,000) = 0.69 ∗ 0.3553 = 0.2452

ssss(10.5) ≈ 0.25 weeks

For an approximate 90-percent confidence interval, compute 1.645 ∗ 0.2452 = 0.4033 ≈ 0.4

weeks. Subtract the number from and add the number to 10.5 weeks to obtain a 95-percent 

confidence interval of 10.1 to 10.9 weeks. 
16

Borrowing αα-ββ parameters from another series

The αα and ββ parameters appearing on the PF tables (PF-1 through PF-16) can be used to 

generate standard errors for CPS estimates that do not appear on tables A1-A16. However, the 

terms in the standard error formulas need to be modified slightly. 

When calculating standard errors for CPS estimates NOT appearing on tables A1-A16 of the 

Employment Situation, it is common to use αα and ββ parameters and factors from an available 

series that is conceptually similar. When borrowing αα and ββ parameters, the implicit assumption 

is that the design effects3 of the lending series and the borrowing series are approximately equal.

There are two primary types of αα and ββ parameters: those for levels and those for rates. 

When borrowing, or generalizing, αα and ββ parameters from a level series, it is important to use 

the total civilian noninstitutional population 16 years and over as NN. When borrowing from a 

rate, ratio, or percentage series, it is important to use the base of that the rate, ratio, or 

percentage, which is referred to as yybbbbbbbbbbbbbbbb, in the formula. Using the wrong population 

estimate or base is a common mistake that can result in inaccurate standard errors.

Note: Parameters for the mean and median duration of unemployment cannot be generalized to 

other means or medians. Additionally, the standard error formulas for the mean and median 

duration of unemployment cannot be used to generate approximate standard errors for other 

means or medians. 

The formulas presented below demonstrate how to borrow αα and ββ parameters from one series 

and apply them to another. 

Guidelines for selecting an appropriate series to lend αα and ββ parameters:

• If possible, αα and ββ parameters should be borrowed from a series of the same type: levelto-level conversions and rate-to-rate conversions. However, sometimes a level-to-rate 

conversion may be necessary.

• For unemployment level or unemployment rate estimates, αα and ββ parameters should 

only be borrowed from an available series that also measures unemployment. Parameters 

for estimates of employed people, the civilian labor force, and people not in the labor 

force can be borrowed from one another, if necessary, as these types of estimates tend to 

have similar design effects. 

• When considering multiple series to borrow from, using the αα and ββ parameters that 

generate the highest standard error is generally advised.

If borrowing factors for changes over time, means, or medians, apply these modifications in the 

same way described below in the formulas to calculate standard errors. 
17

Formulas to calculate standard errors when borrowing αα and ββ parameters and 

factors

A. Borrowing parameters from one level for use by another level

If the α and β parameters and factors are not available for a specific level from tables PF-1 

through PF-16, the approximate standard error se(x) of x can be obtained by borrowing 

parameters and factors from a similar level series. The borrowed parameters and factors would 

be entered into the same formulas displayed above to calculate the standard errors of estimated 

levels. 

Illustration A – calculating a standard error for a level when borrowing 

parameters from one level for use by another level 

Suppose that in a given month, there are 36 million people ages 25 to 34 in the labor force, and 

the civilian noninstitutional population is 250 million. This level estimate does not appear on 

tables A1–A16 of the Employment Situation news release, and thus tables PF-1 through PF-16

do not list corresponding α and β parameters. A standard error can be estimated by borrowing α

and β parameters from another level series and entering them into the standard error formula for 

a single-month level.

Step 1. Choose a level series similar to that for people ages 25 to 34 in the labor force that 

is included in tables PF-1 through PF-16, such as employed people ages 25 to 34 (in row 

15 of table PF-9). Borrow the αα and ββ parameters from that series:

αα = −1410.35

ββ = 0.00000900

Step 2. Use the universe total N, which is the total civilian noninstitutional population 16 

years and over. 

NN = 250,000,000

Step 3. Enter these values into the formula to calculate a standard error for a single-month 

level:

SSSSGGGGGG = �(αα + ββ ) �xx − xx2

NN �

The approximate standard of xx is then calculated as:
18

SSSSGGGGGG = ��−1410.35 + (0.00000900 ∗ 250,000,000)� �36,000,000 − 36,000,0002

250,000,000 �

= 160,856

SSSSGGGGGG ≈ 161,000

B. Borrowing parameters from one rate for use by another rate

If the α and β parameters and factors are not available for a specific rate or percent from tables 

PF-1 through PF-16, the approximate standard error se(p) of p can be obtained by borrowing 

parameters and factors from a similar rate or percent series. The borrowed parameters and factors 

would then be entered into the same formulas displayed above to calculate the standard errors of 

rates, ratios, and percents.

Illustration B – calculating a standard error for a rate or percent when borrowing 

parameters from one rate for use by another rate

Suppose that in a given month, there are 18 million people ages 25 to 29 in the labor force, there 

are 36 million people ages 25 to 34 in the labor force, and the unemployment rate for people ages 

25 to 29 is 3.7 percent. This rate estimate does not appear on tables A1–A16 of the Employment 

Situation news release, and thus tables PF-1 through PF-16 do not list corresponding α and β

parameters. A standard error can be estimated by borrowing α and β parameters from another 

rate series and entering them into the standard error formula for a single-month rate, ratio, or 

percent.

Step 1. Choose a rate series similar to the unemployment rate for people ages 25 to 29 

that is included in tables PF-1 through PF-16, such as the unemployment rate for people

ages 25 to 34 (in row 62 of table PF-10). Borrow the αα and ββ parameters from that series:

αα = −8511.91

ββ = 0.00035280

Step 2. Determine the base or denominator, yybbbbbbbbbbbbbbbb, of the series the parameters are 

borrowed from—in this example, the base for the unemployment rate of people ages 25 

to 34. The base of this unemployment rate is the number of people ages 25 to 34 in the 

labor force, in this case 36 million. This term, yybbbbbbbbbbbbbbbb, is used with the borrowed 

parameters in the numerator of the standard error calculation.

yybbbbbbbbbbbbbbbb = 36,000,000
19

Step 3. Determine the base or denominator, y, of the series being analyzed—the 

unemployment rate for people ages 25 to 29. The base of the unemployment rate is the 

number of people ages 25 to 29 in the labor force, in this case 18 million.

yy = 18,000,000

Step 4. Enter these values into the formula to calculate a standard errors of rates, ratios, 

and percents modified as follows:

SSSSGGGGGG = ��αα + ββyybbbbbbbbbbbb bb�

yy pp(100 − pp)

Where yybbbbbbbbbbbbbbbb = base of the series that the parameters are borrowed from

The approximate standard of p is then calculated as:

SSSSGGGGGG = �(−8511.91 + (0.00035280 ∗ 36,000,000))

18,000,000 ∗ 3.7 ∗ (100 − 3.7)

= 0.288 ≈ 0.3 percent

C. Borrowing parameters from a level for use by a rate

If the α and β parameters and factors are not available for a specific rate or percent from tables 

PF-1 through PF-16, and there is not a similar rate or percent series available in the tables, the 

approximate standard error se(x) of x can be obtained by borrowing parameters and factors from 

a related level series. The borrowed parameters and factors would then be entered into the same 

formulas displayed above to calculate the standard errors of rates, ratios, and percents.

Illustration C – calculating a standard error for a rate or percent when borrowing 

parameters from a level

Suppose that in a given month, the labor force participation rate for people ages 25 to 29 is 81.5 

percent, the civilian noninstitutional population age 16 and over numbers 250 million, and there 

are 22 million people ages 25 to 29 in the population. The labor force participation rate for 

people ages 25 to 29 does not appear on tables A1–A16 of the Employment Situation news 

release, and thus tables PF-1 through PF-16 do not list corresponding α and β parameters. Nor is 

there a similar rate series, such as the employment-population ratio for this age group, in tables

PF-1 through PF-16. A standard error can be estimated by borrowing α and β parameters from 
20

another level and entering them into the standard error formula for a single-month rate, ratio, or 

percent.

Step 1. Choose a level series related to labor force participation rate for people ages 25 to 

29 that is included in tables PF-1 through PF-16, such as the number of employed people

ages 25 to 34 (in row 15 of table PF-9). Borrow the αα and ββ parameters from that series:

αα = −1410.35

ββ = 0.00000900

Step 2: Because the series being borrowed from is a level series and does not have a base 

or denominator, use the universe total N, which is the total civilian noninstitutional 

population 16 years and over. 

NN = 250,000,000

Step 3. Determine the base or denominator, y, of the series being analyzed—the labor 

force participation rate for people ages 25 to 29. The base of the labor force participation 

rate is the population, in this case the 22 million people ages 25 to 29 in the population.

yy = 22,000,000

Step 4. Enter these values into the formula to calculate a standard errors of rates, ratios, 

and percents modified as follows:

SSSSGGGGGG = �(αα + ββNN)

yy pp(100 − pp)

Where NN = the civilian noninstitutional population 16 years and over

The approximate standard of p is then calculated as:

SSSSGGGGGG = �(−1410.35 + (0.00000900 ∗ 250,000,000))

22,000,000 ∗ 81.5 ∗ (100 − 81.5)

= 0.240 ≈ 0.2 percent

1 A June 2022 update of this document introduces information on calculating standard errors for mean and median 

duration of unemployment and slightly revises the parameter and factor values based on standard error models that 

were updated with the release of January 2019 data. A November 2018 update of this document introduced the α

and β parameters and a slightly different methodology for calculating standard errors.
21

2 Beginning with data for January 2011, the Census Bureau incorporated additional safeguards in the CPS public use 

microdata files to ensure that respondent identifying information is not disclosed. In general, respondents' ages were 

altered, or "perturbed," in the public use microdata files to further protect the confidentiality of survey respondents 

and the data they supply. One result of the measures taken to enhance data confidentiality is that labor force and 

other estimates from the public use microdata files will no longer exactly match most estimates published by BLS, 

which are based on internal, nonpublic-use files. Although certain topside labor force estimates for all persons will 

continue to match published data—such as the overall levels of employed, unemployed, and not in the labor force—

estimates below the topside level (such as employment status by age, sex, race, and Hispanic or Latino ethnicity) all 

have the chance of differing slightly from the published data. In addition, estimates calculated using characteristics 

such as industry, occupation, hours worked, duration of unemployment, along with all other characteristics not 

expressly listed above, are subject to such differences. All such differences should fall well within the sampling 

variability associated with CPS estimates.

3 A design effect is a survey methods concept relating the variance of a statistic from a complex survey design (such 

as that used by the CPS) relative to a simple random sample design. CPS does not publish design effects, but the αα-

ββ parameters account for them.