Concept Margin of error

1 concept

1.1 basic concept
1.2 calculations assuming random sampling
1.3 definition
1.4 different confidence levels
1.5 maximum , specific margins of error
1.6 effect of population size
1.7 other statistics

concept

an example 2004 u.s. presidential campaign used illustrate concepts throughout article. according october 2, 2004 survey newsweek, 47% of registered voters vote john kerry/john edwards if election held on day, 45% vote george w. bush/dick cheney, , 2% vote ralph nader/peter camejo. size of sample 1,013. unless otherwise stated, remainder of article uses 95% level of confidence.

basic concept

polls involve taking sample population. in case of newsweek poll, population of interest population of people vote. because impractical poll vote, pollsters take smaller samples intended representative, is, random sample of population. possible pollsters sample 1,013 voters happen vote bush when in fact population evenly split between bush , kerry, extremely unlikely (p = 2 ≈ 1.1 × 10) given sample random.

sampling theory provides methods calculating probability poll results differ reality more amount, due chance; instance, poll reports 47% kerry support high 50%, or low 44%. theory , bayesian assumptions suggest true percentage close 47%. more people sampled, more confident pollsters can true percentage close observed percentage. margin of error measure of how close results be.

however, margin of error accounts random sampling error, blind systematic errors may introduced non-response or interactions between survey , subjects memory, motivation, communication , knowledge.

calculations assuming random sampling

this section briefly discuss standard error of percentage, corresponding confidence interval, , connect these 2 concepts margin of error. simplicity, calculations here assume poll based on simple random sample large population.

the standard error of reported proportion or percentage p measures accuracy, , estimated standard deviation of percentage. can estimated p , sample size, n, if n small relative population size, using following formula:

standard error

≈




p
(
1
−
p
)

n





{\displaystyle {\text{standard error}}\approx {\sqrt {\frac {p(1-p)}{n}}}}

when sample not simple random sample large population, standard error , confidence interval must estimated through more advanced calculations. linearization , resampling used techniques data complex sample designs.

note there not strict connection between true confidence interval, , true standard error. true p percent confidence interval interval [a, b] contains p percent of distribution, , (100 − p)/2 percent of distribution lies below a, , (100 − p)/2 percent of distribution lies above b. true standard error of statistic square root of true sampling variance of statistic. these 2 may not directly related, although in general, large distributions normal curves, there direct relationship.

in newsweek poll, kerry s level of support p = 0.47 , n = 1,013. standard error (.016 or 1.6%) helps give sense of accuracy of kerry s estimated percentage (47%). bayesian interpretation of standard error although not know true percentage, highly located within 2 standard errors of estimated percentage (47%). standard error can used create confidence interval within true percentage should level of confidence.

the estimated percentage plus or minus margin of error confidence interval percentage. in other words, margin of error half width of confidence interval. can calculated multiple of standard error, factor depending of level of confidence desired; margin of 1 standard error gives 68% confidence interval, while estimate plus or minus 1.96 standard errors 95% confidence interval, , 99% confidence interval runs 2.58 standard errors on either side of estimate.

definition

the margin of error particular statistic of interest defined radius (or half width) of confidence interval statistic. term can used mean sampling error in general. in media reports of poll results, term refers maximum margin of error percentage poll.

different confidence levels

for simple random sample large population, maximum margin of error simple re-expression of sample size n. numerators of these equations rounded 2 decimal places.

margin of error @ 99% confidence



≈
1.29

/



n





{\displaystyle \approx 1.29/{\sqrt {n}}\,}




margin of error @ 95% confidence



≈
0.98

/



n





{\displaystyle \approx 0.98/{\sqrt {n}}\,}




margin of error @ 90% confidence



≈
0.82

/



n





{\displaystyle \approx 0.82/{\sqrt {n}}\,}

if article poll not report margin of error, state simple random sample of size used, margin of error can calculated desired degree of confidence using 1 of above formulae. also, if 95% margin of error given, 1 can find 99% margin of error increasing reported margin of error 30%.

as example of above, random sample of size 400 give margin of error, @ 95% confidence level, of 0.98/20 or 0.049 - under 5%. random sample of size 1600 give margin of error of 0.98/40, or 0.0245 - under 2.5%. random sample of size 10 000 give margin of error @ 95% confidence level of 0.98/100, or 0.0098 - under 1%.

maximum , specific margins of error

while margin of error typically reported in media poll-wide figure reflects maximum sampling variation of percentage based on respondents poll, term margin of error refers radius of confidence interval particular statistic.

the margin of error particular individual percentage smaller maximum margin of error quoted survey. maximum applies when observed percentage 50%, , margin of error shrinks percentage approaches extremes of 0% or 100%.

in other words, maximum margin of error radius of 95% confidence interval reported percentage of 50%. if p moves away 50%, confidence interval p shorter. thus, maximum margin of error represents upper bound uncertainty; 1 @ least 95% true percentage within maximum margin of error of reported percentage reported percentage.

effect of population size

the formula above margin of error assume there infinitely large population , not depend on size of population of interest. according sampling theory, assumption reasonable when sampling fraction small. margin of error particular sampling method same regardless of whether population of interest size of school, city, state, or country, long sampling fraction less 5%.

in cases sampling fraction exceeds 5%, analysts can adjust margin of error using finite population correction , (fpc) account added precision gained sampling close larger percentage of population. fpc can calculated using formula:

fpc
=




n
−
n


n
−
1




.


{\displaystyle \operatorname {fpc} ={\sqrt {\frac {n-n}{n-1}}}.}

to adjust large sampling fraction, fpc factored calculation of margin of error, has effect of narrowing margin of error. holds fpc approaches 0 sample size (n) approaches population size (n), has effect of eliminating margin of error entirely. makes intuitive sense because when n = n, sample becomes census , sampling error becomes moot.

analysts should mindful samples remain random sampling fraction grows, lest sampling bias introduced.

other statistics

confidence intervals can calculated, , can margins of error, range of statistics including individual percentages, differences between percentages, means, medians, , totals.

the margin of error difference between 2 percentages larger margins of error each of these percentages, , may larger maximum margin of error individual percentage survey.