Editing Margin of error (section)

== Maximum margin of error at different confidence levels ==
[[File:Empirical Rule.PNG|thumb|350px]]For a confidence ''level'' <math>\gamma</math>, there is a corresponding confidence ''interval'' about the mean <math>\mu\plusmn z_\gamma\sigma</math>, that is, the interval <math>[\mu-z_\gamma\sigma,\mu+z_\gamma\sigma]</math> within which values of <math>P</math> should fall with probability <math>\gamma</math>. Precise values of <math>z_\gamma</math> are given by the [[Normal distribution#Quantile function|quantile function of the normal distribution]] (which the 68–95–99.7 rule approximates).

Note that <math>z_\gamma</math> is undefined for <math>|\gamma| \ge 1</math>, that is, <math>z_{1.00}</math> is undefined, as is <math>z_{1.10}</math>.
{| class="wikitable" style="text-align:left;margin-left:24pt;border:none;"
|-
! <math>\gamma</math>
! <math>z_{\gamma}</math>
| rowspan="8" style="border:none;" |&nbsp;
! <math>\gamma</math>
! <math>z_{\gamma}</math>
|-
| 0.84
| {{val|0.994457883210}}
| 0.9995
| {{val|3.290526731492}}
|-
| 0.95
| {{val|1.644853626951}}
| 0.99995
| {{val|3.890591886413}}
|-
| 0.975
| [[1.96|1.959963984540]]
| 0.999995
| {{val|4.417173413469}}
|-
| 0.99
| {{val|2.326347874041}}
| 0.9999995
| {{val|4.891638475699}}
|-
| 0.995
| {{val|2.575829303549}}
| 0.99999995
| {{val|5.326723886384}}
|-
| 0.9975
| {{val|2.807033768344}}
| 0.999999995
| {{val|5.730728868236}}
|-
| 0.9985
| {{val|2.967737925342}}
| 0.9999999995
| {{val|6.109410204869}}
|}

[[File:Margin of error vs sample size and confidence level.svg|thumb|250px|Log-log graphs of <math>MOE_{\gamma}(0.5)</math> vs sample size ''n'' and confidence level ''γ''. The arrows show that the maximum margin error for a sample size of 1000 is ±3.1% at 95% confidence level, and ±4.1% at 99%.<br/>The inset parabola <math>\sigma_p^2 = p-p^2</math> illustrates the relationship between <math>\sigma_p^2</math> at <math>p=0.71</math> and <math>\sigma^2_{max}</math> at <math>p=0.5</math>. In the example, ''MOE''<sub>95</sub>(0.71) ≈ {{nowrap|0.9 × ±3.1%}} ≈ ±2.8%.]]
Since <math>\max \sigma_P^2 = \max P(1-P) = 0.25</math> at <math>p = 0.5</math>, we can arbitrarily set <math>p=\overline{p} = 0.5</math>, calculate <math>\sigma_{P}</math>, <math>\sigma_\overline{p}</math>, and <math>z_\gamma\sigma_\overline{p}</math> to obtain the ''maximum'' margin of error for <math>P</math> at a given confidence level <math>\gamma</math> and sample size <math>n</math>, even before having actual results. &nbsp;With <math>p=0.5,n=1013</math>

:<math>MOE_{95}(0.5) = z_{0.95}\sigma_\overline{p} \approx z_{0.95}\sqrt{\frac{\sigma_{P}^2}{n}} = 1.96\sqrt{\frac{.25}{n}} = 0.98/\sqrt{n}=\plusmn3.1%</math>
:<math>MOE_{99}(0.5) = z_{0.99}\sigma_\overline{p} \approx z_{0.99}\sqrt{\frac{\sigma_{P}^2}{n}} = 2.58\sqrt{\frac{.25}{n}} = 1.29/\sqrt{n}=\plusmn4.1%</math>

Also, usefully, for any reported <math>MOE_{95}</math>

:<math>MOE_{99} = \frac{z_{0.99}}{z_{0.95}}MOE_{95} \approx 1.3 \times MOE_{95}</math>