The unemployment rate comes from the Household Survey

and the payroll jobs numbers comes from the Establishment Survey. They often widely diverge, and the household survey is much more volatile and lower accuracy.

# Monthly changes for Non-farm payroll employment from the Establishment Survey, thousands
https://data.bls.gov/timeseries/CES0000000001?output_view=net_1mth
2018: 121 406 176 137 278 219 136 244 80 201 134 182
2019: 269 1 147 210 85 182 194 207 208 185 261 184
2020: 214 251 -1373 -20787 2725 4781 1734 1371
Feb and March of 2020 are Corrected. July and August are Preliminary. In thousands
(Yup, that's a "1" in February 2019, meaning only 1,000 net jobs created that month ... I had to look at it again)

# Employed in thousands from the separate Household Survey, Monthly changes:
http://data.bls.gov/timeseries/LNS12000000?output_view=net_1mth
2018: 509 656 49 133 341 85 243 -392 431 450 146 197
2019: -198 239 -125 -45 148 304 198 549 403 246 -8 267
2020: -89 45 -2987 -22369 3839 4940 1350 3756
Jan and Feb of each year are affected by changes in population controls. In thousands

So for August 2020, the nonfarm payroll jobs number increased by 1,371,000 while the "Employed" (Household Survey) increased by a much much larger 3,756,000.

Again, the unemployment rate comes from the Household Survey.

Anyway that's one of the main reasons for the two to diverge big time. Another is the number of people who decide to look for a job (only people who have looked for work in the past 4 weeks are counted as unemployed in the official (U3) unemployment statistics.)

On statistical noise, I found this BLS technical note on sampling error -- http://www.bls.gov/news.release/empsit.tn.htm . Based on what it says, there is a 90% probability that the Establishment Survey's non-farm employment increase is within +/- 120,000 of the stated number. And a 10% chance that it is off by more than 120,000. Again, this is just the sampling error. There are other errors besides sampling error.

Correspondingly, again based on sampling error alone, there is a 50% chance that it is within +/- 49,200. So for example for a reported job gain of 200,000, there is a 50% chance that it is between 150,800 and 249,200, and a 50% chance that it is outside that range based on sampling error alone. Note there are errors other than sampling error that add to the uncertainty {1}

And in the Household Survey, there is a 90% chance that the monthly unemployment change is +/- 300,000 of the stated number (note this is 2.5 times the Establishment Survey's nonfarm employment's sampling error). Also, that there is a 90% chance that the unemployment rate is about +/- 0.2% of the stated number.

The above only covers sampling error. There are also many other sources of error (search the above link for "non-sampling error" ).

Particularly during this pandemic, with some Household Survey surveyors classifying some people on furlough as being employed when they are supposed to be classified as unemployed. For the August report, the BLS claims that the unemployment rate may be understated by a MAXIMUM of 0.7 percentage points due to this problem.

{1} 90% of the area under the normal curve is between +/- 1.645 standard deviations. 50% of the area under the normal curve is between +/- 0.675 standard deviations. Thus if there is a 90% chance that it is within +/- 120,000, then there is a 50% chance that it is within +/- 49,200 (0.675/1.645 * 120,000 = 49,240 , then round to 49,200 ).

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P.S. I'm not "smarter", it's just that I've dug into this particular issue.

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Still a huge huge incredible difference between job gains of 1.371 million (establishment survey) and 3.756 million (household survey)