The convergence statement
For iid observations with a finite first absolute moment, the sample mean converges to the population mean in the law of large numbers. This is a statement about a sampling process.
Probability & statistics / English
Law of Large Numbers Walkthrough
Amanda Gregg · YouTube · 4:01
YouTube · Amanda GreggOriginal video
READ & KEEP
The explanation, unpacked.
Reviewed learning material · Video analysis · English01 02 03
Read the full overview
Use interactive sample-mean simulations to illustrate the law of large numbers for uniform and skewed populations.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Chapters
0:00The convergence statement1:10Small and large samples3:18Shape changes the fluctuationsLearning script
Generated from the video's visuals and explanation; not verbatim speech.
For iid observations with a finite first absolute moment, the sample mean converges to the population mean in the law of large numbers. This is a statement about a sampling process.
The uniform-population simulation compares sample sizes such as 2, 30, 50 and 300. Small samples can fluctuate far from the target mean; larger ones tend to concentrate more closely.
Switching to a skewed chi-squared population changes the visible fluctuations. The law does not guarantee monotone improvement for each additional observation or identical convergence speed for every distribution.
Knowledge cards
Small and large samples
The uniform-population simulation compares sample sizes such as 2, 30, 50 and 300. Small samples can fluctuate far from the target mean; larger ones tend to concentrate more closely.
Shape changes the fluctuations
Switching to a skewed chi-squared population changes the visible fluctuations. The law does not guarantee monotone improvement for each additional observation or identical convergence speed for every distribution.
Explore the knowledge in this video
- Law of large numbers ExplanationAt 0:00
Why this connection?
For iid observations with a finite first absolute moment, the sample mean converges to the population mean in the law of large numbers. This is a statement about a sampling process. The uniform-population simulation compares sample sizes such as 2, 30, 50 and 300. Small samples can fluctuate far from the target mean; larger ones tend to concentrate more closely. Switching to a skewed chi-squared population changes the visible fluctuations. The law does not guarantee monotone improvement for each additional observation or identical convergence speed for every distribution.
Similar explanations
YouTube8:59Probability & statistics·English
Law of large numbers | Probability and Statistics | Khan AcademyFoundationalReviewed notes
Another explanation of Law of large numbers.
Law of large numbers
YouTube3:09Another explanation of Law of large numbers.
Law of large numbers
Connect the next step
YouTube4:22Prerequisite in Probability and statistics: Independence → Law of large numbers.
Independence · View learning sequence
YouTube2:27Probability & statistics·English
Die rolling probability with independent events | Precalculus | Khan AcademyFoundationalReviewed notes
Prerequisite in Probability and statistics: Independence → Law of large numbers.
Independence · View learning sequence
YouTube6:59Prerequisite in Probability and statistics: Expectation → Law of large numbers.
Expectation · View learning sequence
A different approach
YouTube8:59Probability & statistics·English
Law of large numbers | Probability and Statistics | Khan AcademyFoundationalReviewed notes
Whiteboard for Law of large numbers, compared with the current visual explanation.
Law of large numbers