SOCR EduMaterials Activities LawOfLargeNumbers

This is a heterogeneous Activity that demonstrates the Law of large Numbers (LNN)

The Law of Large Numbers (LLN)

Example

The average weight of 10 students from a class of 100 students is most likely closer to the real average weight of all 100 students, compared to the average weight of 3 randomly chosen students from that same class. This is because the sample of 10 is a larger number than the sample of only 3 and better represents the entire class. At the extreme, a sample of 99 of the 100 students will produce a sample average almost exactly the same as the average for all 100 students. On the other extreme, sampling a single students will be an extremely variant estimate of the overall class average weight.

Statement of the Law of Large Numbers

If an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.

SOCR Demonstrations of the LLN

• Exercise 1: Go to the SOCR Experiments and select the Binomial Coin Experiment. Select the number of coints (n=3) and probability of heads (p=0.5). Notice the blue model distribution of the Number of Heads (X), in the right panel. Try varying the probability (p) and/or the number of coins (n) and see how do these parameters effect the shape of this distribution. Can you make sense of it. For example, if p increases, why does the distribution move to the right and become concentrated at the right end (i.e., left-skewed)? Vice-versa, if you decrease the probability of a head, the distribution will become skewed to the right and centered in the left end of the range of X (\(0\le X\le n\)).

Let us toss three coins 10 times (by clicking 10 times on the RUN button of the applet on the top). We observe the sampling distribution of X, how many times did we observe 0, 1, 2 or 3 heads in the 10 experiments (each experiment involves tossing 3 coins independently) in red color superimposed to the theoredical (exact) distribution of X, in blue. The four panels in the middle of the Binomial Coin Applet show:

Coin Box Panel, where all coin tosses are shown	The theoretical (blue) and sampling (observed, red) distributions of the Number of Heads in the series of 3-coin-toss experiments (X)
Summary statistics table that includes columns for the index of each Run, the Number of Heads and the Proportion of heads in each experiment	Numerical comparisons of the Theoretical and Sampling distribution (\(0\le X\le n\)) and two statistics (mean, SD)

---

• SOCR Home page: http://www.socr.ucla.edu

---

---

Translate this page:

(default)	Deutsch	Español	Français	Italiano	Português	日本語	България	الامارات العربية المتحدة	Suomi	इस भाषा में	Norge
한국어	中文	繁体中文	Русский	Nederlands	Ελληνικά	Hrvatska	Česká republika	Danmark	Polska	România	Sverige