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The animation depicts the result of dropping numerous balls on a peg (or Galton) board. As the individual balls fall, they end up randomly distributed at the bottom of the peg board. The current mean position, position squared, and standard deviation are shown in the table, while a graph of the current mean versus the number of balls dropped is also shown. Restart.

In terms of the position of the individual balls that are dropped, x_i, the mean and square of the standard deviation are

<x> = ∑ x_i / N and (Δx)² = ∑ (x_i − <x>)² / N,      [sums are from i = 1 to i = N] (6.1)

where N is the total number of individual balls dropped. The square of the standard deviation can also be written in terms of the mean of the position and the mean of the position squared as:

(Δx)² = <x²> − <x>², (6.2)

since ∑ (x_i − <x>)² / N is the mean of (x − <x>)².

Since there are a finite number of bins (outcomes), as more balls are dropped a pattern emerges from these random outcomes which can be summarized as a discrete probability distribution. On the backdrop of the peg board is a picture of a curve that is a plot of the (continuous) probability distribution function:

P(x) = exp[−(x − <x>)²/2(Δx)²] , (6.3)

that the discrete distribution will approach for a large number of dropped balls. The functional form of P(x) is called a Gaussian. This is an important functional form for random probability distributions, for classical wave packets, and for quantum-mechanical wave packets.³

The Gaussian function is characterized by two parameters: the mean <x>, which tells us where the peak of the curve falls along the x axis, and the standard deviation, Δx, which tells us the width of the curve: the probability density drops to 1/e of its maximum value at x = <x> ± (2^1/2)Δx.

For a continuous distribution, like that of the Gaussian, we no longer sum the individual x_i values. Instead, we integrate the variable of interest over a continuous probability distribution that is normalized, P_N(x), yielding

<x> = ∫ x P_N(x) dx [integrals are from −∞ to ∞] (6.4)

and

<x²> = ∫ x² P_N(x) dx . [integrals are from −∞ to ∞] (6.5)

A continuous probability distribution is normalized when the integral of P_N(x) dx over all space is 1.

³Note that in the future, when we work with Gaussians in quantum mechanics, we will be using the normalized form: (1/ (2π(Δx)²)^1/4) exp[−(x − <x>)²/(2Δx)²].  Since the wave function is the probability amplitude, it is the square of this, (1/(2π(Δx)²)^1/2) exp[−(x − <x>)²/2(Δx)²], which is the normalized probability density, ρ(x), in quantum mechanics.

Original BallDrop Applet by Dave Krider, modified by Wolfgang Christian.