Question

Benford’s law Exercise 9 described how the first digits of numbers in legitimate records often follow a model known as Benford’s law. Call the first digit of a randomly chosen legitimate record X for short. The probability distribution for X is shown here (note that a first digit can’t be 0). From Exercise 9, $E\left(X\right)=3.441$. Find the standard deviation of X. Interpret this value.

Short answer

The standard deviation is 2.4618.

Step-by-step solution

Step 1. Given information.

The given information is:

Step 2. Find and interpret the standard deviation of X.

The given mean is 3.441.

The predicted value of the squared departure from the mean is known as the variance:

$$\begin{gathered} {\sigma }^2=\sum {\left(x-\mu \right)}^{2}P\left(x\right) \\ ={\left(1-3.441\right)}^2\times \left(0.301\right)+{\left(2-3.441\right)}^2\times \left(0.176\right)+{\left(3-3.441\right)}^2\times \left(0.125\right)+ \\ {\left(4-3.441\right)}^2\times \left(0.097\right)+{\left(5-3.441\right)}^2\times \left(0.079\right)+{\left(6-3.441\right)}^2\times \left(0.067\right)+ \\ {\left(7-3.441\right)}^2\times \left(0.058\right)+{\left(8-3.441\right)}^2\times \left(0.051\right)+{\left(9-3.441\right)}^2\times \left(0.046\right) \\ =6.060519 \end{gathered}$$

The standard deviation is:

$$\begin{gathered} \sigma =\sqrt{{\sigma }^2} \\ =\sqrt{6.060519} \\ \approx 2.4618 \end{gathered}$$

We can see that on an average the first digit will vary from a mean of 3.441 by 2.4618.