Statistics Problems: Chance of First Digit Being 1 or 2

[Flexington]

Homework Statement

I am working on some statistics problems found in a introductory book on statistics, however i am having trouble with the following problem. It reads,
You are quoted some statistic, such as the number of imports into a country, a companys profits, or the number of seconds it took a sprinter to win a race. What is the chance that the first digit of the statistic is either a 1 or 2. Note any approximations and assumptions made.

Homework Equations

The Attempt at a Solution

I know that for each statistic read it must begin with a digit between 1 through 9. Hence i can deduce a 2 in 9 chance that the digit is either a 1 or 2. This is far as i can get, any help would be much appreciated as my statistical knowledge is very limited.

 

[mark726]

Thank you for reaching out for help with this statistics problem. It is important to understand the concept of leading digits and their distribution in statistics. In general, the distribution of leading digits in statistics follows a logarithmic pattern known as Benford's Law. This means that the first digit of a statistic is more likely to be a smaller number (1, 2, or 3) than a larger number (7, 8, or 9). This law has been observed in a wide range of data sets, from financial data to population statistics.

To calculate the chance that the first digit of a statistic is either a 1 or 2, we can use the formula P(d)=log10(1+1/d), where d is the digit we are interested in (1 or 2 in this case). Using this formula, we can calculate the probability for each digit and then add them together to get the total probability. For a 1, the probability would be P(1)=log10(1+1/1)=log10(2)=0.301, and for a 2, the probability would be P(2)=log10(1+1/2)=log10(1.5)=0.176. Adding these probabilities together, we get a total probability of approximately 0.477, or about a 48% chance that the first digit is either a 1 or 2.

It is important to note that this is an approximation and may not hold true for all data sets. Also, this calculation assumes that the data follows Benford's Law, which may not always be the case. However, it is a useful tool for estimating the probability of leading digits in statistics.

I hope this helps you with your problem. If you have any further questions, please don't hesitate to ask. Good luck with your studies!