# Expert Answers

Q1: What are the two Branches of Statistics?

Q2: Theorize what the distributions would approximately look like for both the right-hand and left-hand digits of the populations of towns and cities from your selected state. Show a sketch in the rectangles below.

You have to choose one of three options to complete this activity. Your total score depends on the option chosen.

Option 1: Regular score (7.0 points)

Which Option did you choose?

If Option 1, which State?

Analysis: Now with both side-by-side bar graphs completed, answer the questions below.

1. What type of distribution shape would you say fits (models) the following counts?

2. Which of the two types of actual digits fits its theoretical count distribution the best?

Referring to the one you did not circle above, describe in a brief sentence what characteristic(s) of the shape made it not be the best fit.

Discussion: Before addressing the items below, modify the formula for the Left-hand Theoretical Count by using the logarithmic formula from Benford's Law. A brief Internet search on "Benford's Law" and/or "first digit phenomenon" would be very helpful.

Why do you suppose the actual count distributions are not as smooth as the theoretical?

By changing the right-hand theoretical count to reflect the logarithmic formula from Benford's Law, discuss how the shape improved the fit of the actual count

Why would the right-hand digit's distribution be approximately uniform (flat)?

Why would the left-hand digit's distribution be roughly right-skewed? (see #5 below)

To better understand why there is a built-in bias for the lower digits in the left-hand distribution, scan the sorted populations of your state from low to high. Discuss why a city, as it grows in population, would remain with a left-hand digit of a 1 longer than a 2, or why longer with a 2 than a 3, etc. You may come to a better appreciation for the first digit phenomenon that occurs in certain kinds of data by noting how there is a 100% increase from 1 to 2 but then a dramatically tapering percentage thereafter. Fill in the rest of the table and discuss how this might apply to population changes in a town or city.

Based on your Internet research, discuss a practical application of Benford's Law that interested you and why. Also, include what the Benford ratios are for digits 1 through 9.

Q2: Theorize what the distributions would approximately look like for both the right-hand and left-hand digits of the populations of towns and cities from your selected state. Show a sketch in the rectangles below.

You have to choose one of three options to complete this activity. Your total score depends on the option chosen.

Option 1: Regular score (7.0 points)

Which Option did you choose?

If Option 1, which State?

Analysis: Now with both side-by-side bar graphs completed, answer the questions below.

1. What type of distribution shape would you say fits (models) the following counts?

2. Which of the two types of actual digits fits its theoretical count distribution the best?

Referring to the one you did not circle above, describe in a brief sentence what characteristic(s) of the shape made it not be the best fit.

Discussion: Before addressing the items below, modify the formula for the Left-hand Theoretical Count by using the logarithmic formula from Benford's Law. A brief Internet search on "Benford's Law" and/or "first digit phenomenon" would be very helpful.

Why do you suppose the actual count distributions are not as smooth as the theoretical?

By changing the right-hand theoretical count to reflect the logarithmic formula from Benford's Law, discuss how the shape improved the fit of the actual count

Why would the right-hand digit's distribution be approximately uniform (flat)?

Why would the left-hand digit's distribution be roughly right-skewed? (see #5 below)

To better understand why there is a built-in bias for the lower digits in the left-hand distribution, scan the sorted populations of your state from low to high. Discuss why a city, as it grows in population, would remain with a left-hand digit of a 1 longer than a 2, or why longer with a 2 than a 3, etc. You may come to a better appreciation for the first digit phenomenon that occurs in certain kinds of data by noting how there is a 100% increase from 1 to 2 but then a dramatically tapering percentage thereafter. Fill in the rest of the table and discuss how this might apply to population changes in a town or city.

Based on your Internet research, discuss a practical application of Benford's Law that interested you and why. Also, include what the Benford ratios are for digits 1 through 9.

You'll get a 76.3KB .DOCX file.