Benford s Law Lets collect some numbers.

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1 Benford s Law Lets collect some numbers. Have your students pick some numbers out of a science textbook, some from the business section of the newspaper. Pick a few numbers they find on the internet, like the population of 4 large countries the hight of 3 tall mountains, the distance to 2 of the planets, and the cost of building something large like the Golden Gate Bridge. Add in the volume of a large body of water and the depth of the ocean. The units of measure will be different but do not worry. Just make sure that you have picked some smaller numbers and some very large numbers so there are a wide range of values. Keep collecting numbers until you have a large sample size. In a classroom I would have each student collect 2 numbers so you would have over 50 data points. Have each student present their numbers as you record the leading digit (far left digit) for each of the numbers. Create a table to display the numbers of times the first digit of their numbers was a 1, 2, 3,,,,, or 9. With such a random collection you would expect that each of the digits from 1 to 9 will occur about the same number of times. If you have 63 numbers you would not expect each number to occur exactly 7 times each but you would expect that the total number of times each of the digits from 1 to 9 occurs will be close to the same. But real data doesn t always work that way. For lots of the numbers we come across in daily life, especially in the areas of math science and business, chances are about 30% of the time the leading digit will be a 1. About 60% of the time the leading digit will be a 1,2 or 3 and only 15% of the time the leading digit will be a 7,8 or 9. If you have a data set of 50 and the numbers have a wide range in size you will find that the outcome of your data set reflects this 4 to 1 ratio. This outcome is unexpected and will be of interest to your students. How can you turn this unexpected outcome into a classroom activity and why should you consider spending valuable class time doing so? The answer to the second question will be answered later. The answer to the second question will be given later but the answer to the first question is the following activity. Finding Benford Scavenger Hunt Hand out a data collection slip to each student and have them collect 2 numbers that are interesting to them and that also have a numerical measurement. It will help them If you brainstorm some ideas before they start their search. Collect the data slips and then and suggest you make the data collection into a game where you win if the leading digit for a number is a 1,2, or 3 and they win if the leading digit is a 5, 6,7,8 or 9 for each number they handed in. The students will feel they have a good chance to win as there are only 3 leading digits that allow you to win and there are 5 leading digits that allow them to win. There is 1 leading digit that forces a tie game. It would seem that they will win 5/9th of the time and you will win 3/9th of the time. Things look good for the students! Make a table with 2 vertical columns with the left column containing the digits 1 to 9. Read out each description of the numbers then read out the number and state the leading digit. Have a student make a tally in the table next to that digit. When you have completed this add up the total of times the leading digit was a 1,2,3 and 5,6,7,8,9 and see who wins. You will win the game and they will have a lot of questions. Isn t that a great position for a teacher to be in Eitel " Page 1 of 23" amagicclassroom.com

2 Scavenger Hunt Lets collect some numbers. Use an internet search to find 2 numbers that are interesting to you and that also have a numerical measurement. Like the weight of the largest pumpkin and the height of Mt. Everest. Try to be sure to find at least one large number. Search for the population of a large country, the hight of a tall mountain, the distance to a planet, the cost building something like the Golden Gate Bridge, the volume of a large body of water or the depth of the ocean. The units of measure will be different but do not worry. Just make sure that you have picked 2 numbers numbers and one of them is a large number. Description 1 Number 1 Description 2 Number 2 Scavenger Hunt Lets collect some numbers. Use an internet search to find 2 numbers that are interesting to you and that also have a numerical measurement. Like the weight of the largest pumpkin and the height of Mt. Everest. Try to be sure to find at least one large number. Search for the population of a large country, the hight of a tall mountain, the distance to a planet, the cost building something like the Golden Gate Bridge, the volume of a large body of water or the depth of the ocean. The units of measure will be different but do not worry. Just make sure that you have picked 2 numbers numbers and one of them is a large number. Description 1 Number 1 Description 2 Number Eitel " Page 2 of 23" amagicclassroom.com

3 Finding Benford example with 24 data bits Number of dogs in New York Weight of the Titanic Person who lived the longest 85,085 52,310 tons 122 years length of Star Wars 8 movie Height of Pikes Peak Heaviest Pumpkin 155 min 14,114 feet 2145 pounds Length of Golden Gate bridge Height of worlds tallest tree Longest Beard 8,981 feet feet 5.33 meters Tallest building in USA Population of California Highest temperature on earth 1776 ft. 39,250, degree F Minutes in a year Depth of the ocean Longest Book ,994 meters 9,609,000 characters Elevation of Lake Tahoe Fastest Car Tallest person alive 6225 feet miles per hour 251 cm Tallest Mountain Distance to moon Depth of the ocean 23,622 feet 238,900 miles 10,994 Longest River Largest bubblegum bubble Price of gold today 4345 miles (Amazon) 508 mm $1,343,20 1,2, 3 was the leading digit 15 times 5,6,7,8,9 was the leading digit 8 times You have just experienced Benford's Law. Benford's Law is an observation about the probability of the occurrence of the leading digits in many real life sets of dates. The law states that in many naturally occurring collections of numbers the number 1 appears about 30% of the time, while 9 appears less than 5% of the time. It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers and processes described by exponential functions. It tends to be most accurate when values are distributed across a wide range of magnitudes. The actual probabilities for the occurrence of each of the digits will be provided later. Do not expect your sample data to match the probabilities for each digit exactly as listed in that table. To closely approximate the actual probabilities for the occurrence for each of the digits you need a large sample size for the data, a sufficiently wide the range of data sizes and a uniform random sample. It would seem like those requirements would preclude a teacher replicating those numbers with a classroom activity for students. In fact we will not try to replicating the exact numbers with the first few activities. By adding the first 3 probabilities together and then adding the last 5 probabilities together you get the basic outcome from Bedford s las without considering the individual values for each of the 9 probabilities. By grouping the probabilities into 2 sets we ensure a favorable outcome without needing to get the exact outcome for all 9 probabilities. The activities that follow are designed to produce examples of Bedford s Law using several different data sets. Each activity adds more information about the types of data that reflect Bedford s Law. They also provide examples of basic concepts that form a basis for the study of Statistics Eitel " Page 3 of 23" amagicclassroom.com

4 Roll the Die Lets play a game of chance. You will roll 5 normal die with the numbers 1 to 6 on them. You will use a calculator to find the product of the 5 numbers showing on top of the 5 die. If the leading digit (the far left digit) of the product is a 1, 2 or 3 than I win. If the leading digit (the far left digit) of the product is a 4 we will call it a tie If the leading digit (the far left digit) of the product is a 5, 6, 7, 8 or 9 than you win. For example: If your 5 die are a 4, 3, 5, 6, and 2 their product is 720. The leading digit is a 7 so you win. If your 5 die are a 3, 6, 1, 2, and 4 their product is 144. The leading digit is a 1 so I win. There are only 3 leading digits that allow me to win and there are 5 leading digits that allow you to win. There is 1 leading digit that forces a tie game. It would seem that you will win 5/9th of the time and I will win 3/9th of the time. We will have a tie 1/9th of the time. Things look good for you! We will play this game 10 times. The one who wins the most times out of 10 rolls will be the grand champion. Die 1 Die 1 Die 1 Die 1 Die 1 Product Winner (Me or You) Game 1 x x x x = Game 2 x x x x = Game 3 x x x x = Game 4 x x x x = Game 5 x x x x = Game 6 x x x x = Game 7 x x x x = Game 8 x x x x = Game 9 x x x x = Game 10 x x x x = Who is the Grand Champion? 2018 Eitel " Page 4 of 23" amagicclassroom.com

5 The leading digit for any number must be a number from 1 to 9. There are only 3 leading digits (1, 2 and 3) that allow me to win and there are 5 leading digits(5,6,7,8,or 9) that allow you to win. There is 1 leading digit (a 4) that forces a tie game. If things are fair I will win 3/9th of the time, you will win 5/9th of the time and we will have a tie 1/9th of the time. I used an excel spreadsheet to produce one cycle of this game R1 R2 R3 R4 R5 Product Winner You Me Me Me Me You Me Me You Me As you can see I was the winner 70% of the time and the other player was the winner 30% of the time and there was no ties. I was the Grand Champion. Is this example an outlier or rare even? You could use an excel spreadsheet to play this game for a large number of games or with more die. The excel spreadsheet below shows10 rolls of a similar game but we rolled 11 die at one time and found the leading digit of their product. d d occurs times % d occurs R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 Product As you can see, I was the winner 60% of the time and lost only 30%. There was a tie for 10% of the outcomes. I was the Grand Champion again Eitel " Page 5 of 23" amagicclassroom.com

6 Benford's Law Benford's Law is an observation about the frequency distribution of leading digits (left most digit) in many real life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading (left most) digit is more likely to be a lower valued digit like 1 2 or 3 rather that a higher valued digit like 5, 6, 7,8 or 9. For example, in sets that closely obey the law, the number 1 appears as the left most digit about 30% of the time, while 9 appears as the left most digit less than 5% of the time. By contrast, if the digits were distributed uniformly, they would each occur about 11% of the time. The left most digit will be a 1, 2 or 3 about 60% of the time and the left most digit will be a 5, 6 or 7, 8 or 9 only about 30% the time. That means that you have almost twice the chance to win with your 3 numbers as the person with their 5 numbers. d = leading digit for a set of numbers Approximate Percent of time the d will be the leading digit p(d).30 Bedford s Law 1 30% 2 18% 3 12% 4 10% 5 8% 6 7% % 8 5% Leading Digit 9 4% Note: Data sets that reflect Benford's Law most accurately have numbers that that are distributed uniformly across several orders of magnitude. As a rule of thumb, the more orders of magnitude that the data evenly covers, the more accurately Benford's Law applies. For instance, one can expect that Benford's law would apply to a list of numbers representing the populations of local elementary schools because these numbers to not represent a wide range in the size of the numbers. The population of local elementary schools all tend to all be from 300 to 500. Note: The way the game is played you do not need to match the expected table values as close as you might think. Letting the occurrence of a 4 as the leading digit 4 be a tie allows you to have an almost 2 to 1 advantage in the expected probability values from 1 to 3 as the expected probabilities from 5 to 9. When you play the game 10 times it further increases the chance that you will win more than 5 out of 10 times. Letting the occurrence of a 4 as the leading digit 4 be a tie allows you to have 2018 Eitel " Page 6 of 23" amagicclassroom.com

7 A Capital Idea Lets play a game. Use a world map to find 10 countries from around the world. Then use the map to find its capital city. You may need help from your teacher. Try and select ones from around the world. List each country its capital city on the 10 lines below. Use the internet to find a webpage that displays the population of each of the 10 capital cities. I used to find the the distance for the example on the next page If the leading digit (the far left digit) of the product is a 1, 2 or 3 than I win. If the leading digit (the far left digit) of the product is a 4 we will call it a tie If the leading digit (the far left digit) of the product is a 5, 6, 7, 8 or 9 than you win. For example: If the population os is 735,000 people the leading digit is a 7 so you win. If the population is 1,206, 921 people the leading digit is a 1 so I win. There are only 3 leading digits that allow me to win and there are 5 leading digits that allow you to win. Things look good for you! We will play this game 10 times. The one who wins the most times out of 10 rolls will be the grand champion. Country Capitol City Population Winner (Me or You) 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) Who is the Grand Champion? 2018 Eitel " Page 7 of 23" amagicclassroom.com

8 A Capital Idea I used to find many capital cites from countries in the world. It list the countries and capitals but does not list population numbers. It is listed alphabetically so encourage students to look fat countries form all of the list. Country Capital City Population leading digit Winner France Paris 2,241,346 3 Me Ukraine Kiev 2,847,200 2 Me USA Washington DC 658,893 8 You Uganda Kampala 1,659,600 7 Me Saudi Arabia Moscow 4,878,723 1 Tie Russia Moscow 12,197,596 4 Me Brazil Brasilia 2,648,532 2 Me China Beijing 21,516,000 1 Me Mexico Mexico City 8,974,724 5 You India New Delhi 21,678,794 5 Me As you can see I was the winner 70% of the time and the other player was the winner 20% of the time and there was 1 tie. I was the Grand Champion. To help them get started you may search for a list many of the more well known ones such a London (England), Paris (France), Los Angles (USA), San Francisco (USA), Moscow, Sao Paulo (Brazil) New York (USA), Tokyo (Japan), Seoul (South Korea), Moscow (Russia), Mexico City, Beijing (China), Mumbai (India), Lagos (Nigeria), Istanbul (Turkey), Hong Kong (China), Berlin(Germany), Kiev (Ukraine), Havana (Cuba), Manila (Philippines). This is a great way to involve world maps and geography into the lesson. Note: Typing in population of world cities into a web search produced 183 million results. This is a good time in the lesson to talk about search results and search phrases. It also is a good time to discuss that not every site will have the exact same numbers. The date will vary from site to site due to how is was collected and the time the date was collected. Even with that variance the outcome of the contest will almost always be in favor of the test with the 1, 2,and 3 digits Eitel " Page 8 of 23" amagicclassroom.com

9 World Travel Lets play a game. You will pick 2 cities on a world map. Make sure they are far apart. Find some interesting ones like Paris or Hong Kong Need York or Moscow. Use the internet to find a webpage that finds the actual air distance between the 2 cities in miles and write that distance down. I used to find the the distance for the example on the next page If the leading digit (the far left digit) of the product is a 1, 2 or 3 than I win. If the leading digit (the far left digit) of the product is a 4 we will call it a tie If the leading digit (the far left digit) of the product is a 5, 6, 7, 8 or 9 than you win. For example: If the distance is 735 miles the leading digit is a 7 so you win. If the distance is 1,206 miles the leading digit is a 1 so I win. There are only 3 leading digits that allow me to win and there are 5 leading digits that allow you to win. Things look good for you! We will play this game 10 times. The one who wins the most times out of 10 rolls will be the grand champion. City 1 City 2 Distance Winner (Me or You) between cities 1) to = 2) to = 3) to = 4) to = 5) to = 6) to = 7) to = 8) to = 9) to = 10) to = Who is the Grand Champion? 2018 Eitel " Page 9 of 23" amagicclassroom.com

10 I used to find the distances for the data table below. City 1 City 2 Distance Between (miles) leading digit Winner New York London Me LA Maui Me London Rome You Sacramento Vancouver You Denver Atlanta Me Dallas San Francisco Me Seattle Mexico Me Chicago Miami Me Portland Manila You Hong Kong Paris You Note: Be sure that the cities are do not all start and end from the same general areas. For example, most east coast cities in the USA to Europe will all have similar distance between them. This would also be true of travel from the east coast to west coast of US cities. Even with this issue with the data set it is amazing to see how often the 1,2,3 option is more likely to win than the 5,6,7,8,9, option. Very important: They must choose the cities and write them down BEFORE they look up the distances If they do not do this then they will just select the ones that allow them to win. Students may not be able to produce the names of large of cities from around the world. To help them get started you may search for a list many of the more well known ones such a London (England), Paris (France), Los Angles (USA), San Francisco (USA), Moscow, Sao Paulo (Brazil) New York (USA), Tokyo (Japan), Seoul (South Korea), Moscow (Russia), Mexico City, Beijing (China), Mumbai (India), Lagos (Nigeria), Istanbul (Turkey), Hong Kong (China), Berlin(Germany), Kiev (Ukraine), Havana (Cuba), Manila (Philippines). This is a great way to involve world maps and geography into the lesson. Note: Typing in population of world cities into a web search produced 183 million results. This is a good time in a lesson to talk about search results and search phrases. It also is a good time to discuss that not every site will have the exact same numbers. The data will vary from site to site due to how is was collected and the time the date was collected Eitel " Page 10 of 23" amagicclassroom.com

11 Time Tables Lets play a game. Pick 5 one or two digit numbers and place them in the 5 boxes at the top of the grid below. Now pick 5 one or two digit numbers and place them in the 5 boxes at the left side of the grid. Then fill in the times table by multiplying each number at the top by the number at the left and putting the answer in the intersection. Product # # # # # # # # # # Note: It is important that the students chose the numbers to start the activity and then find the products of these numbers before they continue to this part of the activity. After the students have selected the numbers for the top and side row and recorded the products of these rows introduce the next part of the activity. Now look at the 25 answers you used to fill in the times table. If the leading digit (the far left digit) of the product is a 1, 2 or 3 than I win. If the leading digit (the far left digit) of the product is a 4 we will call it a tie If the leading digit (the far left digit) of the product is a 5, 6, 7, 8 or 9 than you win. For example: If the product is 72 the leading digit is a 7 so you win. If the product is 1,20 the leading digit is a 1 so I win. There are only 3 leading digits that allow me to win and there are 5 leading digits that allow you to win. Things look good for you! How many of the 25 numbers in the table have a leading digit (the far left digit) of Total of wins for 1 to 3 = 4 " " " Total of ties for 4 = 5 " 6 " 7 " 8 " 9 Total of wins for 5 to 9 = Who is the Grand Champion? 2018 Eitel " Page 11 of 23" amagicclassroom.com

12 Time Tables Example Product # 4 # 11 # 13 # 5 # 8 # # # # # ,2, 3 was the leading digit 14 times 4 was the leading digit 3 times 5,6,7,8,9 was the leading digit 8 times 1,2, 3 was the the grand champion The product of several numbers follow Benford s Law very well. The larger the numbers the better the numbers will follow Benford s rule but even small numbers will allow the 1,2,3 group to win the game most of the time. X d = leading digit number of times the leading digit occurred % of the time the digit occurred in the data set (decimal) Eitel " Page 12 of 23" amagicclassroom.com

13 Data sets with data developed using multiplicative calculations Many real world examples of data sets that satisfy Benford's Law arise from multiplicative calculations. This is true for many business data sets. For example, if a stock price starts at $100, and then each day it gets multiplied by a randomly chosen factor between 0.95 and 1.05, (increase or decreases by up to 5% each day) then over an extended set of calculations the probability distribution of its price satisfies Benford's Law with higher and higher accuracy. The reason is that over time the probability distribution will get more and more broad and smooth Technically, the central limit theorem says that multiplying more and more random variables will create a log-normal distribution with a larger, so eventually it covers many orders of magnitude almost uniformly. This means it will follow Benford s Law. Data sets developed using additive calculations lead to normal probability distributions which do not satisfy Benford's law. Numbers drawn from normal distributions (IQ scores, human heights) will not follow Benford s because normal distributions do not tend to satisfy Benford's law, since normal distributions can't span several orders of magnitude. However, if you multiply several numbers from those normal distributions together the new data set will follow Benford's Law. You can also take several numbers from unrelated sources and the find the products of those numbers. Repeated selections will produce a data set that closely follows Benford s Law. One example is to select numbers from a magazine, a newspaper and a textbook and find their product. The data set will follow Benford s Law. The times table problem works because you are finding the product of several numbers from those normal distributions. If you chose to multiply 3 or 4 numbers together you would get a data set that is a good approximation of the law. The internet times the Internet activity The Internet times the Internet activity that follows uses the multiplication of 2 numbers found on the internet to create a data set. Students could just make up numbers like the times table activity but this is a lot more fun. Have the students conduct an internet search to find 2 things that they find interesting and that also have a numerical value like the weight of the Titanic and the height of Mt. Everest. The numbers that work best do not end in a series of zeros. For this reason have them select numbers that come from actual measurements. Many numbers in use today are not exact values but are rounded off approximations that have a lot of trailing zeros. If you chose numbers like 100, 560,000 and 300,000 the you will need a larger data set to get a good approximation of the Law Hand out a data slip to each set of students. Try to have 15 or so groups. Let each group find 2 interesting numbers and record the description of the 2 items, the numerical values and their product. Have each group present their findings to the class. Record the numbers as they present them and see if the set or the 5,6,7,8,9 set is the winner and who is the grand champion. The results should follow Benford s Law Eitel " Page 13 of 23" amagicclassroom.com

14 Group Number The internet times the Internet Names Use an internet search to the internet find the numerical values of 2 things that are interesting to you and that also have a numerical measurement. Like the weight of the Titanic and the height of Mt. Everest. Do not select numbers that have been rounded to display trailing zeros like 6,000,000. Write a short description of each of the items on the lines below and then record the numbers in the blanks labeled number 1 and number 2. Use a calculator to find the product of the 2 numbers and record it. Then state if you were a winner (leading digit was a 5,6,7,8, or 9). Description 1 Description 2 X = Number 1 times" Number 2 " Product Winner (yes or no) Group Number Names Use an internet search to the internet find the numerical values of 2 things that are interesting to you and that also have a numerical measurement. Like the weight of the Titanic and the height of Mt. Everest. Do not select numbers that have been rounded to display trailing zeros like 6,000,000. Write a short description of each of the items on the lines below and then record the numbers in the blanks labeled number 1 and number 2. Use a calculator to find the product of the 2 numbers and record it. Then state if you were a winner (leading digit was a 5,6,7,8, or 9). Description 1 Description 2 X = Number 1 times" Number 2 " Product Winner (yes or no) 2018 Eitel " Page 14 of 23" amagicclassroom.com

15 The internet times the Internet example Number of dogs in New York Weight of the Titanic Does the 1,2,3 set win 85,085 52,310 tons 4,450,796,350 (tie) length of starts wars 8 movie height of Pikes Peak 155 min 14,114 feet 2,187,679 (win) Length of Golden Gate bridge Height of worlds tallest tree 8,981 feet feet 3,410,085.7 (win) Tallest building in USA (One world Trade Center) Number of students at Stanford University in ft ,012,736 (win) minutes in a year Depth of the ocean ,994 meters 5,778,446,400 (lose) Elevation of Lake Tahoe Highest temperature recorded 6225 feet 134 degree F 834,150 (lose) Tallest person alive Longest Beard 251 cm 5.33 meters (win) Heaviest Pumpkin Person who lived the longest 2145 pounds 122 years 261,690 (win) Longest River Longest Book (Guinness Records) 4345 miles (Amazon) 9,609,000 characters x (tie) Longest Snake Distance to moon 767 cm 238,900 miles 183,236,300 (win) Tallest Mountain Fastest Car 23,622 feet miles per hour 6,720,459 (lose) Longest Fingernails Largest bubblegum bubble 855 cm 508 mm 434,340 (tie) Data sets with data developed using multiplicative calculations The last 2 activities demonstrated that data sets developed by Benford's Law arise from multiplicative calculations reflect Benford's Law. One topic taught in algebra classes from the 5th grade on is the powers of a given base. The number that represents a simple set of multiplicative calculations so the data set composed will follow Benford s Law Eitel " Page 15 of 23" amagicclassroom.com

16 Powers of a base Power of 2 Number Power of 3 Number x ,2,3 wins 60%" 1,2,3 wins 70% A tie 10%" A tie 20% 5,6,7,8,9 wins 30%" 5,6,7,8,9 wins 10% Power of 7 Number Power of 7 Number x x x ,2,3 wins 50% A tie 10% 5,6,7,8,9 wins 40% 2018 Eitel " Page 16 of 23" amagicclassroom.com

17 Benford s Law with Factorials Factorials are the product of consecutive numbers starting at 1. 5 factorial (written 5!) means 1 * 2 * 3 *4 *5. 7! means 1 * 2 * 3 *4 *5*6 *7. Factorials represent a set of multiplicative calculations so the data set composed will follow Benford s Law. Note: These numbers are very large so excel displays then in scientific notation, The table below displays one contest using 10 different factorials. n nth Fibonacci Number is d from 1 to yes no yes yes yes yes no E+17 yes E+20 no E+27 yes Overall you win 7 of the 10 sets. You are the grand winner. 6 different sets of 10 factorials Win E E E E E E E E E E+25 Lose Win E E E E E E E E E E+48 Lose Win E E E E E E E E E E+68 Lose Win E E E E E E E E E E+138 Lose Win E E E E E E E E E E+170 Lose Win E E E E E E E E E E+203 Lose 5 Overall you win 5 of the 6 sets. You are the grand winner Eitel " Page 17 of 23" amagicclassroom.com

18 A brief history Of Benford s Law Though it is called Benford s Law, the astronomer Simon Newcomb observed this behavior 50 years before Benford. In 1881 Newcomb published an article in the American Journal of Mathematics, Notes on the Frequency of Use of the Different Digits in Natural Number. The article stateed, That the first digits do not occur with equal frequency must be evident to any one making much use of logarithmic tables by noticing how much faster the first pages wear out than the last ones. The first significant figure is oftener 1 than any other digit, and the frequency diminishes up to 9. He provided the table below of the approximate frequency of each leading digit. d = leading digit for a set of numbers Approximate Percent of time the d will be the leading digit 1 30% 2 18% 3 12% 4 10% 5 8% 6 7% 7 6% 8 5% 9 4% Newcomb was an astronomer and he needed to do computations involving large numbers. At this time this was done with the use of logarithms and thus he was constantly using a book of log tables. He wrote that he was led to the law by observing that the pages in logarithm tables corresponding to numbers beginning with 1 were significantly more worn than the pages corresponding to numbers with higher first digit. His explanation for the additional wear and tear was that numbers with a low first digit are more common than those with a higher first digit and thus the logs of those numbers were looked up more often, thus wearing those pages out faster. It was quite fortunate that there were no calculators to perform calculations with large number because otherwise the law may have been ove rlokked for many years. Another key observation of Newcomb s paper is his noting the importance of scale. The numerical value of a physical quantity clearly depends on the scale used, and thus Newcomb suggested that the correct items to study are the ratios of measurements that involve number with a wide range of scales. There is a direct connection between data sets that have exponential relationships and Bedford s Law. Science contains many such measurements so Newcomb was in the right place at the right time to discover the relations involving leading digits. Little interest was expressed in Newcomb s findings and this information was not developed any further at the time. A renewed interest developed with publications by Frank Bedford that set a mathematical basis for the relationship. He also included applications of the rule in Science, Math Statistics and Business. For this reason the relationship would be called Benford s LAw Eitel " Page 18 of 23" amagicclassroom.com

19 Frank Benford s The Law of Anomalous Numbers was published in Benford s paper included the distribution of first digits found in 20 data sets. Topic Count US County 29% 19% 12% 11% 6% 8% 6% 4% 6% 3142 Population Numbers in 33% 19% 12% 8% 7% 6% 5% 5% 4% 308 Readers Digest House Address 29% 19% 13% 9% 8% 6% 6% 5% 5% 342 Factorials 25% 16% 12% 10% 9% 9% 7% 7% 6% 900 Molecular Weight 27% 25% 15% 11% 7% 5% 4% 3% 3% 1800 Atomic Weight 47% 19% 6% 5% 7% 5% 3% 4% 6% 91 Specific Heat 24% 18% 16% 15% 11% 4% 3% 5% 4% 1389 Scientific Constants 41% 14% 5% 9% 11% 6% 1% 3% 11% 104 Numbers in Newspapers 30% 18% 16% 15% 11% 4% 3% 5% 4% 1389 Areas of bodies of water 31% 16% 11% 11% 7% 9% 6% 4% 5% 335 Death Rate 27% 19% 16% 9% 7% 7% 7% 5% 4% other studies : : : : : : : : : : Simple Average 31% 19% 12% 9% 8% 6% 5% 5% 5% 1011 Benford s Law 30% 17% 13% 10% 8% 7% 6% 5% 5% Rounded off to whole percents His studies affirmed the previous probibilities that Newcome published but he also added many insights. He made a clear link between the probability distribution and the logs. This allows him to state distributions that will follow the Law as well as listing many the will not. One of the most important statements is that while individual data sets may fail to satisfy Benford s Law, combining many different sets of data leads to a new sequence whose behavior will be a better approximation to the law. He also presented some justification as to why this is a problem worthy of study. Bedford s Law A set of numbers is said to satisfy Bedford's law if the probability of the occurrence of the leading log(d +1) digit d where(d {1,..., 9} is approximately p(d) log(d +1) log(d) = log(d) d = leading digit for a set of numbers p(d) for Benford s Law p(d) Leading Digit 2018 Eitel " Page 19 of 23" amagicclassroom.com

20 Bedford s Law is independent of the unit of measure Bedford s Law states that the proportion of leading digits should be close to the predicted values even if different units of measurements are used to collect the data. The following graph displays the distribution of leading digits for the areas of 197 countries. The 3 different colored histograms reflect the distributions of the leading digits were recorded in sq. feet, sq. miless and acres. While the numbers vary slightly you can see that all 3distribution of leading digits reflect Benford s Law independent of the unit of measure. Larger data set would produce even closer agreement The table below reflects the leading digit of the heights of the tallest buildings in the world in 51 different categories. (Wikipedia) It confirms that 1 is the most common leading digit, irrespective of the unit of measurement. The exact values of each of the 9 individual probabilities do reflect the general trend of Bedford s Law that the most likely leading digit will be a lower number and the least likely leading digit will be a higher irrespective of the unit of measurement but the smaller sample size causes the individual values to differ more from the expected values. d = leading digit of the height number of times the leading digit occurred in meters % of the time the digit occurred in the data set (decimal) number of times the leading digit occurred in meters % of the time the digit occurred in the data set (decimal) p(d) for Benford s Law In meters p(0 d 3) =.706 and p(5 d 9) =.177 In feet p(0 d 3) =.615 and p(5 d 9) = Eitel " Page 20 of 23" amagicclassroom.com

21 Accounting and Fraud Detection In 1972, Hal Varian suggested that the law could be used to detect possible fraud in data based on the assumption that people who make up figures tend to distribute their digits fairly uniformly. Mark Nigrini showed that Benford's Law could be used in forensic accounting and auditing as an indicator of accounting and expenses fraud. In practice, applications of Benford's Law for forensic accounting use more than the first digit. Many university accounting programs include Bedford s Law as a topic in their programs. A web search for the topic at Universities included, Princeton, University of Washington, University of Pennsylvania, The University of Michigan and many others. A web search for articles on Bedford s Law and fraud has over 135,000 references. A recent article in the Journal Of Accountancy by J. Carlton Collins, CPA was titled Using Excel and Benford s Law to detect fraud: Learn the formulas, functions, and techniques that enable efficient Benford analysis of data sets. Legal status In the United States, evidence based on Benford's law has been admitted in criminal cases at the federal, state, and local levels. In 1993, in State of Arizona v. Wayne James Nelson the accused was found guilty of trying to defraud the state of nearly $2 million by diverting funds to a bogus vendor. He selected payments with the intention of making them appear random. None of the check amounts was duplicated, there were no round-numbers, and all the values included dollars and cents amounts. He did not realize that his seemingly random looking selections violated Benford s Law. Bedford s Law also has fictional-world applications: the main character of the TV series NUMB3RS used Benford s law to catch a criminal Eitel " Page 21 of 23" amagicclassroom.com

22 Data analysis that implies a data set reflects Benford's law The data presented in the tables in this paper can never match the theoretically expected leading digit distributions of Benford s Law. This distribution can never be perfectly realized in any dataset because the values are irrational numbers. These numbers were obtained using tools from calculus including the infinite limits of functions and indefinite integrals, The finite date sets used as examples use whole numbers ratios from a finite set so Benford s Law can only be approached. The purpose of these data sets is to highlight the disproportionate percentage of the occurrence of lower value leading digits over higher value leading digits. The suggestion that the date set reflects Benford s Law can be confirmed by creating the probability distribution of the 9 leading digits or a frequency histogram. Distributions known to reflect Benford's law It has been shown that Benford s Law applies to a wide variety of data sets. It tends to be most predictive when the values in the set are distributed across multiple orders of magnitude. This included electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical constants and mathematical constants (if a large sample size is used). Data sets generated by geometric growth reflect Benford s Law applies: population growth, radioactive decay and bacteria populations are a few examples. Any linear combinations of geometric series will also reflect the law. It applies to any processes described by a power function, which are very common in nature. NASAA satellite measurements show that Sunspots and Solar Winds measurements follow Benford s Las. USGS records show that the depth of earthquake also reflect this law. Benford's law has been empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions and the following distributions have been found to obey Benford s Law.: the uniform distribution, the exponential distribution, the normal distribution and the chi square distribution. In addition, the ratio of 2 uniform distributions, 2 exponential distributions, 2 normal distributions and 2 chi square distribution also obey Benford s Law. Several infinite integer sequences can be proven to satisfy Benford s Law including the Fibonacci Numbers, factorials, powers of 2, and the powers of almost any other number. Functions that reflect exponential growth or decay will have the probabilities distribution of their leading digits approach Benford's s Law as t approaches infinity. That is, Benford s Law acts as an asymptotic limit. Many data sets that contain accounting data reflect Benford's Law including numbers that result from mathematical combinations of numbers like quantity price or any other products. Transaction level data such as disbursements and sales data will also reflect Benford's Law Eitel " Page 22 of 23" amagicclassroom.com

23 Distributions known to NOT reflect Benford's law Most notably, square roots and reciprocals do not obey this law. Benford's law does not apply to telephone numbers. Numbers are assigned sequentially like check numbers and invoice numbers will not obey the law nor will numbers with built in minimum and maximum values. For example, the age of students in your classroom have a limited range with minimum and maximum values that allow little variance in size. Do Prime numbers obey the law? That is a much harder question to answer. This topic is best left for a student of advanced math who is working with infinite sets and density functions. A short answer is yes if the range for the prime numbers are selected in a special way. Central Limit theorem If you pool together a several sets of randomly chosen numbers, each of which have an arbitrary probability distributions, the collective set of numbers will approximate Benford s Law. It s reflects the central limit theorem, that says the values of randomly chosen data from random distributions will approach a bell curve when you combine them all together into one data set. An explanation of Benford s Law Various attempts to explain the mathematical basis for the reason Benford s Law works can be found in documents on the web as well. Formal proofs and extentions of the law into advanced areas of student in math and statistics can also be found with a web search. Many utube videos of exapmles and explanitions can be found on utube. Some of the beat are presented by the Numberphile group Eitel " Page 23 of 23" amagicclassroom.com