A picture can tell a thousand words. Sometimes a number can give a good picture. In the example discussed here, a picture and a number can be combined to make history come alive.

I recently watched a World War II news reels on YouTube called FRONTLINE WWII: Germans Advance into Russia (720p). The video describes the operation called Operation Barbarossa, which was the code name for Nazi Germany’s invasion of the Soviet Union during World War II and was launched on Sunday 22 June 1941. One of the three strategic objectives for the operation was to capture the city of Leningrad, now called Saint Petersburg. The German laid siege to the city for 872 days from September 8, 1941 to January 27, 1944.

Early on in the war, the Germans believed that Leningrad would be taken easily. In fact, it has been reported that Adolf Hitler was so confident of capturing Leningrad that he had invitations printed with the victory celebration party to be held in the city’s Hotel Astoria. Later Hitler made the strategic decision to divert resources to other fronts. The plan for Leningrad was changed from direct attack and capture to a siege with the goal of starving the city into submission.

The siege of Leningrad was one of the longest and most destructive sieges in world history. The destructive impact on the city is detailed in the Wikepedia entry on the siege of Leningrad and in countless other sources. When I watched the YouTube video, one number stands out. During the siege, each soldier or worker doing critical work received 8 ounces of bread a day (and nothing else). The other residents of the city received daily ration of 4 ounces of bread. It did not matter if a resident was young or old, healthy or sick. If the person was not fighting, he or she only had 4 ounces of bread per day for sustenance.

To get an idea how much food is an 8-ounce piece of bread, the following is a picture of a loaf of bread that is found in any grocery store in United States.

The loaf pictured is a 24-ounce loaf. That means that the a Soviet soldier defending Leningrad received about one third of a loaf of bread for an entire day. Here’s the math: 24 oz x 1/3 = 8 oz. But that is only in terms of weight. The quality of the bread that a soldier received could not be compared with the loaf pictured above. The bread during the siege was made up of sawdust and other inedible ingredients (50 to 60%).

The loaf pictured above has 18 slices (you can actually count the slices). The daily ration is then 6 slices of bread for a soldier and 3 slices for everyone else (children and the elderly). Here’s the math: 18 x 1/3 = 6 slices. So a child or an old person subsisted on 3 thin slices of bread that was half saw dust!

The picture was indeed grim. The deaths in Leningrad peaked at 100,000 a month in early 1942, mostly from starvation. Due to the lack of fuel, the trolley service ceased to work for most of the siege. Just to get the meager 4-ounce ration of bread, people would need to walk to a distribution kiosk. In a typical winter in Leningrad, the temperature can drop to minus 30 Celsius (minus 22 Fahrenheit). For many people, the walk to a distribution kiosk would be an insurmountable obstacle.

The Nazi siege of Leningrad that began in 1941 and ended in 1944 was one of the most gruesome episodes of World War II. Nearly three million people endured it. Altogether, the siege lasted nearly 900 days and resulted in the deaths of more than 1 million civilians. The siege of Leningrad was an epic story of sufferings and destruction and ultimately triumph. If the daily ration of bread piques your interest, there are many places to read more. Here’s some links.

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This is a blog about making sense of numbers. The loaf of bread example shows that sometimes it is the other ways round – finding just the right numerical examples to help us understand a complex story or a complex phenomenon. Please feel free to browse the articles in this blog. Here’s are some articles that may be of interest.

Here are two posts on number sense. The first one is on stealth price increases. Some manufacturers do not raise prices but give you less. For example, a pack of peanut may have 16 ounces before and now weights 14 ounces but is charged the same price as a 16-oz pack. This post shows how to calculate the price increase. This post is a plug on quantitative literacy after an encounter with a store clerk.

I had written on lottery, especially how small the odds are. Buying lottery tickets as entertainment is one thing. Anyone buying them as investment or as quick ways to get rich should know that it will take buying thousands of tickets each week since the time of Christ to have a realistic chance of winning the Powerball lottery. The following are some of the most popular posts in this blog.

## How many lottery winners are there in a year?

I have been wanting to answer the questions in the title. I found that statistics on lottery winning is hard to come by. Even when the state lottery commissions are required by law to made the information public, they tend to bury the information and you have to do work to dig it up. I have strong indication that on an annual basis, winning tickets that pay out one million dollars or more only number in the hundreds. In contrast, there were 37,261 people killed in motor vehicle crashes in 2008 in the United States (see the report from the National Highway Traffic Satety Administration). So if you are passionate about winning various state lotteries, it makes sense to be passionate about not winning the negative lottery of fatality in a motor vehicle crash too.

As of November 2010, there were only 247 winning tickets paying one million dollars or more (see the previous post with this discussion). To get this information, I had to look up the winning tickets in each of the 58 California counties in the official site of CalLottery. So about 10 people are made millionaires by CalLottery each year (since its inception 25 years ago).

The state of Iowa is more forthcoming. The official site of the Iowa Lottery actually had a press release listing out the stats. The number of Iowa Lottery tickets that have won prizes of $1 million or more (through August 2010) is 110. Once again in the 25 years history of the Iowa Lottery, only 110 people were made millionaires, on average 4.4 per year. For the Iowa Lottery, the odds for winning$100,000 or more are better for sure (1089 winnings so far in 25 years) but the odds are still small.

The state lotteries are in the business of selling dreams. I suspect that they do not want to provide a picture reflecting the true odds of winning big. With all the state lottery commisions across the United States combined, I cannot see how the number of winning tickets ($1 million or more in each one) in one year can be in the thousands. If someone is forking over hard earned cash each week to play the lottery in the hope of winning big, it also makes sense to pay attention to traffic safety in the hope of not winning the negative lottery of death in a car crash. ## Governor Brown wants to take your cell phone No, this is not a government seizure of private cell phones. Jerry Brown wants your cell phone only if it was issued by the California state government. Even then, the chance of it having to turn it in is only 50%. The newly installed Governor Brown is only proposing to take away government paid cell phones from certain California state employees in an effort to save money. The potential saving is to the tune of$20 million. Trying to close a budget gap in the California state government that is expected to be about $28 billion, the governor needs to find money anywhere he can. On the way to work this morning, I heard a report on the radio about the proposal from Governor Brown (here’s one link). The gist of the report is: 1. Governor Brown does not want all the cell phones back. He just wants half of them back. The total number of phones to be turned in by June 1: about 48,000. 2. The state government currently foots the cell phone bills for about 40% of its workforce. The total number of state employees with free cell phones: about 90,000. 3. The state government is currently paying on average$36 a month per cell phone.
4. Governor Brown’s proposed cell phone reduction order should save the state $20 million. 5. Even with the proposed saving, one fifth of the state workers will still use state-funded cell phones. After I heard this report on the radio, I jotted down the nummbers and I found that these numbers are internally consistent and for the most part accurate. The numbers seem to hang together quite well. Definitely there is no glaring errors. In this one instance of budget cutting at least, Governor Brown and his staff got the numbers right. Here’s how I looked at these numbers. I am going to discuss each of the points listed above. 1. Currently there are about 96,000 state-funded cell phones in the hands of California state givernment employees. Half of these phones are to be turned in by June 1. That leaves 48,000 cell phones still being in the hands of state employees. 2. About 90,000 state employees have government cell phones. The count of 90,000 is about 40% of the total state workforce. This means there are currently about 225,000 state employees (see note 1 below). 3. The average monthly bill per cell phone is$36, making the average annual bill per cell phone $432. 4. The total annual expense for the 48,000 cell phones being cut would be$20,736,000 (see note 2). This amount is slightly over $20 million and is thus in line with the$20 million being reported.
5. Even with the proposed saving, one fifth of the state workers will still use state-funded cell phones. One fifth of 225,000 (from point 2) is 45,000. From point 1, about 48,000 government paid cells phones will still be used by state employees. Though the 45,000 is less than 48,000 by 3,000, the difference is not large.

The only external number that I had to find is the total number of state employees in California. The number I obtained from the California State Controller’s Office is 237,576 (as of October 2010). The estimate obtained from the radio report is 225,000 (point 2 above). Overall, the numbers hang together well.

The proposed cut makes sense. According to Governor Brown, some agency heads and managers need to be in touch with employees 24/7. But most state employees do not. However, not the entire amount of proposed saving will be realized this year since some of the cell phones may be under contract with cell carriers. I hope the savings will work out as planned.

Note 1
To find 40% of 225,000, we multiply 225,000 by 0.40.

$225000 \times 0.40=90000$

But in the radio report, the number 90,000 is given instead. To derive 225,000, we need to do the opposite, i.e. divide 90,000 by 0.40.

$\displaystyle \frac{90000}{0.40}=225,000$

Note 2

$432 \times 48000=20,736,000$.

## Hope there will be no lottery winners this New Year’s Eve

According to a report in npr.org called Road Fatalities Dip Thanks To Safer Cars, Economy, an array of factors are making the road safer. According to a study by the Department of Transportation, the overall number of fatality on American roads has dropped dramatically, fallen by over 20% in the last few years. Two likely reasons for this dramatic drop are safer cars and a slower economy. However, even with the over 20% drop in fatality on the road, there is still one death every 15 minutes on the road.

I always think of dying from a crash involving a drunk driver is a lottery. It is a negative lottery for sure since no one would want to win it. In a previous post (The lottery of drunk driving fatality), I discussed the statistic of one drunk driving fatality every 45 minutes. By comparison, the number of deaths on the roads due to all causes is three times higher than just deaths from drunk driving (in the lottery analogy it is three times more likely to win)! I hope in this holiday season, no one will win this negative lottery.

Be safe on the road. Between drinking and driving, only do one of them!

Now the quantitative stuff. As reported in Road Fatalities Dip Thanks To Safer Cars, Economy, there were almost 44,000 road-related deaths in 2005. In 2009, there were about 34,000 deaths. This is a 22% decrease. There are two ways to see this.

One is to calculate the number of reduction in deaths, which is $44000-34000=10000$. Then divide $10000$ by $44000$. We have:

$\displaystyle \frac{10000}{44000}=0.2273$, which is 22.73%.

Another way to derive the 22.73% is to calculate the following ratio:

$\displaystyle \frac{34000}{44000}=0.7727$

Then subtract one from this ratio and obtain $0.7727-1=-0.2273$, which indicates a 22.73% decrease in road-related deaths.

The 2009 figure for the number of road-related deaths is 34,000. This comes out to be one death every 15 minutes. To derive this rate, we need to calculate the total number of minutes in a year. There are 365 x 24 x 60 = 525,600 minutes in a year. Then divide 525,600 by 44,000 to obtain 15.46 minutes. Then round the answer to 15 minutes.

We can get a perspective of this calculation by looking at an example of taking an exam. For example, if you have two hours (120 minutes) to take an exam and the exam has 10 problems, then on average you have 12 minutes to work one problem. Thus if you can work one problem per 12 minutes, you can expect to finish the exam in the allotted time.

Back to the calculation at hand, there are 525,600 minutes in a year and there are 34,000 events. Thus on average there are 15 minutes allotted for each event.

$\displaystyle \frac{365 \times 24 \times 60}{34000}=15.46=15$

The hope is that the denominator in the above ratio will keep getting smaller in the years to come. From 2005 to 2009, the denominator shrank from 44,000 to 34,000. I have a thought. Supose that in the next 5 years (2009 to 2013), there will be the same percent decrease in the road-related deaths as in the 5-year period from 2005 to 2009. What will be the value of the denominator? In other words, according to the same trend line, what will be the number of road-related deaths in 2013?

The answer to the above question is obtained by reducing the 34,000 deaths in 2009 by 22.73%. Try the following:

$\displaystyle 34000 \times (1-0.2273) = 34000 \times 0.7727=26271.8$

If the same trend that played out between 2005 and 2009 holds, the projection for 2013 would be about 26,000. Whether this is a realistic projection or not, I do not know. I will leave this to the experts who study traffic fatality. Let’s hope that the improvement will be as least no worse than this projection.

## Putting a value on higher education

This post is a continuation of the previous post Putting a value on college education. Here we look at the earnings of various levels of higher education relative to “below high school” and relative to high school graduates. The data are obtained from a report in College Board that is called Education Pays 2010. This reports talks about the benefits of the value of higher education to both individuals and society.

In Figure 1.1 of this report, I found information on the median earnings and tax payments of full-time year round workers ages 25 and older for various levels of education in 2008. The following table shows the median earnings.

$\displaystyle \begin{pmatrix} \text{Education Level}&\text{Median Earnings in Dollars} \\{\text{ }}&\text{ }&\text{ } \\\text{Below High School}&\text{24,300} \\\text{High School Graduate}&\text{33,800} \\\text{Some college but no degree}&\text{39,700} \\\text{Associate Degree}&\text{42,000} \\\text{Bachelor's Degree}&\text{55,700} \\\text{Master's Degree}&\text{67,300} \\\text{Doctoral Degree}&\text{91,900} \\\text{Professional Degree}&\text{100,000}\end{pmatrix}$

To help us compare, I convert the median earnings into ratios using “below high school” as baseline.

$\displaystyle \begin{pmatrix} \text{Education Level}&\text{Ratio}&\text{Percent Increase} \\{\text{ }}&\text{ }&\text{ } \\\text{Below High School}&\text{1.0000}&\text{0 percent} \\\text{High School Graduate}&\text{1.3909}&\text{39.09 percent} \\\text{Some college but no degree}&\text{1.6337}&\text{63.37 percent} \\\text{Associate Degree}&\text{1.7284}&\text{72.84 percent} \\\text{Bachelor's Degree}&\text{2.2922}&\text{129.22 percent} \\\text{Master's Degree}&\text{2.7695}&\text{176.95 percent} \\\text{Doctoral Degree}&\text{3.7819}&\text{278.19 percent} \\\text{Professional Degree}&\text{4.1152}&\text{311.52 percent}\end{pmatrix}$

For a full explanation on how the math is done, see Putting a value on college education. As an example, I show the calculation for the bachelor’s degree level. The following is the ratio of bachelor’s degree to the below high school level.

$\displaystyle \frac{55700}{24300}=2.2922$

Thus, the median earning of college graduates is 2.2922 times the median earning of the below high school level, or equivalently 229.22% of the below high school level. Subtracting one from this ratio gives the percent increase going from below high school to bachelor’s degree. This means that the additional amount a college graduate can expect to earn is 129% of the median earning of the below high school level!

In general, the ratio column in the above table gives information about the median earning at each level as a percentage of the baseline (below high school). The column for percent increase gives the percentage increase going from the baseline to each level, that is, the additional earning power at each level expressed as a percentage. It is clear that the higher the educational level, the higher the additional earning power.

Now the same comparison but using the high school level as the baseline.

$\displaystyle \begin{pmatrix} \text{Education Level}&\text{Ratio}&\text{Percent Increase} \\{\text{ }}&\text{ }&\text{ } \\\text{Below High School}&\text{0.7189}&\text{-28.11 percent} \\\text{High School Graduate}&\text{1.0000}&\text{0 percent} \\\text{Some college but no degree}&\text{1.1746}&\text{17.46 percent} \\\text{Associate Degree}&\text{1.2426}&\text{24.26 percent} \\\text{Bachelor's Degree}&\text{1.6479}&\text{64.79 percent} \\\text{Master's Degree}&\text{1.9911}&\text{99.11 percent} \\\text{Doctoral Degree}&\text{2.7189}&\text{171.89 percent} \\\text{Professional Degree}&\text{2.9586}&\text{195.86 percent}\end{pmatrix}$

With high school graduates as baseline, high school dropouts only earn about 71.89% of the median earning of the high school graduates. The additional earning power of the below high school level is -28.11%, which would actually be a decrease in earning power. Once again, a similar clear pattern here: learning has dollar value. The higher the educational level, the higher the additional earning power. On average, a college graduate can expect to earn 65% (64.79%) more than a high school graduate. The additional earning power for a professional degree is even greater, 195.86%, which is three times the addtional earning power for a college dregree.

## Putting a value on college education

Recently I read a piece in Time.com about high school dropout rates. According to this article, the economic outlook for high school dropouts is bleak. The statistics cited by the article are the average earnings and unemployment rates. In 2009, an average college degree holder earned $1,015 per week and an averaged high school dropout earned just$454 per week. The unemployment rate for the college degree holders is 5.2% and 14.6% for high school dropouts. The figures speak volume. They not only put a dollar value on college education. These numbers also give me an opportunity to talk about the concept of ratio, which is a handy way to compare two quantities.

The numbers I want to focus on are the two weekly earnings of $1,015 and$454. How much larger is $1,015 over$454? In terms of absolute amount, the weekly earning of $1,015 is$561 more than $454 since $1015-454=561$. But the absolute amount does not tell the whole story. We can also look at the ratio of$1,015 over $454, which is obtained by$1,015 divided by $454. $\displaystyle (1) \ \ \ \ \ \ \ \ \ \ \frac{1015}{454}=2.236$ What to make of the ratio 2.236? First, it says that the average earning of a college graduate is 2.236 times of the average earning of a high school dropout. For each one dollar in income for a high school dropout, the average college graduate makes two dollars and twenty four cents. So on average a college degree holder makes over two times more than someone with less than a high school education. In this ratio, the baseline (or denominator) is the average weekly earning of a high school dropout. We can also state the ratio 2.236 in terms of percent. Multiplying it by 100, we get 223.6. So the average earning of a college graduate is 223.6% of the average earning of a high school dropout. This is the same information as in the above paragraph, just that the scale is a little different. For each one hundred dollars in income for a high school dropout, the average college graduate makes two hundred and twenty three dollars and sixty cents. The average college grad’s weekly paycheck is$561 more than the weekly paycheck of an average high school dropout. The additional amount of $561 as a percentage of the paycheck for the average high school dropout is: $\displaystyle (2) \ \ \ \ \ \ \ \ \ \ \frac{561}{454}=1.236$ Note that the answer in (2) is just one less than the ratio in (1). Thus the ratio (1) above has another interpretation. Subtracting one from the ratio (1) gives the percent increase going from$454 to $1,015. The increase is 1.236 (or 123.6% after multiplying by 100). For example, if someone is currently making$454 a week and if his income is to increase to $1,015 a week, this represents a 123.6% raise! This is the average premium of a college education, i.e, the additional amount a college grad can expect to make. We can also flip the ratio in (1) and obtain the following ratio. $\displaystyle (3) \ \ \ \ \ \ \ \ \ \ \frac{454}{1015}=0.4473$ What to make of this ratio? Now the baseline is the college graduate. It tells us that the average earning of a high school dropout is only 0.4473 (or 44.73%) of the average earning of a college grad. For each one dollar earned by a college grad, the high school dropout only makes about 44.73 cents. For each one hundred dollars earned by a college grad, the high school dropout makes about fourty four dollars and seventy three cents. If we subtract one from the ratio in (3), we get the percentage change going from$1,015 to $454 (percentage decrease in this case). Note that $0.4473-1=-0.5527$. Multiplying this by 100, we get negative 55.27%. This says that going from$1,015 to $454 represents a 55.27% decrease in income. So if someone is currently making$1,015 a week, and if his income is to decrease to $454 a week, this represents a 55.27% decrease in income. To summarize, when we compare two quantities $P$ and $Q$ with $Q$ as a baseline, we can calculate the ratio $\displaystyle R=\frac{P}{Q}$. The ratio $R$ gives information about the quantity $P$ as a percentage of the baseline quantity $Q$. The quantity $R-1$ is the percentage change going from the baseline quantity $Q$ to the quantity $P$. If $R-1$ is a positive number, then it is a percentage increase when going from $Q$ to $P$. Otherwise it is a percentage decrease. In our income example in this post, $R-1$ can be interpreted as the additional earning of college graduates expressed as a percentage. ## What is 32 plus 3? What is $32+3$? What is $35-32$? This was a question that apparently stumped a clerk in a Macy’s Department Store. My wife recently had to return a wallet back to Macy’s that was purchased for$32. It was early in the morning and the cash drawer did not have singles. So the clerk did not have the exact amount of cash for $32. However, the cash drawer had$30 in cash plus a $5 bill. My wife offered a solution: taking$35 from the clerk and returning $3 to the clerk. The clerk, a woman possibly in her early 30s, did not believe that this is a correct transaction. After my wife explained to her a few times, the clerk reluctantly agreed to this exchange. This is how the math works in this situation. A customer returned an item back to Macy’s that was worth$32 plus $3 in cash. The total value returned to Macy’s was$35. It was only fair that the clerk gave $35 back to the customer. This is $32+3=35$. We can also look at this as a repurchase. Macy’s Department Store was to purchase an item ($32 in value) from my wife and pay $35 in cash. Then my wife gave the clerk$3 in change. This is $35-32=3$.

This is a telling example. Knowing how to do $32+3=35$ in a math assignment at school, at least for some people, does not mean they know how to use arithemtic to make decisions in real life situations. Some people call this having a number sense. Having a strong number sense is the first step in achieving a larger set of skills commonly called quantitative literacy, which usually includes critical thinking, problem formulation, and written and oral communication.

After I heard this story from my wife, I could not help but wanting to give quantitative literacy a plug. Having a better number sense certainly would help this clerk become more effective at her job. Quantitative literacy can also help open doors. There is a whole host of many exciting, highly in demand and well paying careers that require math and science courses in particular and a comfort level with quantitative reasoning in general. Some of these careers only require training at the community college level (an array of jobs in health sciences and computer technology comes to mind). Being proficient quantitatively will give one the confidence and ability to pass the necesary course work to pursue these career options.

How to become more proficient quantitatively? There is no magic bullet. It will take time and effort. There are books that one can read and courses to take. I would also recommend to start with one’s own situation. Start paying attention to the aspects in one’s life that involve quantitative concepts (e.g. work, buying of consummer goods, personal finance, household buget). In the case of the clerk at the Macy’s store, learning how to make changes correctly is a good place to start. Learning the concept of interest (with respect to saving as well as borrowing) and learning how credit card works are a couple of worthwhile examples. Working with quantitative examples in one’s personal life is a good first step.