Category Archives: misc

Anime worth watching: Silver Spoon and school on the farm

Continuing my intermittent series of recommendation posts, today we’ll put aside more intellectual topics and focus on entertainment, albeit with a bit of a serious side. A few weeks ago I had the pleasure of taking a young fan to Otakon at the Baltimore Convention Center. Like Comic-Con in San Diego, Otakon features lots of people dressing up in unusual costumes (the technical term is “cosplay”); however unlike Comic-Con, which at this point is dominated by the promotion of high-profile Hollywood blockbusters, Otakon and its sister conventions (including CHS Otaku Fest, right here at Centennial High School in Ellicott City) focus on the less well-known parallel world of anime (Japanese animated movies and TV series) and manga (Japanese comics).1

American comics, and the movies adapted from them, focus primarily on superheroes.2 Japanese manga have a somewhat wider range of subjects, and so do the anime that they spawn. I’ve recently started watching more anime, and since I’m not a pre-teen or teenager my interests are not in the kid-oriented fantasy or science fiction anime that typically show up on American TV (“Pokemon”, “Dragonball Z”, “Sailor Moon”, and so on). Much of what I watch is oriented more for Japanese domestic consumption than for export; it can be quite interesting but not necessarily a great choice for those new to anime, especially if you’re also allergic to the typical anime style of art (most notably the “big eyes” look). I also wanted to recommend something other than the movies of Hiyao Miyazaki; Miyazaki has made some great movies (“My Neighbor Totoro”, “Kiki’s Delivery Service”, “Spirited Away”, and so on), but looking only at his work would be like watching only movies from Pixar: rewarding but unnecessarily limiting.

Silver Spoon: Yugo Hachiken and his classmates

Yugo Hachiken and his classmates

Hence my first recommendation: the current TV series Silver Spoon (in Japanese Gin no Saji, or 銀の匙 to be pendatic). Silver Spoon follows Yugo Hachiken, a junior high school student who for various reasons (more fully revealed over the course of the series) decides to drop out of the exam-driven Japanese academic rat race and leave his home in the city to enroll in Ōezo Agricultural High School, a boarding school with a farming “voc-tech” program far out in the countryside.

Young Yugo, or “Hachiken-kun”3 as everyone calls him, has a series of “fish out of water” adventures as he encounters the realities of farm work and life, in contrast to his classmates who’ve been born to it and consider it their destiny. Silver Spoon is not action-packed and plot-driven, but more of what’s referred to as a “slice of life” series—a surprisingly popular genre of anime, and one I very much enjoy.

I think Silver Spoon is worth recommending for lots of reasons, including the fact that it uses a relatively realistic style of animation, has a lot of humorous touches, and features good character development, not just of Hachiken-kun but of his classmates as well. But the primary attraction of the series is as a fascinating and thought-provoking window into the world of agriculture. Hiromu Arakawa, the author of the manga on which the anime is based, grew up on her family’s dairy farm in Hokkaido (the northernmost island of Japan, roughly equivalent to Ireland in size, population, and rural character), and thus knows whereof she writes and draws. The series is perhaps best characterized as an ongoing meditation on the sometimes harsh realities of how food ends up in our groceries and on our tables, and on the lives of those who deal with those realities every day.

I’ve written previously of the historical tensions between the residents of Columbia and the rural inhabitants of western Howard County. Few of us in suburbia are likely to uproot ourselves to live and work on a farm in western Howard; the closest we might get is a visit to the Howard County Fair in which we venture beyond the rides to the livestock sheds, or a trip with the kids to the petting zoo at Clark’s Elioak Farm. In some ways watching an animated TV series in a foreign setting and a foreign language might be a much better way to gain a deeper understanding of the lives and concerns of our fellow Howard Countians just a few miles to the west.

A final note on logistics: Thanks to the miracle of the Internet you can watch Silver Spoon for free on either Hulu or Crunchyroll (a streaming service dedicated to anime). The series is subtitled, not dubbed in English, but if you watch any foreign films at all you should be able to deal with that. Hulu Plus subscribers can watch the series ad-free in high definition, as can Crunchyroll Premium subscribers; premium subscribers can also watch episodes almost immediately after they air in Japan.

Since this is my first anime recommendation post (and hopefully not my last), I’ll kick things off with a special offer: I’ll provide a Crunchyroll guest pass to the first person to comment on this post, good for 48 hours of ad-free HD viewing of any show on the site, so that you can try out Silver Spoon for yourself. If you’re a binge watcher of TV you can see the entire first season of the series; the last episode just aired and is not yet available on the free streaming services.


1. Incidentally, the Japanese pronunciations of “anime” and “manga” are approximately “ah-nee-may” and “mahn-gah” respectively, with all syllables equally stressed. Americans of course are free to pronounce them any way they want, so you’ll also hear “a-ni-may” and “mayn-ga” with the stress on the first syllable.

2. There are American comics not about superheroes, for example the comic series on which the TV show The Walking Dead is based. There are also alternative comics or “art comics” (or “comix”), the comics equivalent of literary novels, arthouse films, and indie rock; the recently-concluded Small Press Expo (SPX) event held annually in North Bethesda is a great place to get introduced to them.

3. “-kun” is the Japanese honorific typically used for young men, compared to the more well-known “-san” used between adults of equal status. Watching lots of anime is a good way to pick up on these and other fine points of Japanese culture.

A new way to blog

As you’ll know if you read my last post, I decided to stop blogging for a while to save wear and tear on my shoulder. However as it happened I got a phone call from Nuance, the makers of Dragon Dictate for Mac, who were trying to get me to upgrade to the new version of the product that I purchased a few years ago. It was a reasonably good deal and I thought it was worth trying out, so I took advantage of the offer.

I was quite pleasantly surprised. It turns out that voice recognition software has gotten a lot better since I last used it a few years ago. The recognition rate is quite high although not perfect; it’s certainly good enough to do informal blog posts.

So I’m going to try doing some blogging using this new technology. It will be interesting, because it’s harder to compose one’s thoughts when dictating than it is when writing by hand. That may make for more spontaneous posts, or it may make for more incoherent ones. I guess we’ll just have to wait and see.

P.S. This entire post was composed using Dragon Dictate, except for my having go back in a few places and correct mistakes by hand.

Calculating growth rates (for Howard County or otherwise), part 5

In part 4 of this series I discussed the general problem of estimating growth rates for periods less than a year, and using Howard County’s population in the 21st century as an example calculated estimated monthly, week, daily, and even hourly growth rates for the county based on the Census population figures for 2000 and 2010.

The problem with those calculations is that it’s hard to get a sense for the relative magnitude of the growth rates. For example, how much different is a growth rate of 0.12256% per month from a growth rate of 1.4807% per year? It would be nice to express the growth rates according to a common time period, just as (for example) we use “miles per hour” to refer to the speed of our cars even when we’re just driving 2 minutes to the grocery.

However we have to be careful about this, since a percentage increase that occurs (say) every hour on the hour leads to different results than a percentage increase that is conceived to occur but once a month or once a year. (This is exactly what tripped me up when doing my original estimates of growth rates.)

The solution is to multiply the various rates appropriately to express them as an annual rate, but then to qualify the result by referencing the period over which the growth is assumed to occur. The standard way to do this is to speak of growth being “compounded” at particular intervals. For example, we can take the monthly growth rate of 0.12256% estimated in part 4, multiply it by 12 (the number of months in a year), and express it as an annual growth rate of 1.4707% “compounded monthly”. The following table does this for all the growth rates I calculated in part 4:

Period Per-period growth rate Annual growth rate (with compounding)
Decade 15.384% 1.5384% (compounded per decade)
Year 1.4807% 1.4807% (compounded yearly)
Month 0.12256% 1.4707% (compounded monthly)
Week 0.028271% 1.4701% (compounded weekly)
Day 0.0040271% 1.4699% (compounded daily)
Hour 0.00016779% 1.4698% (compounded hourly)

There are several points worth noting here. First, the idea of compounding in this context is exactly the same as that used in financial calculations: For example, if you have a savings account, your bank will periodically credit you with whatever interest you’ve earned on the money in your account; this is the compounding period.

Second, as compounding periods get shorter the same amount of growth can be produced with a lower nominal rate: In our example a 1.4698% rate compounded hourly produced the same growth over a decade as a 1.4807% rate compounded yearly. Or to put it another way, in a financial context you are better off having a shorter compounding period for a given nominal rate: If the rate on your savings account is nominally 2% per year then you are better off with daily compounding than with monthly.

However there is a limit to how much shorter compounding can affect growth: In the table above moving from yearly to monthly compounding reduced the needed growth rate from 1.4807% to 1.4707%, a difference of 0.01%, but moving from monthly to weekly compounding reduced it only to 1.4701%, a difference of 0.0006%.

As compounding periods get shorter and shorter (per minute, per second, and so on), it appears as if the estimated growth rate will reach some sort of limit, around 1.4698% or 1.4699% in our example. We can refer to this as an annual growth rate compounded continuously or, more simply, as a continuous growth rate. A continuous growth rate isn’t that applicable to your savings account (since your bank isn’t going to credit you with new interest earned every nanosecond) or even to Howard County’s population (since it doesn’t grow every nanosecond either, and you can’t add fractional people like you can fractional dollars).

However continuous or near-continuous growth is very common in nature: Think of bacteria multiplying in a fresh petri dish or (more ominously) in your body after you’re infected with something. The mathematics of continuous growth is also simpler and more elegant than that used for growth compounded on a periodic basis. For example, to compute a continuous growth rate for our Howard County population example I don’t need to do calculations for compounding per minute, per second, and so on in order to find the limit. I can do a quick calculation on my iPhone and tell you that the continuous growth rate in our example is 1.4698689% expressed to 8 significant figures.1

How did I do that? If anyone out there is still reading and (more important) if I can find a good way to explain it, I’ll address that question in a possible part 6.


1. Note that this is higher than the figure of 1.4698% we calculated for hourly compounding, when we would expect the limit to be slightly smaller. As it turns out we’re the victims of rounding in computing the growth rate per hour; expressed to 8 digits the estimated hourly growth rate is actually 0.00016779340%, which corresponds to an annual rate of 1.4698701% compounded hourly, slightly larger than the continuous growth rate of 1.4698689%.

Calculating growth rates (for Howard County or otherwise), part 4

In part 3 of this series I recapped the method derived in part 2 for estimating growth rates (using Howard County’s population in the 21st century as an example) and discussed how to use such estimates to project growth in future years.

Now let’s go back to a question I asked at the end of part 2: Can we calculate a more accurate estimate for the growth rate? We can begin exploring this question by going back to my original inaccurate estimate in part 1 and considering where I went wrong. To get that estimate I simply took the final population in 2010, divided it by the initial population in 2000, then divided that by 10 to get an annual growth rate (which I then converted to a percentage value). That initial estimate was too high: When I used that value to estimate the population in 2001, 2002, and so on, it produced a final population estimate for 2010 that was well in excess of the actual 2010 population.

In making my original estimate I saw that the population in 2010 was 15.834% higher than the population in 2000. The additional population didn’t get added all at once; some population growth occurred in each of the 10 years. I tried to account for that ongoing growth by dividing 15.384% by 10 and assuming 1.5384% growth per year. But that corresponded to adding 1.5384% of the 2000 population each year, and that was my mistake. In actuality an annual growth rate as applied to (say) estimating the 2006 population produces a percentage increase relative to the 2005 population, not the 2000 population.

The 2005 population was larger than the 2000 population because it reflected population growth in the years since 2000. Thus using our initial estimated growth rate of 1.5384% (based on a percentage of the 2000 population) produced too high an estimate of the population growth when we computed population growth year by year (and as part of that process applied that growth rate to the 2005 population). Or, to put it another way, a smaller growth rate than 1.5384% was able to produce 15.384% growth from 2000 to 2010 when applied on a year-by-year basis. In fact, a growth rate of about 1.4807% (vs. 1.5384%) is sufficient to produce 15.384% growth over the 10-year period, as I showed in part 2.

Let’s now turn to a new but (as we’ll see) related question: What if instead of projecting population growth on a year-by-year basis, we wanted to project it on a month-to-month basis? For example,  the 2010 population figure of 287,085 was for April of that year (actually April 1). How could we project the population in May 2010, June 2010, and so on? Could we simply divide our estimated annual growth rate of 1.4807% by 12 to calculate a monthly growth rate?

Based on the discussion above, we should suspect the answer is no. Let’s work out the numbers though just to be sure: We divide 1.4807% by 12 to obtain an initial estimate of 0.12339% growth per month. Using this estimate the population for May 2010 (i.e., on May 1) would be 287,085 times 1.0012339 (converting 0.12339% to non-percentage form and adding 1), or 287,439. Per our discussion in part 3, the population for April 2011 (12 months later) would be 287,085 times 1.0012339 raised to the 12th power, which equals 287,085 times 1.014908, or 291,365.

But wait: according to our estimated annual growth rate of 1.4807% the population for April 2011 (1 year later) should be 287,085 times 1.014807 or 291,336. It’s not a big difference (29 people), but it’s still significant. Again our initial estimated growth rate produced estimated population figures that are too high, and for a similar reason as previously: Our estimated monthly growth rate assumed that for each month we’re adding a given percentage of the population as of April 2010, but in actuality the increase in each month is based on the prior month’s population, which in our example is always higher that that (since we’re making estimates for later in 2010 and 2011).

How can we get a better estimate? We simply go back to our approach in part 2 for computing a growth rate using the actual Census population figures for 2000 and 2010, this time computing everything on a monthly (rather than yearly) basis:1

  1. We again start by dividing the population in (April) 2010 by the population in (April) 2000. This gives 287,085 divided by 247,842, or 1.158339.
  2. Since there are 120 months between the starting and ending population figures, this time we take the 120th root of the result from step 1 to find the growth factor. The 120th root of 1.158339 is 1.0012256.
  3. Again we subtract 1 from the growth factor to find the growth rate, which this time is a monthly growth rate. This gives 1.0012256 minus 1, or 0.0012256.
  4. Again we multiply the growth rate by 100 to convert it into percentage form. This gives 100 times 0.0012256, or 0.12256% per month.

So the more correct estimate for a monthly growth rate is 0.12256% instead of 0.12339%.

Can we go further, and estimate weekly growth rates or even daily growth rates? Of course we can: It’s simply a matter of finding the number of time periods (days, weeks, months, or years) between the initial and final populations, and then using that number when we take the root in step 3 of our general method.

The results are shown in the following table, each expressed to 5 significant figures; just for fun I’ve added entries for a growth rate per decade and growth rate per hour. As an example, the daily growth rate is computed by dividing the population on April 1, 2010 (287,085) by the population on April 1, 2000 (247,842), taking the 3650th root of the result to get the daily growth factor (since there are 10 years of 365 days each), subtracting 1 to get the daily growth rate, and then multiplying by 100 to put the daily growth rate in percentage terms.2

Growth Rate Period Number of periods Estimated Growth Rate
Decade 1 15.384%
Year 10 1.4807%
Month 120 0.12256%
Week 520 0.028271%
Day 3,650 0.0040271%
Hour 87,600 0.00016779%

The above is all well and good, but the way the growth rates are expressed makes it hard to compare them. What would be nice would be to express all rates as annual rates, just as (for example) we talk of driving 50 miles per hour whether our trip lasts for 2 minutes, 2 hours, or 2 days. However we’ve seen enough thus far to know we have to be careful in how we do this, and since this post is long enough as it is I’ll postpone discussion of this topic until part 5.


1. I’m implicitly assuming that each month has equal length. This is not true (at 31 days the month of January is more than 10% longer than February in non-leap years), but it doesn’t affect my overall argument.

2. For simplicity I’ve assumed that each year is exactly 52 weeks (actually a year is about 52.1 weeks), and that there are no leap years (actually there was an extra day in both 2004 and 2008). Correcting these would change the weekly, daily, and hourly growth rates very slightly.

Calculating growth rates (for Howard County or otherwise), part 3

In part 2 of this series I discussed a more correct approach to the problem of estimating growth rates, using Howard County’s population in the 21st century as an example. Given the population figures for the 2000 and 2010 censuses, we can estimate an annual growth rate as follows:

  1. Divide the final population in 2010 by the initial population in 2000.
  2. Take the 10th root of the result from step 1 to find the growth factor. (We use 10 because the period we’re considering is 10 years long.)
  3. Subtract 1 from the growth factor to find the growth rate.
  4. Multiply the growth rate by 100 to convert it into percentage form.

Recall that you can take roots using a scientific calculator app for your smartphone, tablet, or PC, as described in the last post; you can also compute roots in a application like Microsoft Excel or Google Spreadsheets.1

Picture of iPhone scientific calculator

iPhone scientific calculator

Using the technique above I estimated the growth rate of Howard County from 2000 to 2010 as 1.4807% per year, or 0.014807 in non-percentage form. I then asked how we could estimate the future population of Howard County, say in 2020.

One approach to do this is similar to how we computed estimated populations from 2001 through 2009: We could add 1 to the non-percentage form of the growth rate to get the growth factor, and then multiply the growth factor by the Census population in 2010 to get an estimated population for 2011. We could then multiply the estimated 2011 population by the growth factor to get an estimated population for 2012, multiply that value again by the growth factor to get an estimate for 2013, and continue year by year until after ten multiplications we had an estimate for 2020.

Photo of y-to-power-of-x key

y to the x-th power

However we can simplify this calculation as follows: Since we started with the population in 2010, multiplied by the growth factor each time, and estimated the population for 10 years out (2011 through 2020), this is the same as raising the growth factor to the power of 10 and then multiplying the resulting value by the population in 2010. But do we still have to compute the 10th power of the growth factor by doing all the multiplications ourselves?

THe answer is no. Just as the scientific calculator app on your iPhone or other device can compute roots for you, it can also compute powers. Let’s try it out: Suppose we want to find the value of 5 raised to the 4th power (in other words, 5 times 5 times 5 times 5). On the iPhone’s calculator we enter 5, press the “y to the x-th power” key (pictured), enter 4, then press the “=” key. The answer should be 625, since 5 to the 4th power is 625.2

We can now try the suggested approach to estimating Howard County population in 2020, given our estimated annual growth rate:

  1. Divide the growth rate by 100 to convert it into non-percentage form. This gives us 1.4807% divided by 100, or 0.014807.
  2. Add 1 to the growth rate to find the growth factor. This gives us 1 plus 0.014807 or 1.014807.
  3. Compute the growth factor raised to the power of 10. (We use 10 because we’re estimating the population in 2020, 10 years after 2010.) In our scientific calculator app we enter 1.014807, press the “y to the x-th power” key, enter 10, then press the “=” key; the result is 1.158336.
  4. Multiply the population in 2010 by the value just computed. This gives us 287,085 times 1.158336 or 332,541 for the estimated population in 2020.

Note that we could actually bypassed this computation by noting that if the population grew by 15.84% in the 10 years from 2000 to 2010 (as noted in part 1) and the growth rate didn’t change, we’d expect the population to grow another 15.84% in the next 10 years from 2010 to 2020. This is exactly what we found in the computation above: We calculated the population in 2020 as 1.158336 times the population in 2010, which corresponds to a percentage increase over 10 years of 15.834% (rounding off to five significant digits).

However what if we had an interval that wasn’t a multiple of 10? For example, what if we want to estimate the Howard County population in 2035? Assuming we use the 2010 population as our starting point we need to calculate the 25th power of the growth factor 1.014807: Enter 1.014807, press the “y to the x-th power” key, enter 25, then press the “=” key; the result is 1.444064. Multiplying this by the 2010 population of 287,085 gives us 414,569 for the estimated population in 2035.

So far so good. In my next post I’ll go back and explore further how I got my initial estimate in part 1 so wrong, as a prelude to discussing growth rates in a financial context and how to obtain better estimates of growth rates.


1. Microsoft Excel has a built-in function SQRT to take square roots, but uses more cryptic formulas for taking higher roots. In our example we had to take the 10th root of 1.158339; in Excel this would be expressed with either of the formulas “=POWER(1.158339,1/10)” or “=1.158339^(1/10)”. In general if we have values in two spreadsheet cells A1 and A2 then either of the formulas “=POWER(A1,1/A2)” or “=A1^(1/A2)” would take the A2-th root of A1.

These formulas will also work in Google Spreadsheets. To avoid a lengthy digression I’ll skip for now any explanation of why these formulas are written the way they are.

2, In Microsoft Excel or Google Spreadsheets the equivalent operation of computing 5 to the power of 4 can be done using either of the formulas  “=POWER(5, 4)” or “=5^4”. In general if we have values in two spreadsheet cells A1 and A2 then either of the formulas “=POWER(A1,A2)” or “=A1^A2” would produce A1 raised to the A2-th power.

Calculating growth rates (for Howard County or otherwise), part 2

In my last post I introduced the problem of estimating growth rates, using Howard County’s population in the 21st century as an example. I took a simpleminded approach:

  1. Take the difference between the county’s population in 2010 and 2000.
  2. Divide that difference by the population in 2000 and multiply by 100 to get the percentage growth increase from 2000 to 2010.
  3. Divide that percentage by 10 to get an estimate of the population growth per year.

As we saw in the last post, the simpleminded approach produces an incorrect answer: the estimated growth rate is too large. In this post I’ll show a more correct way to estimate the growth rate. As before, I’ll avoid mathematical notation and restrict myself to operations you can do on a calculator or in a program like Microsoft Excel or Google Spreadsheets.

The key to finding a better approach is to look at the method we used to prove the simpleminded approach incorrect: taking the supposed growth rate, estimating year by year what growth it would produce, and then comparing it to the actual final population figure. The twist this time is to assume that we don’t know the growth rate initially, and instead use the method to estimate it. So, we start with the population of Howard County in 2000 (247,842), and we then look at how populations in 2001, 2002, and so on, would be calculated for a given growth rate:

  1. We assume that the (as yet unknown) growth rate is expressed in the non-percentage form. (For example, we’d express a 5% growth rate as 0.05.)
  2. To get the population in 2001 we add 1 to the (non-percentage) growth rate and then multiply that value times the initial population in 2000. To save some words we’ll refer to the sum of 1 plus the growth rate as the growth factor. (For example, if the growth rate were 5% or 0.05 then the growth factor would be 1.05.) Another way to express this is that the population in 2001 is calculated as the population in 2000 times the growth factor.
  3. To get the population in 2002 we multiply the population previously calculated for 2001 by the growth factor. But wait: The population in 2001 was in turn calculated as the 2000 population times the growth factor. So another way to calculate the 2002 population is to multiply the population of 2000 by the growth factor (which gives us the 2001 population), and then to multiply by the growth factor again. The result is that the 2002 population is calculated as the 2000 population times the growth factor times the growth factor.

This is a key point, so let’s stop here and look at an example. If the growth rate were actually 5% then the growth factor would be 1.05. The 2001 population would then be 247,842 times 1.05 or 260,234, and the 2002 population would be 260,234 times 1.05 or 273,246. We could get the same answer by multiplying 1.05 by 1.05 to get 1.1025, and then multiplying 1.1025 times 247,842 to get 273,246.

Now let’s continue:

  1. The population in 2003 is calculated as the population in 2002 times the growth factor. Since we can calculate the population in 2002 as the population in 2000 times the growth factor multiplied by itself (growth factor times growth factor), we can calculate the population in 2003 as the population in 2000 times the growth factor multiplied by itself and then multiplied again by itself again. In other words, we calculate the 2003 population as the 2000 population times the growth factor times the growth factor times the growth factor.
  2. The population in 2004 is calculated as the population in 2003 times the growth factor. Following from the previous item we can calculate the population in 2004 as the population in 2000 times the growth factor times the growth factor times the growth factor times the growth factor.

Let’s stop again here to introduce some new terminology: You may know that the product of the growth factor times itself (growth factor times growth factor) is referred to as the square of the growth factor, and that the growth factor times the growth factor times the growth factor is referred to as the cube of the growth factor. What do we call the value calculated as the growth factor times the growth factor times the growth factor times the growth factor (i.e., where the growth factor appears four times in the product)?

The standard term is for this value is the growth factor raised to the power of 4 or (less verbosely) the 4th power of the growth factor, because in calculating the product the growth factor appears four times. As an example, if the growth factor is 1.05 then the 4th power of the growth factor would be 1.05 times 1.05 times 1.05 times 1.05, or 1.2155 (to five significant digits). Continuing on…

  1. We can now rephrase how we estimate the 2004 population: It’s calculated as the 2000 population times the 4th power of the growth factor.
  2. The population in 2005 is then the 2004 population times the growth factor, or equivalently the 2000 population times the 4th power of the growth factor, times the growth factor again. How do we express the product of the 4th power of the growth factor and the growth factor itself? In this product the growth factor appears 5 times (4 times from the 4th power and one time when multiplying by the growth factor once more), so we refer to it as the 5th power of the growth factor. The 2005 population is thus the 2000 population times the 5th power of the growth factor.
  3. The 2006 population is then the 2005 population times the growth factor, which is equal to the 2000 population times the 5th power of the growth factor, times the growth factor again, which is equal to the 2000 population times the 6th power of the growth factor.
  4. The 2007 population is then the 2006 population times the growth factor, which is equal to the 2000 population times the 7th power of the growth factor.
  5. Do you see the pattern here? The 2007 population is equal to the 2000 population times the 7th power of the growth factor. The 2008 population is equal to the 2000 population times the 8th power of the growth factor. Finally, the 2009 population is equal to the 2000 population times the 9th power of the growth factor.

We now come to 2010. On the one hand, we can calculate the population in 2010 as the population in 2000 times the 10th power of the growth factor. On the other hand, we actually know the population in 2010. If we get the growth rate (and thus the growth factor) correct then those two numbers should be the same. In particular, we can plug in the population values for 2000 (247,842) and 2010 (287,085) and see that 287,085 should equal 247,842 times the 10th power of the growth factor. This in turn means that the 10th power of the growth factor should be equal to 287,085 divided by 247,842, or 1.158339 (to seven significant digits). We now know what the 10th power of the growth factor is. How do we calculate the growth factor itself?

The short answer is that we do what’s called “taking a root”. What is a root? Some examples: We know that 5 times 5 is 25, so we refer to 25 as 5 squared; alternatively we could say that 5 is the “square root” of 25. Similarly, 5 times 5 times 5 is 125, so 125 is 5 cubed and 5 is the cube root of 125. Finally, 5 times 5 times 5 times 5 is 625, so 625 is the 4th power of 5 and 5 is the 4th root of 625. In other words, taking a power and taking a root are inverse operations: If one number is (say) the 10th power of a second number then the second number is the 10th root of the first.

Picture of iPhone scientific calculator

iPhone scientific calculator

In this case the 10th power of the growth factor is 1.158339, so the growth factor is the 10th root of 1.158339. How do we find this value? We use a calculator or a computer. For example, the standard calculator app on the iPhone can be turned into a so-called “scientific calculator” by turning the phone on its side (like you would do when watching a YouTube video). On the left are keys for special functions, among which is one to take the xth root of a number y.1

Picture of x-root-y key on iPhone scientific calculator

Take the xth root of y

Let’s try it out: Suppose we want to find the 4th root of 625. On the iPhone’s calculator we enter 625, press the “xth root of y” key (pictured), enter 4, then press the “=” key. The answer should be 5, since 5 to the 4th power is 625.2

Now we’re almost there. The growth factor is the 10th root of 1.158339, which our calculator tells us is 1.014807. (On the iPhone’s calculator enter 1.158339, press the “xth root of y” key, enter 10, then press the “=” key.) Recall that the growth factor is 1 plus the growth rate, so the growth rate is 1 minus the growth factor, or 0.014807. Multiplying by 100 to convert this into a percentage, the estimated growth for Howard County’s population from 2000 to 2010 is 1.4807% per year.

Before we go on, we can summarize this new method for calculating the growth rate as follows:

  1. Divide the population in 2010 by the population in 2000. This gives 287,085 divided by 247,842, or 1.158339.
  2. Take the 10th root of the result from step 1 to find the growth factor. The 10th root of 1.158339 is 1.014807.
  3. Subtract 1 from the growth factor to find the growth rate. This gives 1 minus 1.014807, or 0.014807.
  4. Multiply the growth rate by 100 to convert it into percentage form. This gives 100 times 0.014807, or 1.4807%.

We can check this estimate as we did before, by calculating the estimated populations from 2001 to 2009, multiplying the previous year’s population by the growth factor each time:

Year Population (Actual) Population (Estimated)
2000 247,842
2001 251,512
2002 255,236
2003 259,015
2004 262,850
2005 266,742
2006 270,692
2007 274,700
2008 278,767
2009 282,895
2010 287,085

As the final step we calculate an estimated population for 2010 using the method we used in computing the values for 2001 and 2009: We take 282,895 (the estimated value for 2009) and multiply it by 1.014807. The result is an estimate of 287,084 for the population in 2010, within 1 of the actual value of 287,085. (This difference is due to rounding error.)

Unlike the previous method from part 1, this new method produces a good estimate for the growth rate. Can we project future Howard County population using this estimate? Can we use this method of estimating growth rates in other contexts, for example in financial calculations? And can we produce an even better estimate? I’ll answer these questions in part 3.

UPDATE: Corrected the values in step 4 of the new method (changed 1.014807 to 0.014807) and in the computation of the estimated population for 2010 (changed 287,083 to 287,084).


1. I’ve never owned an Android smartphone or tablet, so I don’t know if Android devices typically have a similar scientific calculator app built in. However I suspect there are plenty of scientific calculator apps, including some free ones, in the app store for whatever smartphone or tablet you have.

2. You may have noticed a key to the left of the “xth root of y” key that has a similar symbol but no “x” or “y”. This is the square root key. For example, if you enter 25 and then press the square root key you’ll get the answer 5, the square root of 25. This is exactly the same as entering 25, pressing the “xth root of y” key, entering 2, and pressing the “=” key.

Calculating growth rates (for Howard County or otherwise), part 1

[I'm interrupting my series of “weekend reading” posts to bring you an actual blog post.]

Last week at work one of my tasks was estimating growth rates for a particular quantity (never mind exactly what). I found that doing this was not exactly trivial, as there are multiple ways to calculate growth rates, some of them more mathematically complicated than others. I think I now understand how this all works, and to test my understanding I’m going to try to explain it here.

If you ever have to deal with growth rates in your own life (and remember, they come up a lot in financial matters) this may be useful to you as well. To make things simpler I’ll avoid mathematical notation and restrict myself to operations you can do on a calculator or in a program like Microsoft Excel or Google Spreadsheets.

As a local example of the problem, let’s try to calculate the rate of growth of Howard County’s population from 2000 to 2010. (Perhaps we’re interested in forecasting future population growth, or we’re just curious about the rate of past growth.) Here are the two numbers we’ll start with, the total population of Howard County as counted in the 2000 and 2010 censuses:1

Year Population
2000 247,842
2010 287,085

We’ll start with the simplest calculation possible; feel free to follow along using your own calculator:

  1. Take the difference between the final population in 2010 and the initial population in 2000. This difference is 287,085 minus 247,842, or 39,243.
  2. Divide this difference by the initial population. This gives us 39,243 divided by 247,842, or 0.15834 (rounded off to five significant digits).
  3. Express this value as a percentage of the initial population by multiplying it by 100. This gives us 100 times 0.15834 or 15.834. So the population of Howard County increased 15.834% from 2000 to 2010.
  4. Finally, divide the percentage increase from 2000 to 2010 by 10, to calculate an estimated growth rate per year. This gives us 15.834% divided by 10 or 1.5834%, our first (but not last!) estimated annual growth rate for Howard County’s population during the decade.

So we’re done, right? Not so fast! Let’s check this estimate by calculating estimated populations for Howard County for 2001 through 2009. For example, if the true annual growth rate is 1.5834%, then from 2000 to 2001 Howard County should have added a total number of people equal to 1.5834% of 247,842. We can estimate the population in 2001 as follows:

  1. Convert the growth rate expressed as a percentage into the non-percentage form, by dividing it by 100. This gives us 1.5834 divided by 100 or 0.015834.
  2. Multiply the (non-percentage) growth rate times the initial population in 2000. This gives us 0.015834 times 247,842 or 3,924 (rounding to a whole number) for the estimated number of people added to the population from 2000 to 2001.
  3. Add the number of people added from 2000 to 2001 to the initial population in 2000. This gives us 3,924 plus 247,842 or 251,766.

Before we go on to estimate the population for the other years, note that we can simplify the computation above: In step 2 we’re calculating 0.015834 times 247,842, and in step 3 we’re adding to it 1 times 247,842. Since we’re multiplying by 247,842 in both steps, we can factor it out and just multiply 247,842 by the sum of 0.015834 plus 1, or 1.015834.2

The calculation above then becomes the following (again, feel free to follow along at home):

  1. Convert the growth rate expressed as a percentage into the non-percentage form, by dividing it by 100. This gives us 1.5834 divided by 100 or 0.015834.
  2. Add 1 to the (non-percentage) growth rate, and then multiply that value times the initial population in 2000. This gives us 1.015834 times 247,842 or 251,766 for the estimated population in 2001 (again, rounding down to a whole number).

Now that we have an estimated population for 2001, we can calculate the estimated population for 2002 by following the steps above and just substituting 251,766 as our initial population in 2001. This gives us 1.015834 times 251,766 or 255,752 for the estimated population in 2002. We can continue doing this for the other years; the general process is as follows:

  1. Start with the initial population in 2000 (247,842) and the estimated growth rate (1.5384% per year, or 0.015384 in non-percentage form).
  2. For each year from 2001 to 2009, multiply the previous year’s population by the sum of 1 plus the (non-percentage) growth rate, or 1.015384 in our example.

Doing this gives us the following table; I’ve included the actual initial and final populations in 2000 and 2010 for comparison:

Year Population (Actual) Population (Estimated)
2000 247,842
2001 251,766
2002 255,752
2003 259,802
2004 263,916
2005 268,095
2006 272,340
2007 276,652
2008 281,033
2009 285,483
2010 287,085

So far so good. Now let’s do one final step and calculate an estimated population for 2010 using the method we used in computing the values for 2001 and 2009: We take 285,483 (the value for 2009) and multiply it by 1.015834. The result is an estimate of 290,005 for the population in 2010.

But wait—this seems way too high. The actual population in 2010 was 285,075 or almost five thousand less than our estimate. Maybe we shouldn’t have rounded off the estimated growth rate and the estimated populations we used in the calculations? No, if you do the calculations without rounding then the estimated 2010 population is almost exactly the same.3

We have to conclude that we’ve overestimated the growth rate, and that the actual growth rate is somewhat less than 1.5834%. How can we produce a better estimate? I’ll take up that question in part 2.


1. Population figures are taken from the “Total Population” value in the “Profile of General Demographic Characteristics” for Howard County, Maryland for 2000 and 2010.

2. This is a application of what they teach in elementary school as the distributive law of arithmetic. For example, if you multiply 5 times the sum of 3 plus 7 that’s the same as multiplying 5 times 3 and adding it to 5 times 7, and vice versa. (3 plus 7 is 10, and 5 times 10 is 50. Similarly, 5 times 3 is 15, 5 times 7 is 35, and the sum is again 50.)

3. More specifically, when I used the full precision of my calculator the estimated population for 2010 was 290,003, a difference of only 2 from the estimate above.