How Would You Estimate The Number Of American Teenagers

Probability Level 4

In the population census of 2010 , USA reported that her population was 309 million in 2010. The median age was 37, and the lower quartile was 18.

Which of the following is the best estimate for the number of teenagers aged 13-19 in USA today?

Details and assumptions

You may need to look up further information.

If you are interested in Population statistics, you can look at What Is The Difference Between USA And India .

Image credit: Flickr user x-ray delta one

75 million 40 million 30 million 20 million

3 solutions

Calvin Lin Staff
Jan 24, 2014

We know that America is a developed country, and the population graph that we expect is to be pretty vertical at the base, and tapers off at the top due to old age. This is supported by having the lower quartile being (close to) half of the median.

As such, a reasonable assumption is that there are $\frac{ 309}{4} \div 19$ million people in each of the age groups from 0 to 18 (and also 19), which implies that the number of teenagers is $\frac{ 309}{4} \times \frac{7}{19} = 28.5$ million. As such, we choose the option of 30 million.

Would problems you would find in an introductory statistics class be appropriate for Brilliant? (I am assuming they would also be in Combinatorics)

Sean Elliott - 7 years, 4 months ago

Certainly, all problems are welcome.

We currently group statistics (probability, expected value, variance, etc) under combinatorics.

Calvin Lin Staff - 7 years, 4 months ago

Sam Thompson
Jan 22, 2014

Using various population graphs (Wikipedia, UNICEF, etc.), I concluded that the population growth has been rather steady for the past 80 years. Since the average life expectancy in America is around 80 and the current population is 310 million (give or take several million), it is clear that teenagers make up a little less than $\frac{1}{10}$ of the current population, or around 30 million.

I don't like this problem. How should I solve it? I think the information needed for this problem should be included here...........

Partho Kunda - 7 years, 4 months ago

Another similar one was posted too. I don't know if there's been a merge between Combinatorics and Data Searching or something but this is definitely not the math we're used to here

Michael Tong - 7 years, 4 months ago

For this question, there is enough information.

You are given that the median is 37, and the lower quartile is 18. What does that tell you about the population structure of those aged <37? It's going to be an almost vertical block, with a slight bias to the bottom due to deaths at older ages. (Of course, this doesn't rule out that 24% of people are 1, 2% are 18, 24% of people are 36, but those scenarios are very unlikely.)

As such, a good approximation would be $\frac{ \text{number of years from 13 to 19 }} { \text{number of years from 0 to 18 }} \times \frac{ 309}{ 4} = 28.5$ . Hence we go with the option of 30.

Calvin Lin Staff - 7 years, 4 months ago

Given your information, is that a valid conclusion?

Why is "steady" population growth important? When you say steady, do you mean in absolute terms or in relative terms? Does it matter?

Are you saying that the number of people aged 3 to 9 is the same as those aged 13 to 19 and is the same as those aged 53 to 59?

Calvin Lin Staff - 7 years, 4 months ago

The derivative of the population graph experiences very little change (the graph turns out to be more linear than exponential in the recent years (1940-2010)

Sam Thompson - 7 years, 4 months ago

S L
Jan 26, 2014

1/4 are under 18 = 77million

10% more are 19 and under ~ 85million

Of these 7/20 are in the 13-19 range ~ 30million

How Would You Estimate The Number Of American Teenagers

Image credit: Flickr user x-ray delta one

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