Proportional Predictions: How to Use Proportions to Estimate Population Size

1. Proportions: A Quick Refresher

A proportion is a mathematical statement showing that two ratios are equal. For instance, if there are 5 apples and 15 oranges, the ratio of apples to oranges is 5:15, which simplifies to 1:3. If you were to say, “For every apple, there are 3 oranges,” you’d be expressing a proportion.

2. Using Proportions to Estimate Population Sizes

So, how can we use proportions to estimate population sizes? Suppose you want to estimate the number of fish in a lake. It’s not practical (or particularly nice for the fish) to try and count each one. Instead, you could catch a small sample of fish, mark them in a harmless way, and release them back into the lake. After giving them time to mix back into the population, you’d catch another sample. The proportion of marked fish in the second sample can give you a good estimate of the total fish population.

Step-By-Step Guide to Using Proportions to Estimate Population Size

Here’s how you might use proportions to estimate population size, step by step:

Step 1: Take a Sample and Mark

Catch a small sample from the population and mark or tag them. For instance, let’s say we catch and mark 50 fish from the lake.

Step 2: Release and Resample

Release the marked sample back into the population. Wait for a period to let them mix back in. Then, take a new sample from the population. Let’s say we catch 80 fish on our second go.

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Step 3: Count the Marked Sample in the Second Sample

Count the number of marked fish in your second sample. Suppose 20 of our 80 fish have marks.

Step 4: Set Up a Proportion

Now you can set up a proportion. The ratio of marked fish to total fish in the second sample should be approximately equal to the ratio of marked fish to total fish in the entire population. This gives us:

(marked fish in second sample) ÷ (total fish in second sample) = (marked fish in total) ÷ (total fish in the lake)

20/80 = 50/x

Step 5: Solve for the Unknown

Solving this equation for x gives us an estimate of the total fish population in the lake. Cross multiplying gives us 20 × x = 80 × 50 = 4,000, so x = 4,000 ÷ 20 = 200 fish in the lake.

And voila! We’ve used proportions to estimate the total population size.

Remember, this is an estimation method. The actual population size could be somewhat more or less than our estimate, especially with smaller or less well-mixed samples. However, this method still provides a valuable tool for population estimation in many fields.

So, keep honing your math skills, and who knows? You might be the next to make a significant statistical discovery in the real world!

This blog post introduces the concept of using proportions to estimate population sizes, providing a straightforward explanation along with a step-by-step guide to understanding and applying the concept.

The Capture-Recapture Formula and More Examples

The proportion in Step 4 can be rearranged into a short formula. Let M be the number you marked first, C the size of the second sample, and R the number of marked ones you recaptured. Then:

Estimated population N = (M × C) ÷ R

This is the same as cross multiplying R/C = M/N. Scientists call it the Lincoln-Petersen estimate.

When the estimate is reliable

The answer is only as good as the sampling. The method works best when:

• No animals are born, die, arrive or leave between the two samples.
• Marks do not fall off and do not make an animal easier or harder to catch.
• Marked animals have time to mix back in before the second sample.
• Both samples are large enough. If you recapture only one or two marked animals, a small counting change makes the estimate jump a lot.

For example, with M = 40 and C = 60, finding 8 marked gives 300, but finding 6 marked gives 400. That is why larger samples give steadier answers.

Estimating a Population from a Survey Sample

Proportions also help when you cannot count everyone in a group of people or things. The idea is the same: what is true of a fair sample is about true of the whole group.

Example 1: In a random sample of 50 students, 12 say they walk to school. The school has 900 students. About how many walk?

• Set up the proportion: 12/50 = x/900.
• Cross multiply: 50x = 12 × 900 = 10,800.
• Divide: x = 10,800 ÷ 50 = 216 students.

Example 2: A factory checks 200 light bulbs and finds 3 that do not work. About how many faulty bulbs are in a batch of 10,000?

• 3/200 = x/10,000
• 200x = 30,000, so x = 150 bulbs.

Example 3: A jar holds red and blue marbles. You draw 40 marbles at random and 10 are red. If the jar holds 600 marbles, about how many are red? 10/40 = x/600, so 40x = 6,000 and x = 150 red marbles.

The key word is random. Asking only your friends, or testing only the first bulbs off the line, can give a sample that does not look like the whole group. A biased sample gives a biased estimate no matter how carefully you do the math.

Practice Quiz: 10 Questions

Frequently Asked Questions

What is the formula to estimate population size with capture-recapture?

Multiply the number marked in the first sample by the size of the second sample, then divide by the number of marked animals found in the second sample. If 50 fish are marked, 80 are caught later, and 20 of those are marked, the estimate is 50 × 80 ÷ 20 = 200 fish.

Why is a population estimate from proportions only approximate?

A sample never matches the whole population perfectly. Random chance decides which animals or people end up in the sample, so a second sample could give a slightly different ratio. Animals may also move in or out, or marked ones may be easier to catch. Larger random samples usually give closer estimates.

What if no marked animals are recaptured?

Then the proportion has a zero in it and you cannot divide by zero, so the formula gives no estimate. It usually means the population is much larger than your samples, or the marked animals had not mixed in yet. Take a larger second sample, or wait longer before sampling again.

What grade learns to use proportions to estimate population size?

This skill is usually taught in grade 7, when students use data from a random sample to draw conclusions about a whole population. It builds on ratio and proportion work from grade 6. It also appears again in high school statistics and biology lessons on sampling wildlife.