Choose a mode for what you want to find, then enter the results of your survey. "Capture-recapture" estimates the total number, such as the fish in a pond. "Count in a population" estimates how many in the whole match a condition (such as the total number of defective items).
Enter counts as whole numbers. The answer from a sample is an estimate, so both the exact result of the division (a fraction) and a rounded "about ..." number are shown.
Result and graph
Enter the results of your survey in the fields on the left and press "Calculate". The estimate and a bar diagram will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formula and diagram
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Estimate the total size of a group you cannot count one by one, such as the fish in a pond, with the capture-recapture method (mark some, release them, catch a second group and check the share that is marked)
- When you know the size of the whole population (for example, 3,000 products), estimate how many in it match a condition (the total number of defective items) from a random sample (for example, 2 defective out of 100)
- Shows the steps with a proportion, just as in a middle school textbook (the product of the means equals the product of the extremes). The answer is shown both as the exact result of the division (a fraction) and as a rounded "about ..." number
- A bar diagram lets you compare the share in the sample with the share in the population and see that they are the same
Estimates from a sample assume that the sample was chosen without bias, like drawing names from a hat (random sampling). The result is only an estimate and will differ somewhat from the real number.
What is this calculation used for?
Counting fish and wildlife populations (ecology and conservation)
To count animals that cannot all be counted, such as fish in a pond, deer in the mountains or invasive fish in a lake, researchers really use the capture-recapture method on this page. In fisheries management and invasive species control, the population estimate is the basic data for deciding catch limits and how big a control effort should be.
The method assumes that marks do not fall off, that few animals come or go (are born, die or move) during the study, and that the marked animals mix well with the rest. Real studies choose the timing and the method carefully to come close to these conditions.
Sample inspections in a factory (quality control)
Tests that destroy the product or make it unsellable, such as battery life tests or checking the contents of canned food, cannot be done on every item. Instead, a random sample is checked, and the total number of defective items is estimated from the defective share in the sample.
Estimating "about 60 in all" from 2 defective out of a sample of 100 is one of the most basic tools of quality control.
Opinion polls and TV ratings (social surveys)
Opinion polls such as the presidential approval rating do not ask every voter in the country. They estimate the share for the whole from a random sample of one thousand to a few thousand people. TV ratings, too, are shares for the whole audience estimated from a sample of households.
The power of sampling is that, as long as the sample is chosen at random, even a small sample can estimate the whole quite accurately (how big the error is depends on the sample size). Real surveys also add steps to avoid bias by region or age group, such as weighting, but the base is this idea of estimating the whole from the share in a sample.
Election night projections and exit polls
News networks can call a race when only a small share of the votes have been counted because they estimate each candidate's share of the total vote from samples: exit polls (interviews with people who just voted) and the part of the votes already counted.
Applying the share in a sample to the whole is the same idea as the proportion on this page. When calling a race, networks also combine data from past elections and statistical theory and decide carefully.
Formula and diagram
Capture-recapture (estimating the total)
Bar diagram
Standard notation
(the usual math form)
\(x\)
\(=\)
\(m\)
\(\times\)
\(n\)
\(\div\)
\(k\)
In words
(symbols replaced with words)
④
\(x\): total in the pond
\(=\)
①
\(m\): number marked
\(\times\)
②
\(n\): number in the 2nd catch
\(\div\)
③
\(k\): marked in the 2nd catch
The formula in words
①
Take the
\(m\): number caught first and marked
②
multiply it by the
\(n\): number in the 2nd catch
③
divide by the
\(k\): number marked in the 2nd catch
④
and you get the estimate of the
\(x\): total in the pond
Quick example
If you mark 50 fish, put them back in the pond, and later catch 30 fish of which 6 are marked, the total number in the pond is
\(x\): total in the pond
\(=\)
number marked (50 fish)
\(\times\)
number in the 2nd catch (30 fish)
\(\div\)
number marked in it (6 fish)
\(x = \dfrac{50 \times 30}{6} = \dfrac{1500}{6} = 250\)
Key idea
This formula works because "the share of marked fish in the whole pond" and "the share of marked fish in the second catch" can be assumed to be about the same. If the marked fish mix evenly through the pond, the mix should be about the same wherever you scoop.
So let \(x\) be the total in the pond, set up the proportion \(x : m = n : k\) (total : marked), and solve it with "the product of the means equals the product of the extremes" in the next card. This formula comes out.
The answer is only an estimate. Counting every fish could give a different number, but the value of this method is that it gives you a good idea of the total without draining the pond.
Property of proportions (product of the means = product of the extremes)
Standard notation
(the usual math form)
\(a\)
\(:\)
\(b\)
\(=\)
\(c\)
\(:\)
\(d\)
\(b \times c\)
\(=\)
\(a \times d\)
In words
(symbols replaced with words)
extreme \(a\)
\(:\)
mean \(b\)
\(=\)
mean \(c\)
\(:\)
extreme \(d\)
①
product of the means \(b \times c\)
\(=\)
②
product of the extremes \(a \times d\)
The formula in words
①
When a proportion holds, the two inside terms (\(b\) and \(c\)) multiplied together, the
product of the means \(b \times c\)
,
②
and the two outside terms (\(a\) and \(d\)) multiplied together, the
product of the extremes \(a \times d\)
, are always equal
Quick example
Solving the proportion \(x : 50 = 30 : 6\) from the pond example gives
product of the means \(50 \times 30\)
\(=\)
product of the extremes \(x \times 6\)
\(50 \times 30 = x \times 6\)
\(6x = 1500\)
\(x = \dfrac{1500}{6} = 250\)
Key idea
The proportion \(a : b = c : d\) says that "the ratio of \(a\) to \(b\) equals the ratio of \(c\) to \(d\)". Written as fractions, it is the same as \(\dfrac{a}{b} = \dfrac{c}{d}\). Multiply both sides by \(b \times d\) and the denominators cancel, giving \(a \times d = b \times c\): the product of the extremes equals the product of the means. In fraction form this is the familiar "cross multiplication" (the cross products are equal).
When a proportion has just one unknown number \(x\), this property turns it into an equation in \(x\) that you can solve. Estimating from a sample is an application of this property.
Estimating the matching count in a population
Bar diagram
Standard notation
(the usual math form)
\(x\)
\(=\)
\(N\)
\(\times\)
\(k\)
\(\div\)
\(n\)
In words
(symbols replaced with words)
④
\(x\): matching count in the population
\(=\)
①
\(N\): population size
\(\times\)
②
\(k\): matching in the sample
\(\div\)
③
\(n\): sample size
The formula in words
①
Take the
\(N\): population size
②
multiply it by the
\(k\): number matching in the sample
③
divide by the
\(n\): size of the sample you took
④
and you get the estimate of the
\(x\): matching count in the population
Quick example
If you take 100 of 3,000 products at random, check them, and find 2 defective, the total number of defective items is
\(x\): total defective items
\(=\)
population size (3,000)
\(\times\)
defective in the sample (2)
\(\div\)
sample size (100)
\(x = \dfrac{3000 \times 2}{100} = \dfrac{6000}{100} = 60\)
Key idea
The idea is the same as capture-recapture: you assume that "the matching share in the population" and "the matching share in the sample" are about the same. You can also read it as multiplying the whole population \(N\) by \(\dfrac{k}{n}\) (the share in the sample).
Let \(x\) be the matching count, set up the proportion \(N : x = n : k\) (total : matching), and solve it with "the product of the means equals the product of the extremes". This formula comes out.
This estimate only works when the sample is chosen without bias, like drawing names from a hat (random sampling). For example, if a factory checks only "the first 100 items made", any quirk of the machine that day shows up in the result, and the estimate will be off.
Both modes rest on the same idea: "the share in the population and the share in a random sample are about the same". Just by solving a proportion (product of the means = product of the extremes), you can estimate the whole without counting everything. The answer is only an estimate, so the standard way is to give a rounded "about ..." number. Planning a survey, that is, how large a sample you need to keep the error within a limit, is covered by the sample size calculator in the related pages.
Symbols and terms
Symbols
| \(x\) | ex | The letter used for a number you do not know yet (an unknown). On this page, the estimate you want, such as "the total in the pond" or "the matching count in the population", is called \(x\) to set up the proportion. The custom of using \(x\) for an unknown goes back to math books of the 1600s. |
| \(m\) | em | On this page, "the number caught first and marked". Think of it as m for "marked". It is also often used as a count together with \(n\). |
| \(n\) | en | The number in the sample (the second catch, or the number taken out and checked). It comes from "number" and is the standard letter for a count. In statistics, the sample size is written \(n\) by custom. |
| \(k\) | kay | The number in the sample that matched (the marked ones, the defective ones, and so on). Along with \(n\) and \(m\), it is a common letter for counts and positions. |
| \(N\) | capital N | The size of the whole population. By custom in statistics, the larger group is written with a capital letter to tell it apart from the sample size \(n\) (lowercase). |
| \(a : b\) | a to b | A way of writing a ratio. ":" is the ratio sign and is read "to". An equation that joins two ratios with an equals sign, such as \(a : b = c : d\), is a proportion. |
| \(\approx\) | approximately equal to | The sign for "almost equal". It can also be read "about". An answer from a sample is an estimate, so it is written with this sign or the word "about" instead of an equals sign. |
Terms
| sample survey | A survey that studies only part of the population (a sample) and estimates what the whole looks like. It is used for things you cannot count completely or that are destroyed by testing. Students learn it in Grade 7 as random sampling and making inferences about a population. |
| census | A survey that checks every member of the population. The US Census, taken every 10 years, is an example. Checking everything is accurate, but when it takes too much time or money, or when testing destroys the product, a sample survey is used instead. |
| population | The whole group you want to study, such as all the fish in a pond, all the products a factory made, or all the voters in the country. |
| sample | The part taken from the population that you actually study. |
| sample size | The number of items in the sample. The larger the sample size, the smaller the error of the estimate becomes. |
| random sampling | Taking a sample so that every member has the same chance of being chosen, like drawing names from a hat. No one picks on purpose. It is the basic condition for estimates from a sample to work. |
| capture-recapture | A method that marks (tags) the animals caught, releases them, catches a second group after some time, and estimates the total from the share that is marked. Also called mark and recapture, it is really used to count fish and wildlife. |
| proportion | An equation that says two ratios are equal, such as \(a : b = c : d\). The product of the means equals the product of the extremes, and this property lets you find an unknown number. |
| means | In the proportion \(a : b = c : d\), the inside terms \(b\) and \(c\). The product of the means equals the product of the extremes. (This is not the average.) |
| extremes | In the proportion \(a : b = c : d\), the outside terms \(a\) and \(d\). The product of the means equals the product of the extremes. |
| share | How much of the whole a part takes up, found by dividing the part by the whole. Sample surveys rest on the idea that the share is the same in the population and in the sample. |
| rounded number | An approximate number, cut off at a convenient place value by rounding. An answer from a sample is an estimate, so the standard way is to give a rounded number such as "about 250". |
| rounding | To round to a place value, look at the digit just to its right. If it is 4 or less, round down; if it is 5 or more, round up. |
| estimate | Judging what the whole looks like from the part of the information you have. An answer found from a sample is called an estimate in this sense. |
| error | The gap between the estimate and the real value. Results from a sample always have some error, and the larger the sample size, the smaller the error becomes. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over the topics in this table is the fastest way forward.
| Percents and shares (Grade 6) |
|
| Ratios and proportions (Grades 6–7) |
|
| Expressions and one-step equations (Grades 6–7) |
|
| Rounding (Grades 3–4) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table for capture-recapture (estimating the total)
| Marked first m | 50 |
| 2nd catch n | 30 |
| Marked in 2nd catch k | 6 |
| Estimated total x = m×n÷k | =B1*B2/B3 |
| Rounded to a whole number | =ROUND(B4,0) |
Table to check the property of proportions (product of the means = product of the extremes)
| Extreme a (total; the answer to the pond example) | 250 |
| Mean b (number marked) | 50 |
| Mean c (2nd catch) | 30 |
| Extreme d (marked in 2nd catch) | 6 |
| Product of the means b×c | =B2*B3 |
| Product of the extremes a×d | =B1*B4 |
Table to estimate the matching count in a population
| Population size N | 3000 |
| Sample size n | 100 |
| Matching in sample k | 2 |
| Estimated count x = N×k÷n | =B1*B3/B2 |
| Rounded to a whole number | =ROUND(B4,0) |
After pasting, the upper rows (your survey results) are the inputs and the lower rows are calculated automatically.
The first table is the pond example (50 marked, 6 marked out of 30 in the second catch), and the answer is 250. When the division does not come out even, B4 shows a decimal, so read B5, which goes through the ROUND function, as the "about ..." answer.
The second table puts the answer 250 in for x in the proportion x : 50 = 30 : 6 and checks that the product of the means (B5) and the product of the extremes (B6) are both 1500, so they are equal.
The third table is the example of 3,000 products with 2 defective out of a sample of 100, and the answer is 60.
The first table is the pond example (50 marked, 6 marked out of 30 in the second catch), and the answer is 250. When the division does not come out even, B4 shows a decimal, so read B5, which goes through the ROUND function, as the "about ..." answer.
The second table puts the answer 250 in for x in the proportion x : 50 = 30 : 6 and checks that the product of the means (B5) and the product of the extremes (B6) are both 1500, so they are equal.
The third table is the example of 3,000 products with 2 defective out of a sample of 100, and the answer is 60.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table for capture-recapture (estimating the total)
| Marked first m | 50 |
| 2nd catch n | 30 |
| Marked in 2nd catch k | 6 |
| Estimated total x = m×n÷k | =B1*B2/B3 |
| Rounded to a whole number | =ROUND(B4,0) |
Table to check the property of proportions (product of the means = product of the extremes)
| Extreme a (total; the answer to the pond example) | 250 |
| Mean b (number marked) | 50 |
| Mean c (2nd catch) | 30 |
| Extreme d (marked in 2nd catch) | 6 |
| Product of the means b×c | =B2*B3 |
| Product of the extremes a×d | =B1*B4 |
Table to estimate the matching count in a population
| Population size N | 3000 |
| Sample size n | 100 |
| Matching in sample k | 2 |
| Estimated count x = N×k÷n | =B1*B3/B2 |
| Rounded to a whole number | =ROUND(B4,0) |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the survey results with your own numbers.
How to calculate it in Python
from fractions import Fraction
# Capture-recapture: estimating the total number in the pond
marked_first = 50 # number caught first and marked, m
caught_second = 30 # number in the second catch, n
marked_in_second = 6 # number marked in the second catch, k
estimated_total = Fraction(marked_first * caught_second, marked_in_second)
# Adding Fraction(1, 2) and dropping the fraction part rounds 0.5 up every time
approx_total = int(estimated_total + Fraction(1, 2))
print(f"Estimated total (fraction): {estimated_total}")
print(f"Estimated total (rounded): about {approx_total}")
# Estimating the matching count in a population
population = 3000 # population size, N
sample_size = 100 # size of the sample taken, n
sample_hits = 2 # number matching in the sample, k
estimated_count = Fraction(population * sample_hits, sample_size)
approx_count = int(estimated_count + Fraction(1, 2))
print(f"Estimated count (fraction): {estimated_count}")
print(f"Estimated count (rounded): about {approx_count}")
With the fractions module from the standard library, you can calculate with exact fractions and no rounding error, even when the division does not come out even. The first half is the pond example (50 marked, 6 marked out of 30 in the second catch) and shows 250. The second half is the example of 3,000 products with 2 defective out of 100, and shows 60. Replace the survey results and run it.
How to write it in LaTeX and other math languages (copy and paste)
Capture-recapture (estimating the total)
x = m × n ÷ k
x = \dfrac{m \times n}{k}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow>
<mi>k</mi>
</mfrac>
</mrow>
</math>
x = (m*n)/k
(m*n)/k
x := m*n/k;
x = m*n/k;
x = (m×n)/k
Property of proportions (product of the means = product of the extremes)
a : b = c : d ⇔ a × d = b × c
a : b = c : d \iff ad = bc
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mo>:</mo><mi>b</mi>
<mo>=</mo>
<mi>c</mi><mo>:</mo><mi>d</mi>
<mo>⇔</mo>
<mi>a</mi><mi>d</mi>
<mo>=</mo>
<mi>b</mi><mi>c</mi>
</mrow>
</math>
a : b = c : d <=> a*d = b*c
a*d == b*c
a*d = b*c;
a*d == b*c
a : b = c : d ⇔ a×d = b×c
Estimating the matching count in a population
x = N × k ÷ n
x = \dfrac{N \times k}{n}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>N</mi><mo>×</mo><mi>k</mi></mrow>
<mi>n</mi>
</mfrac>
</mrow>
</math>
x = (N*k)/n
(N*k)/n
x := N*k/n;
x = N*k/n;
x = (N×k)/n
How to have ChatGPT do the calculation
You are a math calculation assistant (middle school sampling). Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Estimate the total number of fish in a pond with the capture-recapture method. First, 50 fish were caught, marked and put back in the pond. Later, 30 fish were caught, and 6 of them were marked. Show each of the following: 1. The proportion with x as the total (x : number marked = number in the 2nd catch : number marked in it) 2. The steps to solve it with "the product of the means equals the product of the extremes" 3. The estimated total in the pond (if the division does not come out even, give both the fraction in lowest terms and the number rounded to the nearest whole number) In Python, calculate exactly with the fractions module from the standard library, and show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
DataChef Features
Easy and Free
Unlimited conversions for free.
No technical knowledge required.
Intuitive and user-friendly operation.
No technical knowledge required.
Intuitive and user-friendly operation.
No Registration Required
Available immediately after access.
Can be used without registering personal information.
Can be used without registering personal information.
Safe and Secure
Fully SSL encrypted communication.
Automatic file deletion by clicking "download".
Automatic file deletion by clicking "download".
Fast
High-speed site access
and rapid file conversion.
and rapid file conversion.
No Watermark
No watermark.
No attribution required.
No attribution required.
Commercial Use Available
Free for commercial use.
No need to contact us for commercial use permission.
No need to contact us for commercial use permission.
Related Features
Sample Size Calculator (Surveys and Margin of Error)
Find the sample size you need to keep a survey's error within a limit. It is for planning a survey before you take the sample (this page estimates the population from a sample you already took)
Ratio and Proportion Calculator (Solve A:B = C:D, Scale a Ratio)
Solve for the unknown x in a proportion a : b = x : c (cross products) or simplify a ratio, step by step
Percentage Calculator (Percent, Part and Whole)
Work with the whole, the part and the percent - enter any two to find the third
Confidence Interval Calculator for a Population Mean
Estimate the population mean from a sample mean as a range (an interval estimate). It takes the idea of estimating the whole from a sample on to high school and college statistics
Ad 
