Population Size

Lessons

Senior 3 · Biology

Population Size

5 lessons

Lesson 1 of 5

Population, Population Density and Quadrat Sampling

25 min

Counting a country, and counting a field

In August 2022 the National Institute of Statistics of Rwanda (NISR) counted every person in the country in the fifth Population and Housing Census. The count was 13,246,394 people. Enumerators visited every household over two weeks, because a census tries to count everyone.

Ecologists who want to know how many grass plants are in a field, or how many beetles live under a forest's dead leaves, cannot visit every individual. They count a small, fair sample and use it to estimate the whole population. This unit shows how, and then asks what makes populations grow and stop growing.

Definition

Population: all the organisms of one species living in one area at the same time. Example: all the impalas in Akagera National Park in 2026.

Population size: the number of individuals in the population, written $N$ .

Population density: the number of individuals per unit area (or per unit volume, for water): $\text{population density} = \frac{\text{number of individuals}}{\text{area}}$ Units: individuals per m², per hectare (ha) or per km². $1\ \text{ha} = 10\,000\ \text{m}^2$ and $1\ \text{km}^2 = 100\ \text{ha}$ .

Quadrat: a square frame of known area, often $1\ \text{m} \times 1\ \text{m}$ or $0.5\ \text{m} \times 0.5\ \text{m}$ , placed on the ground to count the plants or slow-moving animals inside it.

Random sampling: choosing sample positions by chance, so that every part of the area has the same chance of being sampled and the person sampling cannot bias the result.

Worked example 1: population density from the census

NISR reported for 2022: Rwanda, 13,246,394 people and a density of 503 people per km²; Kicukiro District, 491,731 people and a density of 2,944 people per km². (a) Estimate the area of Rwanda. (b) Estimate the area of Kicukiro. (c) How many times more crowded is Kicukiro than the national average?

Rearrange the formula: $\text{area} = \frac{\text{number of individuals}}{\text{density}}$ .
(a) $\frac{13\,246\,394}{503} = 26\,335\ \text{km}^2$ , about $26\,300\ \text{km}^2$ .
(b) $\frac{491\,731}{2944} = 167\ \text{km}^2$ .
(c) $\frac{2944}{503} = 5.9$ : Kicukiro is almost six times as densely populated as Rwanda as a whole.

Note: the population is the same whatever the area, but the density depends on the area chosen. Always state the area and its unit.

Sampling plants with quadrats

A quadrat works for plants and for animals that move slowly or not at all, such as snails and limpets. The method:

Measure the study area and mark two edges as axes, for example a field 50 m by 40 m.
Use random numbers (from a calculator, a phone or numbered papers drawn from a bag) to choose the coordinates of each quadrat.
Place the quadrat at each position and count the individuals of the species inside it.
Sample enough quadrats, usually at least 10, until the mean count stops changing much.
Calculate the mean number per quadrat, the density, and the estimated population size.

Figure 1: Ten quadrats placed at random coordinates in a 50 m × 40 m field (quadrats drawn larger than scale), and the counting rule for one quadrat.

Counting rule for plants on the edge: count a plant that touches the top or the left side, and do not count one that touches the bottom or the right side. Then no plant is counted twice or missed on average.

Worked example 2: estimating a weed population

Students placed ten 1 m² quadrats at random in the 50 m × 40 m field of Figure 1 and counted a weed, blackjack (Bidens pilosa). Their counts were: 3, 0, 5, 2, 4, 1, 0, 6, 3, 2. (Illustrative data.) Estimate (a) the density and (b) the population of blackjack in the field.

Total counted: $3 + 0 + 5 + 2 + 4 + 1 + 0 + 6 + 3 + 2 = 26$ .
Mean per quadrat: $\bar{x} = \frac{26}{10} = 2.6$ plants per quadrat.
(a) Density: each quadrat is $1\ \text{m}^2$ , so the density is $\frac{2.6}{1\ \text{m}^2} = 2.6$ plants per m².
Area of the field: $50\ \text{m} \times 40\ \text{m} = 2000\ \text{m}^2$ .
(b) Population: $N = \text{density} \times \text{area} = 2.6 \times 2000 = 5200$

Answer: about 2.6 plants per m², and about 5,200 plants in the field. It is an estimate: a different set of random quadrats would give a slightly different answer.

Worked example 3: a smaller quadrat

In a school garden, a $0.5\ \text{m} \times 0.5\ \text{m}$ quadrat was used. The mean count of snails was 1.5 per quadrat. Calculate the density per m².

Area of one quadrat: $0.5 \times 0.5 = 0.25\ \text{m}^2$ .
Density: $\frac{1.5}{0.25} = 6$ snails per m².

Check: four quadrats of 0.25 m² fit in 1 m², and $4 \times 1.5 = 6$ .

Warning

Common mistakes. (1) Throwing the quadrat or placing it where the plants look interesting: that biases the sample. Use random coordinates. (2) Forgetting to divide by the quadrat area when the quadrat is not 1 m². (3) Using quadrats for animals that run or fly away: use capture–recapture instead (Lesson 2). Safety: wear shoes, watch for stinging plants and insects, and wash your hands after fieldwork.

Sampling with Quadrats: GCSE Biology Required Practical (Malmesbury Education). Watch how random coordinates are chosen and how the mean count is scaled up to the whole area.

Tip

Exam technique: the three-line estimate. (1) mean per quadrat = total ÷ number of quadrats; (2) density = mean ÷ area of one quadrat; (3) population = density × total area. Write the units at every step.

✦

Practice: population density and quadrats

Define population and population density.
Why is a quadrat not suitable for estimating the number of rats on a farm?
The City of Kigali had 1,745,555 people in 2022 and a density of 2,401 people per km². Estimate its area.
Eight 1 m² quadrats gave these counts of a grass species: 12, 9, 15, 10, 8, 14, 11, 9. Calculate the mean per quadrat.
Using your answer to question 4, estimate the number of plants in a 25 m × 20 m plot.
A 0.25 m² quadrat gave a mean of 3 plants. Calculate the density per m².
Explain why a student should use at least ten quadrats, not just two.
A student placed all her quadrats in the greenest part of a field. Explain how this would affect her estimate.

Show answersHide answers

A population is all the organisms of one species in one area at the same time. Population density is the number of individuals per unit area.
Rats move about and run away; a quadrat only works for organisms that stay still long enough to be counted, such as plants.
$\frac{1\,745\,555}{2401} = 727$ km².
Total $= 88$ ; mean $= \frac{88}{8} = 11$ plants per quadrat.
Area $= 25 \times 20 = 500$ m²; $N = 11 \times 500 = 5500$ plants.
$\frac{3}{0.25} = 12$ plants per m².
Plants are not spread evenly. Two quadrats might both land on a bare or a crowded patch; more quadrats give a mean that is closer to the true mean.
It would be too high (an overestimate), because the greenest part probably has more plants than the field as a whole; the sample was biased, not random.

Key points

A population is one species in one area at one time; density = number ÷ area.
Quadrats placed at random estimate plant populations: mean per quadrat → density → population.
Random placement and enough quadrats reduce bias and chance error.

Sources and credits

19.1: Population Demographics and Dynamics, from Concepts of Biology by Samantha Fowler, Rebecca Roush and James Wise (OpenStax), via LibreTexts Biology, licensed under CC BY 4.0. Adapted by Ganzaa. (section "Population Size and Density")
Census figures: National Institute of Statistics of Rwanda, Fifth Rwanda Population and Housing Census, 2022: Main Indicators Report, Table 1.
Photo: 08Biocrust (6517406791), SonoranDesertNPS, CC BY 2.0, via Wikimedia Commons.
Video: Sampling with Quadrats - GCSE Biology Required Practical, Malmesbury Education, embedded from YouTube.

Written by Ganzaa: Figure 1, the worked examples (the weed and snail data are illustrative) and the practice questions.

1Counting a country, and counting a field

2Sampling plants with quadrats

Counting a country, and counting a field

Sampling plants with quadrats