Sampling and Simple Probability

Why scientists use samples

Counting every plant in a large field, or testing every object in a large batch, may take too long or be impractical. Scientists therefore investigate a sample: a smaller selection from the whole population. Here, population means the entire group being investigated, not necessarily people or organisms.

The sample provides an estimate rather than an exact measurement of the whole population. For that estimate to be useful, the sample should be representative: it should reflect the variation within the population. Sampling only the most convenient places can give a misleading picture.

Random sampling reduces selection bias

Suppose you want to estimate the number of plants in a field. Plants may grow in patches, so choosing only places where many plants are visible would overestimate their abundance. Choosing only bare patches would underestimate it. This systematic favouring of particular locations is sampling bias.

Random sampling removes personal choice from the selection of locations. A standard method uses a quadrat, a frame enclosing a known area:

1. Mark out the study area using measuring tapes and a coordinate grid.
2. Use a random number generator to choose coordinate pairs within the area.
3. Place the quadrat at each selected position using the same positioning rule each time.
4. Count the plants of the chosen species inside the quadrat and record the results.
5. Repeat at many randomly selected positions, keeping the quadrat area and counting method the same.

Check for uneven ground and other hazards before sampling, avoid damaging organisms unnecessarily, and wash your hands after handling soil or plants. Sampling is complete when the planned set of locations has been counted and recorded.

A larger, well-selected sample usually gives a more reliable estimate because an unusually crowded or empty quadrat has less influence on the overall result. However, taking more samples from the same biased locations does not make the method representative.

Scaling a sample up to the whole area

First calculate the mean number of plants per unit area:

$$\text{mean number per m}^2 = \frac{\text{total number counted}}{\text{total sampled area in m}^2}$$

Then scale this density up:

$$\text{estimated population} = \text{mean number per m}^2 \times \text{total area in m}^2$$

For example, five quadrats, each with an area of $0.5\text{ m}^2$, contain 20, 25, 47, 12 and 5 daffodils. The total count is 109 and the total sampled area is $2.5\text{ m}^2$. The density is therefore $109÷2.5=$ daffodils per square metre. In a field, this gives an estimated of , or about 2200 daffodils.

The estimate assumes the sampled areas represent the field as a whole. It does not mean that every square metre contains the same number of plants.

Sampling to investigate a change across an area

Not every investigation needs randomly scattered locations. To study how plant abundance changes with distance from a tree, quadrats can be placed at regular intervals along a measuring tape. This is systematic sampling using a belt transect.

Record the distance and plant count at each position. Measuring light intensity at the same positions can help investigate whether the distribution is associated with changing light conditions. Random sampling is useful for estimating abundance across an area; a transect is useful for investigating a pattern along a route.

Probability describes chance

Probability is the likelihood of an event occurring. An outcome is a possible result; an event is the result, or group of results, you are interested in.

Probability runs from 0 to 1. A probability of 0 means impossible, 1 means certain, and $0.5$ means an even chance. Values between 0 and $0.5$ describe unlikely events; values between $0.5$ and 1 describe likely events. Probabilities have no units and can be written as fractions, decimals or percentages.

When individual outcomes are equally likely:

$$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$$

A fair six-sided die has six equally likely outcomes. Only one is a 2, so $P(2)=\frac{1}{6}$. The outcomes greater than 2 are 3, 4, 5 and 6, giving a probability of $\frac{4}{6}=\frac{2}{3}$.

An event either happens or does not happen, so these probabilities add to 1:

$$P(\text{not happening}) = 1-P(\text{happening})$$

More generally, probabilities add to 1 for a complete set of outcomes that cannot occur together in the same trial.

Using repeated trials to estimate probability

In scientific investigations, probability may be estimated from observed results rather than calculated from equally likely outcomes. The relative frequency is:

$$\text{relative frequency} = \frac{\text{number of times the event occurred}}{\text{total number of trials}}$$

For example, if a coin lands on heads 325 times in 500 flips, the relative frequency of heads is $325 \div 500=0.65$. This provides an estimate of its probability of landing on heads. More trials generally make a probability estimate less affected by chance variation, provided the conditions remain comparable.

To predict how often an event will occur, multiply its probability by the number of trials. For a fair die rolled 24 times, the expected number of 2s is $24 \times \frac{1}{6}=4$. The actual number may differ: probability predicts patterns over repeated trials, not the result of each individual trial.

Sampling

• Use a representative sample to estimate properties of a whole population.
• Random coordinates reduce selection bias; many samples reduce the influence of chance variation.
• Keep quadrat area and counting rules consistent.