Sampling & Data Collection (Edexcel A-Level Mathematics): Revision Notes

1.1.1 Sampling & Data Collection

Data and Sampling

Types of Data

• Data can be divided into two main types: qualitative and quantitative.
  • Qualitative data is non-numerical.
  • Example: Somebody's written opinion about the state of the roads.
  • Quantitative data is numerical.
  • Example: The number of different colour cars that pass a certain point.

Discrete, Continuous, and Categorical Data

• Quantitative data can be divided into three types:
  • Continuous data:
  • Numerical in nature.
  • Can take an infinite number of values.
  • Example: Heights of trees. This must be a scale with no gaps.
  • Discrete data:
  • Data that can only take certain values.
  • These values are countable.
  • Example: Number of goals scored in a game, height of trees to the nearest metre.
  • Categoric data:
  • Data that describes but the variables are non-numeric.
  • Example: Apples - $12$, Bananas - $6$, Carrots - $4$.

Populations and Samples

• A population is the entire set of objects from which we can draw a sample.
• A sample is a subset of the population.
• When a sample is the same size as the population, it becomes a census.

Advantages of Taking a Census

• You get data about the characteristics of the entire population.

Disadvantages of Taking a Census

• Costly
• Time-consuming Taking a sample mitigates these disadvantages provided the sample is representative of the population. A sample that does not represent the population is referred to as biased.

Sampling Techniques

Simple Random Sampling

• This method ensures that each member of the population has an equal chance of being chosen.
• A possible method for choosing such a sample:
  1. Assign each member of the population an integer $1, 2, \ldots, n$, where $n$ is the size of the population.
  2. Determine the sample size $s$.
  3. Using a random number generator, generate integers, and if an integer chosen corresponds to a number assigned to a member of the population, choose them to be part of the sample. Ignore repeated numbers and numbers out of range.
  4. Stop when we have a sample of $s$ objects.

Stratified Sampling

• This is a sampling technique in which each subcategory of the population is proportionally represented within the sample.

• Once the size of subcategories is chosen for the sample, use simple random sampling on each subcategory.

Systematic Sampling

• This involves coming up with a rule to apply to the population to generate your sample.
  • Example: Choosing every $10$th person from a list of the entire population. All of the above sampling methods assume that participants are willing. This is rarely the case. In such circumstances, where willing participants are scarce, the following methods could be used:

Opportunity Sampling

• Ask the first people you see until you have enough data.

Quota Sampling

• Have an idea of how many of each subcategory of the population are to be surveyed (e.g., like in stratified sampling), then use opportunity sampling to fill these quotas.

Cluster Sampling

• Choose a cluster of people from a population, then ask them all.