Terminology (Leaving Cert Mathematics): Revision Notes

Statistics Terminology

Understanding populations and samples

In statistics, we need to distinguish between the group we want to learn about and the group we actually study.

Population refers to the complete group of individuals or items that we want to investigate. This could be all Leaving Cert students in Ireland, all cars manufactured in 2023, or all trees in a particular forest. The population represents everyone or everything we're interested in understanding.

Sample means a smaller subset taken from the population for practical study. Since we usually cannot examine every member of a population (imagine trying to survey every Leaving Cert student!), we select a manageable group to represent the whole.

A representative sample accurately reflects the characteristics of the entire population. This means the sample should have similar proportions of different groups as found in the population.

Types of data

Understanding how to classify data helps us choose appropriate analysis methods. Data falls into two main categories, each with important subcategories.

Categorical data consists of information sorted into distinct categories or groups.

• Nominal data has no natural ordering between categories. Examples include colours, gender, or favourite subjects. You cannot rank these meaningfully.
• Ordinal data has categories with a natural order or ranking. Examples include satisfaction ratings (poor, fair, good, excellent) or finishing positions in a race.

Numerical data represents measured or counted quantities.

• Discrete data consists of countable values, often whole numbers. Examples include number of students in a class, goals scored in a match, or books owned.
• Continuous data can take any value within a range and is typically measured rather than counted. Examples include height, weight, time, or temperature.

Sampling methods and errors

Different methods exist for selecting samples, including simple random, systematic, stratified, and quota sampling. Each method has specific advantages and applications depending on the research context.

Sampling error represents the natural difference between what we find in our sample and the true population value. This occurs simply because we're studying part of the population, not all of it.

Non-sampling error includes mistakes unrelated to sample selection, such as measurement errors, recording mistakes, or response bias.

Measures of central tendency

These statistics help us identify the "typical" or "average" value in our data. Each measure has specific uses and limitations.

Mean calculates the arithmetic average by adding all values and dividing by the number of observations. The mean can be significantly affected by extreme values (outliers), which may make it less representative of typical values.

Median identifies the middle value when data is arranged in order. With an even number of values, we take the average of the two middle numbers. The median remains stable even when extreme values are present.

Mode represents the most frequently occurring value in the dataset. A dataset might have no mode, one mode, or multiple modes.

Measures of spread

These statistics describe how much the data varies around the central value, giving us insight into data consistency and reliability.

Range provides the simplest measure of spread by calculating the difference between the maximum and minimum values: $Range = Max - Min$.

Interquartile range (IQR) measures the spread of the middle 50% of data by finding the difference between the third quartile and first quartile: $IQR = Q\_3 - Q\_1$.

Variance calculates the average of the squared deviations from the mean, providing a measure of overall variability:

$\text{Variance} = \frac{\sum(x\_i - \bar{x})^2}{n}$

Standard deviation represents the typical distance of data points from the mean. It's calculated as the square root of variance and uses the same units as the original data:

$\text{Standard Deviation} = \sqrt{\text{Variance}}$

Generalisability and data presentation

Generalisability refers to our ability to apply sample findings to the broader population. This is only valid when our sample is representative and bias is minimised.

Probability concepts

Basic probability terminology forms the foundation for understanding statistical inference and decision-making under uncertainty.

Events are specific outcomes we're interested in, while outcomes are possible results. The sample space contains all possible outcomes.

Mutually exclusive events cannot occur simultaneously, while independent events mean one outcome doesn't affect another.

Correlation and regression

Correlation describes the relationship between two variables, which can be positive (both increase together), negative (one increases as the other decreases), or show no relationship.

The correlation coefficient (r) quantifies this relationship on a scale from -1 to +1, where $r = -1$ indicates perfect negative correlation, $r = 0$ indicates no linear relationship, and $r = +1$ indicates perfect positive correlation.

Line of best fit represents the straight line that best describes the relationship between two variables, minimising the sum of squared residuals.

Common statistical misuse

Summary