IB Maths AI HL: Descriptive Statistics and Bivariate Data

Vocabulary flashcards detailing key terminology, sampling techniques, measures of central tendency, spread, outlier rules, and bivariate analysis for IB Maths AI HL.

Qualitative Data

Data expressed as words or categories (such as eye colour or favourite sport) rather than numerical values.

Quantitative Data

Numerical data, which is classified into two distinct types: discrete and continuous.

Discrete Data

Quantitative data consisting of specific distinct numerical values that are counted (such as number of siblings, test scores, or shoe size).

Continuous Data

Quantitative data that can take any value within a given range and is obtained by measuring with tools such as a ruler, scale, or stopwatch.

Population

All members of a defined group (for example, every student in an entire school).

Sample

A subset of the population that is actually surveyed or measured.

Biased Sample

A non-random sample where not every member of the population has an equal chance of being selected, which may result in a non-representative sample and misleading conclusions.

Simple Random Sampling

A probability sampling method where every member of the population has an equal chance of being chosen, typically by numbering each member and selecting them via a random number generator.

Systematic Sampling

A sampling method in which every $k^{\text{th}}$ member is chosen from a list after a random starting point, where the interval is $k = \frac{N}{n}$.

Stratified Sampling

A random sampling method where the population is divided into subgroups and a random sample is chosen from each subgroup in the exact proportion it appears in the population.

Quota Sampling

A non-random sampling method where the researcher decides the target quantity needed from each subgroup and selects individuals until each quota is filled.

Convenience Sampling

A non-random sampling method that involves selecting whichever individuals are easiest to reach until a sufficient sample size is gathered.

Mean ($\bar{x}$)

The measure of central tendency found by adding all data values and dividing by the total count: $\bar{x} = \frac{\sum x}{n}$, or $\bar{x} = \frac{\sum fx}{\sum f}$ for a frequency table.

Median

The middle value in an ordered data set, located at the $\left(\frac{n+1}{2}\right)^{\text{th}}$ position.

Mode

The data value that occurs with the highest frequency in a data set.

Range

A measure of spread calculated as the difference between the maximum and minimum values in a data set: $\text{Range} = \text{maximum} - \text{minimum}$.

Interquartile Range (IQR)

The measure of dispersion representing the spread of the middle $50\%$ of data, calculated as $\text{IQR} = Q_3 - Q_1$.

Standard Deviation ($\sigma$)

A measure of spread roughly representing the typical distance of data values from the mean, calculated as $\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$.

Variance ($\sigma^2$)

The square of the standard deviation ($\sigma^2$), providing a measure of the spread of values around the mean.

Outlier

An extreme value falling beyond the boundaries defined by $x < Q_1 - 1.5 \times \text{IQR}$ or $x > Q_3 + 1.5 \times \text{IQR}$.

Linear Coding Rules ($+k$ and $\times k$)

Adding $k$ translates the mean ($\bar{x} \pm k$) but leaves standard deviation unchanged; multiplying by $k$ scales the mean ($k\bar{x}$) and scales the standard deviation by $|k|\sigma$.

Modal Class

The class interval that contains the highest frequency in a grouped frequency distribution.

Five-Number Summary

The five summary values used to construct a box-and-whisker plot: minimum, lower quartile ($Q_1$), median, upper quartile ($Q_3$), and maximum.

Bivariate Data

Data involving pairs of measurements on two variables for each item, typically plotted on a scatter diagram.

Pearson's Correlation Coefficient ($r$)

A statistical value ranging between $-1$ and $1$ that measures the strength and direction of a linear relationship between two quantitative variables.

Interpolation

Making an estimate from a regression line within the range of the given data values, which is generally reliable.

Extrapolation

Using a regression line to predict values outside the range of observed data, which is unreliable.