PY2RMP 2) Getting to Know Data - Distributions, Transformations, and Relationships

Thirty practice vocabulary flashcards covering key statistical concepts from Week 2 lecture notes, including distributions, skewness, spurious associations, Simpson's paradox, measurement effects, linear transformations, z-scores, and non-linear transformations.

Looking at Data

The exploratory phase of analysis aimed at checking assumptions, understanding what is occurring within the dataset, and determining how to proceed.

Skewness

A statistical metric used to evaluate the symmetry of a distribution; a value of 0 indicates that the distribution is symmetrical or normal.

Symmetrical Distribution

A distribution where skewness equals 0, meaning scores are balanced equally on both sides of the center.

Negative Skew

An asymmetrical distribution where the data clusters to the right, leaving a longer tail pointing toward the lower values on the left.

Positive Skew

An asymmetrical distribution where the data clusters to the left, leaving a longer tail pointing toward the higher values on the right.

68.3% Rule (Normal Distribution)

The percentage of observations within a normal distribution that lie within one standard deviation of the mean, spanning from $\mu - 1\sigma$ to $\mu + 1\sigma$.

95.5% Rule (Normal Distribution)

The percentage of observations within a normal distribution that fall within two standard deviations of the mean, spanning from $\mu - 2\sigma$ to $\mu + 2\sigma$.

99.7% Rule (Normal Distribution)

The percentage of observations within a normal distribution that fall within three standard deviations of the mean, spanning from $\mu - 3\sigma$ to $\mu + 3\sigma$.

Spurious Relationship

An apparent association between two variables that vanishes once a relevant grouping variable (such as grade level) is taken into account.

Simpson's Paradox

A phenomenon wherein an observed association between two variables either vanishes or completely reverses direction when a grouping variable is applied.

Ceiling Effect

A condition in which scores cluster at the maximum possible value, indicating that a test is too easy or that responses reflect a desirability bias.

Floor Effect

A condition in which scores cluster at the minimum possible value, indicating that a test is too difficult or that responses reflect a desirability bias.

Desirability Bias

A response bias in attitude measures where participants avoid reporting negative feelings toward a topic, thereby producing ceiling or floor effects.

Variance Impact of Ceiling and Floor Effects

A reduction in the variance of a variable caused by data points bunching up at either the extreme upper or lower boundary of a measure.

Mean Distortion from Ceiling Effects

The artificial upward shift in the sample mean, causing it to be higher than it should be if the measurement were completely unbiased.

Linear Transformation

An operation on data expressed generally by $X' = aX + b$, which rescales or shifts values without changing the underlying distribution shape.

Adding a Constant to All Scores

A linear transformation ($X' = X + b$) in which the new mean becomes $M + b$, while the standard deviation ($SD$) remains unchanged.

Multiplying All Scores by a Constant

A linear transformation ($X' = aX$) in which the new mean becomes $M \times a$ and the standard deviation ($SD$) becomes $S \times a$.

Standardisation

A linear transformation process converting raw scores into standard units ($z$-scores); it shifts and rescales the data but cannot alter a non-normal distribution into a normal one.

z-Score (Standard Score)

A metric indicating how far and in what direction a data point lies from the mean of a distribution, measured in standard deviation units.

z-Score Formula

The mathematical formula $z = \frac{(X - \bar{X})}{S}$, where $X$ is the original score, $\bar{X}$ is the mean of the original scores, and $S$ is the standard deviation.

Positive z-Score

A standard score greater than zero, signifying that an individual's score falls above the group average.

Negative z-Score

A standard score less than zero, signifying that an individual's score lies below the group average (e.g., a score of $-1.5$ is $1.5$ standard deviations below the mean).

Score Comparison via z-Scores

The analytical advantage of standard scores that allows researchers to directly compare values from distributions possessing different means and standard deviations.

Non-linear Transformation

Mathematical operations applied to variables to deliberately change the shape of the data distribution, which linear transformations cannot do.

Square Root Transformation

A non-linear transformation expressed as $X' = \sqrt{X}$, utilized to alter distribution shapes and reduce skewness.

Inverse (Reciprocal) Transformation

A non-linear transformation expressed as $X' = \frac{1}{X}$, used to change the shape and properties of a distribution.

Power Transformation

A non-linear transformation calculated as $X' = X^2$, employed to modify the distribution shape of a dataset.

Log Transformation

A non-linear transformation defined as $X' = \log_{10}(X)$, commonly used to compress extreme values, reduce positive skew, and reshape data.

Objectives of Non-linear Transformations

The practical goals for applying non-linear adjustments to data: making data easier to understand, reducing skew, minimising outliers, and making relationships more linear.