Statistics Flashcards: Fitting Linear Functions To Data

What this deck covers

This deck focuses on Fitting Linear Functions To Data, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Choose the correct statement: correlation $r$ is unitless or depends on measurement units?

Answer: $r$ is unitless. Correlation is a pure number, independent of measurement scales.

Flashcard 2: What does $r=-1$ mean for a scatterplot and linear fit?

Answer: Perfect negative linear association. All points lie exactly on a downward-sloping line.

Flashcard 3: What is the range of possible values for the correlation coefficient $r$?

Answer: $-1 \le r \le 1$. Correlation is bounded between perfect negative and perfect positive.

Flashcard 4: Identify the correlation coefficient if a linear fit reports $R^2=0.81$ and the slope is negative.

Answer: $r=-0.9$. Since $r^2=0.81$ and slope is negative, $r=-\sqrt{0.81}$.

Flashcard 5: Identify the correlation coefficient if a linear fit reports $R^2=0.36$ and the slope is positive.

Answer: $r=0.6$. Since $r^2=0.36$ and slope is positive, $r=+\sqrt{0.36}$.

Flashcard 6: Identify the stronger linear association: $r=-0.20$ or $r=0.05$.

Answer: $r=-0.20$. Compare absolute values: $|-0.20|=0.20 > |0.05|=0.05$.

Flashcard 7: What does a correlation of $r=-0.93$ indicate about the linear fit quality?

Answer: Very strong negative linear fit. Near -1 means points cluster tightly around downward line.

Flashcard 8: What does $r=1$ mean for a scatterplot and linear fit?

Answer: Perfect positive linear association. All points lie exactly on an upward-sloping line.

Flashcard 9: Find $|r|$ if the correlation coefficient is $r=-0.34$.

Answer: $|r|=0.34$. Absolute value removes the negative sign.

Flashcard 10: What does the sign of $r$ indicate about the direction of a linear relationship?

Answer: $r>0$ positive; $r<0$ negative. Positive $r$ means upward trend; negative $r$ means downward trend.

Flashcard 11: Which value indicates a stronger linear association: $|r|=0.2$ or $|r|=0.8$?

Answer: $|r|=0.8$. Closer to $|r|=1$ means stronger linear relationship.

Flashcard 12: What is a key limitation of using $r$ to describe a relationship?

Answer: $r$ measures only linear association. Cannot detect curved or other nonlinear patterns.

Flashcard 13: What does $r=-1$ mean about the scatterplot and linear fit?

Answer: Perfect negative linear association. All points lie exactly on a line with negative slope.

Flashcard 14: What does the sign of the correlation coefficient $r$ indicate about the association?

Answer: Positive: increasing; negative: decreasing. Positive $r$ means $y$ increases with $x$; negative means $y$ decreases.

Flashcard 15: What is the range of the correlation coefficient $r$ for any data set?

Answer: $-1 \le r \le 1$. Correlation is bounded between perfect negative and perfect positive.

Flashcard 16: What is the correlation coefficient $r$ if $R^2=0$ for a linear regression with an intercept?

Answer: $r=0$. If $R^2=0$, then $r^2=0$, so $r=0$.

Flashcard 17: What happens to $r$ if all $x$-values are multiplied by $-3$?

Answer: The sign of $r$ flips; magnitude unchanged. Negative scaling reverses direction but preserves strength.

Flashcard 18: Which statement is correct if $r=0.12$: the linear association is strong or weak?

Answer: Weak. Values close to 0 indicate weak linear relationship.

Flashcard 19: Which option is correct: for data with a positive slope, can $r$ ever be negative?

Answer: No, the sign of $r$ matches the slope. Correlation and slope always have the same sign in linear regression.

Flashcard 20: What does $r=0$ mean about linear association between two quantitative variables?

Answer: No linear association. Variables have no linear relationship (may still have nonlinear).

Flashcard 21: Identify the main limitation: does a large $|r|$ guarantee a cause-and-effect relationship?

Answer: No, correlation does not imply causation. High correlation shows association, not that one variable causes the other.

Flashcard 22: Which value shows a stronger linear association: $r=-0.81$ or $r=0.63$?

Answer: $r=-0.81$. Compare absolute values: $|-0.81|=0.81 > |0.63|=0.63$.

Flashcard 23: What happens to $r$ if all $y$-values are multiplied by a positive constant $k$ (use $y'=ky$, $k>0$)?

Answer: $r$ is unchanged. Scaling by positive constant preserves pattern and correlation.

Flashcard 24: What does a correlation of $r=0.97$ indicate about the linear fit quality?

Answer: Very strong positive linear fit. Near 1 means points cluster tightly around upward line.

Flashcard 25: State the formula that relates $r$ and the coefficient of determination $r^2$.

Answer: $r^2=(r)^2$. Square the correlation to get coefficient of determination.

Flashcard 26: Identify what an outlier typically does to correlation $r$ in a scatterplot with an otherwise linear trend.

Answer: It can greatly change $r$. Outliers pull the line toward them, affecting correlation strength.

Flashcard 27: State the relationship between $r$ and $R^2$ for a linear regression with an intercept.

Answer: $R^2=r^2$. Coefficient of determination equals correlation squared.

Flashcard 28: What does $r=0$ mean about linear association in the data?

Answer: No linear association. Variables have no linear relationship, though other patterns may exist.

Flashcard 29: Which value shows a stronger linear association: $r=0.92$ or $r=0.48$?

Answer: $r=0.92$. Compare absolute values: $|0.92|=0.92 > |0.48|=0.48$.

Flashcard 30: Which statement is correct if $r=-0.76$: the association is positive or negative?

Answer: Negative. Negative $r$ indicates downward linear trend.

Flashcard 31: Find $r^2$ if the correlation coefficient is $r=-0.8$.

Answer: $r^2=0.64$. Square the correlation: $(-0.8)^2 = 0.64$.

Flashcard 32: What happens to $r$ if all $x$-values are shifted by a constant $c$ (use $x'=x+c$)?

Answer: $r$ is unchanged. Shifting doesn't change the pattern or strength of association.

Flashcard 33: What is the symbol for the correlation coefficient of a linear fit?

Answer: $r$. Standard notation for Pearson's correlation coefficient.

Flashcard 34: What does $r=1$ mean about the scatterplot and linear fit?

Answer: Perfect positive linear association. All points lie exactly on a line with positive slope.

Flashcard 35: Find the correct interpretation: if $r=-0.85$, is the linear association strong or weak, and positive or negative?

Answer: Strong negative linear association. $|r|=0.85$ is close to 1, and negative sign shows decreasing trend.

Flashcard 36: Which situation can make $|r|$ misleadingly small even if a strong pattern exists: linear or curved trend?

Answer: Curved (nonlinear) trend. Correlation only measures linear relationships, not curves.

Flashcard 37: What happens to $r$ if all $y$-values are multiplied by a negative constant $k$ (use $y'=ky$, $k<0$)?

Answer: $r$ changes sign. Negative scaling reverses the direction of association.

Flashcard 38: Find the correct interpretation: if $r=0.12$, is the linear association strong or weak, and positive or negative?

Answer: Weak positive linear association. $|r|=0.12$ is close to 0, and positive sign shows increasing trend.

Flashcard 39: What is a common effect of an outlier on the correlation coefficient $r$?

Answer: It can greatly increase or decrease $r$. Outliers pull the line toward them, changing $r$ substantially.

Flashcard 40: What happens to $r$ if a constant $5$ is added to every $y$-value?

Answer: $r$ is unchanged. Adding constants doesn't affect correlation strength or direction.