Bivariate Data and Scatter Plots

The independent variable is the one that is controlled or naturally varies; the dependent variable responds to it. Time naturally progresses (independent) and the coffee's temperature responds (dependent). (Temperature is the dependent variable; the cup and the scientist are not measured variables.)

Correlation measures how closely the points follow a straight-line relationship, and that does not depend on which variable is plotted on which axis. Swapping the axes reflects the plot but keeps the points equally tight and equally upward-trending, so it stays strong and positive.

Each point is one observation with coordinates $(x, y) = (\text{hours}, \text{score})$. So $(5, 80)$ means a student who studied $5$ hours scored $80\%$. (Swapping gives "80 hours, score 5%"; counting students misreads the point.)

Positive correlation: as one variable increases, so does the other. Taller children generally have bigger feet, so height and shoe size rise together. (TV vs scores and temperature vs jumpers are negative; speed vs time is negative; day vs rainfall shows none.)

Substitute $y = 45$ and solve for $x$: $45 = 2.5x + 10$, so $2.5x = 35$ and $x = 14$. (Reading the line backwards turns a prediction into an equation to solve, linking bivariate data with algebra.)

No correlation means there is no apparent trend: the x-value tells you nothing about the y-value. (Upward and downward trends are positive and negative; a straight line going up is strong positive; a curve is a non-linear relationship.)

Because the mean point lies on the line, substitute $\bar{x} = 8$: $\bar{y} = 3(8) + 5 = 29$.

Use $y = mx + c$ with $m = 4$ and the point $(6, 30)$: $30 = 4(6) + c$, so $c = 30 - 24 = 6$. (Finding a line from a gradient and a point is coordinate geometry applied to data.)

$R(5) = 8(5) + 40 = 80$ and $R(10) = 8(10) + 40 = 120$. The increase is $120 - 80 = 40$, a percentage rise of $\dfrac{40}{80} = 50\%$. (Combines reading a linear model with percentage change.)

An outlier off the trend line spreads the points away from a straight line, weakening the correlation. Removing it tightens the remaining points around the line, so the correlation becomes stronger. (It cannot flip to negative, since the underlying trend is still increasing.)

The line of best fit passes through the middle of the data so that roughly half the points lie above and half below; it need not pass through any particular point or the origin. (Passing through every point is only possible if the data is exactly collinear.)

Gradient $= \dfrac{19 - 7}{5 - 1} = \dfrac{12}{4} = 3$. Substituting $(1, 7)$: $7 = 3(1) + c$, so $c = 4$ and $y = 3x + 4$. ($y = 3x + 7$ used $7$ as the intercept instead of solving for $c$.)

$y = 1.5(20) + 4 = 34$. Since $x = 20$ lies beyond the observed range $0$ to $10$, this is extrapolation, which is less reliable because the trend may not continue.

Gradient $= \dfrac{21 - 9}{6 - 2} = 3$, and using $(2, 9)$: $9 = 3(2) + c$ gives $c = 3$, so $y = 3x + 3$. Setting $y = 30$: $30 = 3x + 3$, so $x = 9$. (Build the equation from two points, then solve it backwards for $x$.)

The gradient $4$ is the rate of change: each extra hour adds $4$ marks, so $3$ extra hours add $4 \times 3 = 12$ marks. ($42$ computed $y$ at $x = 3$ instead of the change; the intercept $30$ is not involved in a change.)

Substitute $\text{temp} = 25$: $\text{sales} = 8(25) - 20 = 200 - 20 = 180$. Since $25^\circ\text{C}$ lies inside the observed range, this is reliable interpolation. (200 forgot to subtract the $20$.)

Set the two models equal: $2x + 10 = 5x + 1$, so $9 = 3x$ and $x = 3$. (Finding where two predictions agree is a pair of simultaneous equations.)

The y-intercept is the value of the dependent variable when the independent variable is $0$. At $\text{days} = 0$, $\text{height} = 0.5(0) + 2 = 2$ cm, the initial height. (Daily growth is the gradient $0.5$, not the intercept.)

The residual is the actual $y$ minus the predicted $y$: $39 - 45 = -6$. The negative sign means the point lies $6$ units below the line. (A residual measures how far a point sits from the model.)

This is a "diminishing returns" pattern: a steep rise that flattens out, which is a curve rather than a straight line. (A straight line up would mean growth continues forever; a scattered cloud shows no pattern.)

The symbol $\sum$ ("sigma") means "the sum of", so $\sum x = 60$ is the total of all six $x$-values. Each mean is the sum divided by the number of points, giving the mean point $\left(\dfrac{60}{6}, \dfrac{180}{6}\right) = (10, 30)$. Substituting into $y = 2x + c$: $30 = 2(10) + c$, so $c = 10$. (Compute the means from the sums, then apply coordinate geometry.)

Gradient $= \dfrac{124 - 100}{9 - 5} = \dfrac{24}{4} = 6$ cm/year. Using $(5, 100)$: $100 = 6(5) + c$, so $c = 70$ and $y = 6x + 70$. At age $12$: $y = 6(12) + 70 = 142$ cm. ($124$ just repeated a data value.)

This is the classic correlation-versus-causation puzzle: both variables changed over time (piracy declined for social reasons; temperature rose with industrialisation), but neither causes the other. Always ask whether a hidden third variable, such as time, explains both. (The correlation is real and the data is not faulty; the error would be inferring causation.)

A scatter plot with a line of best fit is the standard tool for the relationship between two numerical variables. (Histograms and box plots describe a single variable; bar and pie charts are for categories.)

An outlier should be acknowledged and investigated, but the line of best fit should follow the main pattern of the bulk of the data rather than be dragged toward a single point. (Letting it determine or pass through the line skews the fit; ignoring it silently is poor practice.)