Time Plot vs Scatterplot

A time plot puts time on the horizontal axis and the measured quantity on the vertical, one point per period, usually joined in order. The connecting segments are a drawing convention rather than data: the line between January and February does not claim that any value was observed between them. You read the plot for trend, for repeating seasonal patterns, and for sudden breaks.

Monthly rainfall from January through April is 2.1, 2.4, 3.0 and 3.6 inches. The rise averages $(3.6 - 2.1) / 3 = 0.5$ inch per month. The mean of the four values is 2.775 inches and the sample standard deviation is 0.665. Shuffle the four points into any other order and both of those summaries are unchanged while the trend disappears, which is exactly the information a time plot carries and a histogram of the same four numbers does not.

The misreading is "rainfall is rising 0.5 inch per month, so December should be near $2.1 + 11(0.5) = 7.6$ inches." Two errors in one sentence. The 0.5 is an average over three observed gaps, not a mechanism, and rainfall is seasonal, so whatever produced the rise will turn over later in the year. Running a short time plot past its own range is extrapolation, and it is where time plots do the most damage.

What the display itself gets wrong is scale. The eye reads slope, and slope depends on the aspect ratio and on the range of the vertical axis. Those same four points look nearly flat on an axis running 0 to 20 inches and steep on one running 2 to 4. A squeezed time axis or a cropped vertical axis can manufacture or erase a trend without a single number changing, so check the axes before describing the pattern.

A time series is one unit measured repeatedly, not a random sample of separate individuals, so treat its mean and standard deviation as descriptions of the observed stretch rather than estimates of anything wider.

A scatterplot shows one point $(x, y)$ per individual, with the explanatory variable on the horizontal axis and the response variable on the vertical. Both numbers must come from the same individual: two lists of equal length measured on different units do not make a scatterplot. You read it for direction, form, strength, and any point that departs from the pattern.

Five students study 1, 2, 3, 4 and 5 hours and score 62, 68, 74, 76 and 85. The cloud rises from left to right and looks close to linear. The least-squares line is $\hat{y} = 56.8 + 5.4x$ (y-hat) and the correlation is $r = 0.986$.

The misreading is "the points look tight, so $r$ must be about 0.99." Strength cannot be read off reliably by eye, because how tight the cloud looks depends on the axis scaling and $r$ does not. Multiply every score by 3 and the plot stretches into a steeper, apparently tighter band, and $r$ is still 0.9859 to four places. Record hours in minutes instead and the plot flattens, and $r$ is unchanged again. Correlation is computed from standardized values, so multiplying either variable by a positive constant, or adding one to it, leaves the number alone while changing the picture completely.

A scatterplot also cannot report how many observations sit at one spot. Add four more students who each studied 3 hours and scored 74 and the plot looks identical, one dot at $(3, 74)$, while $r$ moves from 0.986 to 0.982. Overplotting like that can bury a whole subgroup in a dense region.

A correlation of zero is not the same as no relationship. The five points $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$ and $(2, 4)$ lie exactly on $y = x^2$ and give $r = 0$. The scatterplot shows that curve plainly, which is the reason to look at it before trusting any single summary number. Scatterplots are topic 5.1.