Time plot

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.