Statistics · Data representation
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Time series
Measurements plotted against time and joined in order — why joining is legitimate here but not on a scatter graph, how to describe a trend so it earns the mark, how to spot seasonal variation and use it to predict the next quarter, and why predicting far beyond the data is worthless.
Statistics · Data representation
Time series
Measurements plotted against time and joined in order — why joining is legitimate here but not on a scatter graph, how to describe a trend so it earns the mark, how to spot seasonal variation and use it to predict the next quarter, and why predicting far beyond the data is worthless.
Why it works
A time series is the same quantity measured over and over at regular times: a shop's sales each quarter, a town's rainfall each month, a country's population each year. You plot each measurement against the time it was taken — and then, unlike on any other scatter of dots, you join them up in order with straight lines.Why joining is allowed here. That is worth questioning, because on a scatter graph joining the dots is flatly wrong, and a scatter graph looks like the same object: dots on a pair of axes. The difference is what one dot means. On a scatter graph each dot is a different individual — pupil 1, pupil 2, pupil 3 — and those individuals have no natural order. Plot them in a different order and the zigzag you would draw changes completely, so the zigzag carries no information at all. Worse, two individuals can share the same (two pupils the same height), so there is not even a single path to draw.
On a time-series graph the horizontal axis is time. Time has exactly one order, each time has exactly one measurement, and the segment from one point to the next shows how the quantity changed over that gap: uphill means it rose, downhill means it fell, steeper means it changed faster. That is real information, and it is the same whoever draws the graph. The line does not claim to know the values in between — nobody thinks the café below sold exactly the halfway number of cups in mid-February — it is a guide for the eye that makes the order and the direction of change visible.
Reading it. Go up from the time on the horizontal axis to the point, then straight across to the vertical axis. A change is the difference between two readings; a total is the sum of several.
The trend, and why "it goes up and down" earns nothing. Here are a café's quarterly sales of hot chocolate, in hundreds of cups.
| Quarter | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 2023 | 6 | 3 | 2 | 8 |
| 2024 | 7 | 4 | 3 | 10 |
| 2025 | 8 | 5 | 4 | 12 |
Plotted, that graph zigzags violently. Write "it goes up and down" and you have described every individual segment and said nothing whatever about the café, so it scores nothing. The trend is the overall direction across the whole graph once you look past the zigzag — and here it is upwards.
A fall that means nothing. From quarter 4 of 2024 ( hundred cups) to quarter 3 of 2025 ( hundred cups), sales dropped by hundred — they more than halved. Is the café collapsing? Compare like with like instead. Quarter 3 in each year runs , , : quarter 3 sales have risen every single year. The big fall is the summer arriving, not the business failing.
That is the whole trap. Comparing quarter 4 with quarter 3 mixes two different things — the trend and the season — and here the season is much the larger of the two, so it drowns the trend out completely. Comparing the same quarter in different years cancels the season, because both readings carry the same summer effect, and leaves the trend on its own.
Seasonal variation is exactly that repeating within-year pattern. You may only claim it once you have watched it repeat: at least two full years of quarterly data, so the second cycle confirms the first. Here every year runs high–low–lowest–highest (winter, spring, summer, Christmas), three times over. That is a seasonal pattern, and it is what turns a good description into a full-mark one: "the sales go up and down" earns nothing, while "an upward trend, with sales peaking in quarter 4 every year and lowest in quarter 3" names the trend, names the season and gives its period.
Using both halves to predict. A sensible estimate needs the trend and the season. To estimate quarter 4 of 2026: quarter 4 has gone , , — up about a year — so about hundred cups. Notice what was not done. Continuing the last segment of the graph (, a rise of ) would predict hundred cups for the next quarter, which is nonsense: that segment is season, not trend.
Why predicting far ahead is worthless. All the graph shows is what happened between 2023 and 2025. Extending the pattern to 2040 assumes that both the trend and the season survive fifteen years of new cafés, new tastes and new prices, and the data says nothing at all about that. The further beyond the data you go, the less a prediction is worth — and a falling trend extended far enough will eventually predict a negative number of cups, which settles the argument.