Time Series and Frequency Polygons
Joining points can show change over time or the shape of a frequency distribution. Learn what each horizontal position means before deciding how to draw and interpret the graph.
Foundation & Higher. Frequency polygons are included for the combined course; OCR does not require them. Use grouped frequency tables and clear chart scales.
Plot values in time order
A time series records a quantity at successive times. Put time on the horizontal axis and the measured quantity, with units, on the vertical axis. Use distances that match the time intervals, plot each value accurately and join successive points with straight lines.
| Quarter | Year 1 | Year 2 |
|---|---|---|
| Q1 | 20 | 25 |
| Q2 | 35 | 40 |
| Q3 | 50 | 55 |
| Q4 | 30 | 35 |
Worked example — construct and read the graph
Quarter 3 has value 50 hundreds, so plot (3, 50): this means 5000 sales. Quarter 7 has 55 hundreds, or 5500 sales. The rise from quarter 2 to quarter 3 is 15 hundreds = 1500 sales.
The connecting segments help show the pattern. They do not prove that sales changed steadily at every instant between the observations. If a time is missing, leave its correct space on the time scale.
Separate a general trend from a seasonal pattern
A trend is the general direction over time. A seasonal pattern is a repeated rise or fall at similar points in a cycle. Look at the whole series, not just one increase.
Worked example — interpret the sales pattern
Q3 is the highest quarter in both years; Q1 is the lowest. This suggests a seasonal pattern. Each Year 2 quarter is 5 hundreds above the matching Year 1 quarter, suggesting an upward trend as well.
A simple prediction for Year 3 Q3 is 55 + 5 = 60 hundreds, or 6000 sales, if the same annual change continues. It is an estimate: weather, prices or a new competitor could change the pattern.
Compare like periods when seasonality matters. Comparing a summer quarter with a winter quarter alone may exaggerate a long-term change. Two years give limited evidence that a pattern will continue. Moving-average calculations are not needed for this GCSE Mathematics course.
Construct a frequency polygon from midpoints
A frequency polygon displays a grouped frequency distribution. Plot class midpoint against frequency, then join consecutive points with straight lines. The horizontal axis is the measured variable, not time unless time itself is the variable being grouped.
| Time (min) | Midpoint | f |
|---|---|---|
| 0–10 | 5 | 3 |
| 10–20 | 15 | 8 |
| 20–30 | 25 | 6 |
| 30–40 | 35 | 3 |
Worked example — find and place a midpoint
The 10–20 class has midpoint (10 + 20)/2 = 15 and frequency 8, so plot (15, 8). Repeat for each class. Do not plot at its upper boundary 20; that would shift the representation.
Read a plotted frequency as the count in its whole class. Eight journeys are in 10–20 minutes; they are not all exactly 15 minutes. There is no need to invent extra observations beyond the end classes. Use the class widths and instructions given; unequal-width histograms use a different height rule.
Compare distributions on the same axes
Use the same class intervals and scales, and label the two lines clearly. If sample sizes differ, raw frequencies alone may give an unfair impression; compare suitable proportions or summary statistics.
Worked example — turn a visual comparison into evidence
Blue A has frequencies 3, 8, 6, 3; orange B has 1, 5, 9, 5. Both total 20. B has more journeys in the longer-time classes. Using midpoints, A has estimated mean (15 + 120 + 150 + 105)/20 = 19.5 min; B has (5 + 75 + 225 + 175)/20 = 24 min.
B therefore takes about 4.5 minutes longer on average by these estimates. The polygon alone does not show every journey or guarantee that each B journey is longer.
Think Like a Mathematician
A rising segment in a time series means a quantity increased over time. A rising segment in a frequency polygon means the next class has a greater frequency. The shape can look similar while its meaning is different.
Common Mistakes
Unequal time gaps: horizontal distances must reflect elapsed time.
Midpoints versus boundaries: frequency polygons use class midpoints.
Trend versus certainty: a repeating pattern supports a cautious prediction, not a guarantee.
Scale units: 50 hundreds means 5000, not 50.
Practice
1. In the time-series example, what does quarter 6 represent and what is its value?
Year 2 Q2: 40 hundreds = 4000 sales.
2. Find the change from Year 1 Q3 to Year 1 Q4.
30 − 50 = −20 hundreds: a decrease of 2000 sales.
3. Give evidence for both a seasonal pattern and an upward trend.
Q3 is highest in both years; each matching Year 2 quarter is 5 hundreds higher than Year 1.
4. Using the same annual increase, predict Year 3 Q1 and state a limitation.
25 + 5 = 30 hundreds = 3000 sales. This assumes the pattern continues; two years and changing conditions limit its reliability.
5. Classes 2–6, 6–10 and 10–14 have frequencies 3, 7 and 5. Give the polygon coordinates.
Midpoints are 4, 8, 12, so plot (4,3), (8,7), (12,5) and join successive points. Total frequency is 15.
6. A polygon point is (25,6) for the class 20–30 minutes. Does it mean six journeys lasted exactly 25 minutes?
No. Six journeys lie in that class; 25 is its plotting midpoint.
7. A and B have frequencies 8 and 12 in a class, with total samples 20 and 40. Which has the greater proportion in that class?
A: 8/20 = 40%. B: 12/40 = 30%. A has the greater proportion despite its smaller count.
8. Why should a time-series plot at 09:00, 10:00 and 12:00 not use equal gaps?
The second time gap is two hours, twice the first. Equal spacing would distort the rate and timing of change.
Frequently Asked Questions
Must a frequency polygon meet the horizontal axis at both ends?
Follow the question or supplied diagram. Plot and join the required midpoint-frequency pairs; do not treat invented end points as extra data.
Is joining scatter-graph points the same method?
No. A scatter graph shows paired measurements. Do not join each observation to the next; a line of best fit describes its overall relationship.