Time series graphs
🎯 In this topic you will
- Draw and interpret time series graphs
🧠 Key Words
- time series graph
- trend
Show Definitions
- time series graph: A line graph that shows how a set of data changes over time.
- trend: The general direction in which data points are moving over time, such as upward or downward.
A time series graph is a series of points, plotted at regular time intervals and joined by straight lines.
Time series graphs are used to show trends, which tell you how the data changes over a period of time.
When you draw a time series graph, make sure you:
- put time on the horizontal axis
- use an appropriate scale on the vertical axis
- plot each point accurately
- join the points with straight lines
- give the time series graph a title and label the axes
❓ EXERCISES
1. The time series graph shows the profit made by a company each year for a six-year period.
a. How much profit did the company make in: i) 2006 ii) 2007?
b. In which year did the company make the largest profit?
c. Between which two years was the greatest increase in profit?
d. Between which two years was the greatest decrease in profit?
e. Describe the trend in the company profits over the six-year period.
👀 Show answer
1a. i) 2006 ≈ $1.0$ million ii) 2007 ≈ $1.5$ million
1b. The largest profit was in 2008 (≈ $2.8$ million).
1c. Greatest increase was between 2007 and 2008.
1d. Greatest decrease was between 2008 and 2009.
1e. Profits rose steadily from 2006 to 2008, then fell gradually from 2008 to 2011.
2. The time series graph shows the value of a house over a ten-year period.
a. What was the value of the house in: i) 2000 ii) 2010?
b. In which year did the house reach its greatest value?
c. Between which two years was the greatest increase in the value of the house?
d. Describe the trend in the value of the house over the ten-year period.
e. Use the graph to estimate the value of the house in: i) 2003 ii) 2009.
👀 Show answer
2a. i) 2000 ≈ $120,000 ii) 2010 ≈ $170,000
2b. Greatest value in 2008 (≈ $190,000).
2c. Greatest increase between 2004 and 2006.
2d. Overall upward trend from 2000 to 2008, then a decline to 2010.
2e. i) 2003 ≈ $135,000 ii) 2009 ≈ $185,000
🧠 Think like a Mathematician
Task: Interpret the time series graph of average crude oil prices and evaluate Marcus and Sofia’s statements.
Scenario: The graph shows the average price of crude oil, per barrel (to the nearest dollar), every ten years since 1965. Marcus says: “The average price of crude oil was at its highest in 2005.” Sofia says: “You can’t tell from this graph in which year the average price of crude oil was at its highest.”
Questions:
👀 show answer
- The graph shows the highest plotted average is in 2005, at about $50 per barrel.
- So Marcus is correct that, based on the given data points, 2005 shows the highest average.
- However, Sofia also has a point: the graph only shows values every ten years. It’s possible that in between (e.g. 2007–2008) the average price was even higher, but that is not displayed here.
- Conclusion: Marcus is right for the data shown, but Sofia is right that the graph cannot prove which exact year had the overall highest price, since intermediate years aren’t included.
❓ EXERCISES
4. The table shows the number of people staying in a guest house each month for one year.
| Month | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Number of people | 8 | 6 | 11 | 15 | 17 | 20 | 24 | 26 | 18 | 14 | 8 | 7 |
Between which two months did the number of people at the guest house change the most?
👀 Show answer
5. The table shows the average price of silver and copper, per ounce, every four years since 1990. The prices are rounded to the nearest $0.10.
| Year | 1990 | 1994 | 1998 | 2002 | 2006 | 2010 | 2014 | 2018 |
|---|---|---|---|---|---|---|---|---|
| Average price of silver $ | 4.80 | 5.30 | 5.50 | 4.60 | 11.60 | 20.20 | 19.10 | 15.70 |
| Average price of copper $ | 1.20 | 1.10 | 0.80 | 0.70 | 3.10 | 3.10 | 3.20 | 3.10 |
a) Draw a time-series graph to show both sets of data.
b) Write true (T) or false (F) for each statement:
i. Between 1990 and 2002 the prices did not change very much.
ii. Silver increased by the greatest amount between 2002 and 2006.
iii. We can use the graph to predict an accurate price of copper and silver in 2022.
c) Use your graphs to estimate the price in 2008 of:
i. silver
ii. copper
d) Marcus says ‘The price of both silver and copper went up in 2002.’
Explain why Marcus may not be correct.
👀 Show answer
a) Graph required (not shown here).
b)
i. True – between $1990$ and $2002$, silver changed from $4.80$ to $4.60$ and copper from $1.20$ to $0.70$, only small changes compared to later years.
ii. True – silver rose from $4.60$ in 2002 to $11.60$ in 2006, an increase of $7.00$, which is the largest rise.
iii. False – future prices cannot be predicted accurately, as trends change unpredictably after 2010.
c)
i. Estimated silver price in 2008: about $16$ (midpoint between 2006: $11.60$ and 2010: $20.20$).
ii. Estimated copper price in 2008: about $3.10$ (since 2006 and 2010 values are both $3.10$).
d) Marcus is incorrect because silver fell in 2002 (from $5.50$ in 1998 down to $4.60$ in 2002), while only copper fell slightly from $0.80$ to $0.70$. Not both metals went up in 2002.
🧠 Think like a Mathematician
Task: Decide whether Zara should plot all her cycling distances or summarise them when making a time series graph.
Scenario: - Zara goes to a one-hour cycling class every Monday, Wednesday, and Friday. - She records the distance she cycles at each class. - Over a year this gives more than 150 data points. Zara says: “I think I will plot every distance I have recorded over the year.” Sofia says: “If you do that, you’ll have more than 150 points to plot! I think I would plot fewer points.”
Questions:
👀 show answer
- Zara’s idea: Plotting every point would give the most accurate record of her cycling distances. However, with 150+ points, the graph might be cluttered and hard to read.
- Sofia’s idea: Summarising the data (e.g. using weekly averages, or plotting just one point per week) would reduce the number of points, making the trend easier to see.
- Best approach: It depends on the purpose: - If Zara wants detailed analysis → plot all points. - If she wants to spot long-term trends → summarise (weekly or monthly averages).
- Conclusion: Sofia’s suggestion of plotting fewer points (by summarising) is usually better for showing a clear trend, while Zara’s method is useful if maximum detail is required.
❓ EXERCISES
7. A sports shop sells the rugby shirts of two teams, Scarlets and Dragons.
The time series graph shows the number of rugby shirts the shop has in stock each week over an 8-week period.
a) Describe the trend in the sales of:
i. Scarlets rugby shirts
ii. Dragons rugby shirts
b) Do you think the shop has enough Scarlets rugby shirts in stock for week 9?
Explain your answer.
c) Do you think the shop has enough Dragons rugby shirts in stock for week 9?
Explain your answer.
👀 Show answer
a)
i. Scarlets rugby shirts steadily decrease each week, from about $70$ down to near $10$ by week 8.
ii. Dragons rugby shirts also decrease, but from about $20$ down to almost $0$ by week 8.
b) No. At week 8, Scarlets stock is about $10$, so by week 9 there may be none left. Demand is too high.
c) No. Dragons shirts are already close to $0$ by week 8, so there will be none left for week 9.
8. The time series graph shows the number of hotel rooms booked in a seaside town. It shows the number booked in spring, summer, autumn and winter from 2018 to 2020.
a) Describe how the number of hotel rooms booked changes over the seasons during 2018.
b) Do similar changes over the seasons that you have noticed in 2018, also happen in 2019 and 2020? Explain your answer.
c) Describe the yearly trend in the number of hotel rooms booked.
d) Use your graph to predict the number of hotel rooms that will be booked in Autumn 2021.
e) Explain why your answer to part d may be incorrect.
👀 Show answer
a) In 2018, bookings increased from spring to summer, peaked in winter, and dipped again in autumn.
b) Yes. The same seasonal pattern repeats in 2019 and 2020: higher bookings in summer/winter, lower in spring/autumn.
c) Overall, bookings rose from 2018 to 2019, then peaked around 2020 winter, showing an upward trend across the years.
d) Prediction: about $60{,}000$ rooms in Autumn 2021 (following the repeating pattern, between summer and winter peaks).
e) The prediction may be incorrect because external factors (e.g., economy, travel restrictions, unusual demand) can disrupt the seasonal pattern.
⚠️ Be careful!
- Time on the x-axis: always put time on the horizontal axis with equal time intervals (no squeezed or stretched gaps).
- Choose a sensible y-scale: use equal steps that cover all values; avoid needlessly tiny steps that hide changes or huge steps that flatten trends.
- Zero baseline? Line graphs don’t always need to start at zero. If you truncate the axis, clearly show it to avoid exaggerating changes.
- Plot accurately: place each point at the correct time and value; then join points with straight segments in order.
- Don’t join across missing data: leave gaps or mark breaks; a straight line implies values existed between points.
- Trend vs wiggles: a downward trend means “generally decreasing,” even if some years tick up slightly.
- Compare like-for-like: when comparing two series, keep the same y-axis scale and label lines with a clear key.
- Rate vs total: the area under a simple value-over-time line is not a total unless the vertical axis is a rate.
- Interpolation ≠ exact: reading between plotted years is an estimate; state that values are approximate.
- No over-extrapolation: don’t extend the pattern far beyond the last point; trends can change.