Plotting Time Series - Mr-Mathematics.com - Free Printable

Let’s solve this step by step.

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## i) Draw a time series graph to illustrate the trend of this data.

We are given quarterly sales data for 2012, 2013, and 2014.

Step 1: Organize the data with time points

We’ll treat each quarter as a point in time:

| Time Point (Quarter) | Year | Quarter | Sales |
|----------------------|------|---------|-------|
| 1 | 2012 | Q1 | 85 |
| 2 | 2012 | Q2 | 79 |
| 3 | 2012 | Q3 | 74 |
| 4 | 2012 | Q4 | 68 |
| 5 | 2013 | Q1 | 91 |
| 6 | 2013 | Q2 | 82 |
| 7 | 2013 | Q3 | 76 |
| 8 | 2013 | Q4 | 71 |
| 9 | 2014 | Q1 | 103 |
| 10 | 2014 | Q2 | 86 |
| 11 | 2014 | Q3 | 81 |
| 12 | 2014 | Q4 | ? |

> We’ll plot these 11 points on the graph (Q4 2014 is unknown — we’ll predict it later).

Step 2: Plot the points on the grid

On the provided graph:

- X-axis: labeled “Year and Quarter” — so mark 1 to 12 along the x-axis (each number = one quarter).
- Y-axis: “Sales” from 60 to 110.

Plot the following points:

- (1, 85)
- (2, 79)
- (3, 74)
- (4, 68)
- (5, 91)
- (6, 82)
- (7, 76)
- (8, 71)
- (9, 103)
- (10, 86)
- (11, 81)

Connect them with line segments to form a time series graph.

✔ Graph drawn — you should see a pattern: sales dip each year in Q4, then jump up again in Q1 of next year.

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## ii) Draw a trend line.

A trend line is a straight line that best fits the overall direction of the data — not necessarily passing through every point, but showing the general upward or downward movement.

How to draw it:

Look at the overall pattern:

- Start: Q1 2012 → 85
- End: Q3 2014 → 81

But notice: there’s a seasonal pattern — sales rise in Q1, then fall through the year.

To draw a trend line, ignore the seasonal ups and downs and look at the long-term direction.

Compare Q1 values:

- 2012 Q1: 85
- 2013 Q1: 91
- 2014 Q1: 103 → increasing

Compare Q2:

- 2012 Q2: 79
- 2013 Q2: 82
- 2014 Q2: 86 → increasing

Compare Q3:

- 2012 Q3: 74
- 2013 Q3: 76
- 2014 Q3: 81 → increasing

Compare Q4 (only 2 known):

- 2012 Q4: 68
- 2013 Q4: 71 → increasing

➡️ So overall, sales are increasing over time, even though they dip each year in Q4.

Drawing the trend line:

Draw a straight line that starts near the first point (Q1 2012, 85) and slopes gently upward to end near the last point (Q3 2014, 81), but adjusted to reflect the *general upward trend*.

You can approximate by connecting the average of early points to the average of later points.

Or use two anchor points:

- Early: Q1 2012 = 85
- Late: Q1 2014 = 103 → slope = (103 - 85)/2 years = 18/2 = 9 per year → about 2.25 per quarter.

So trend line might go from (1, 85) to (9, 103), and extend to (12, ~110).

✔ Trend line drawn — a gentle upward-sloping line.

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## iii) Comment on the trend of sales over time.

> ✔ Answer:
The overall trend of car sales at AW Cars is upward over time. Although there is a clear seasonal pattern — sales peak in Q1 and decline through Q2, Q3, and Q4 each year — the level of sales in corresponding quarters is increasing year-over-year. For example, Q1 sales rose from 85 in 2012 to 91 in 2013 to 103 in 2014. This indicates growth in demand or improved performance despite seasonal dips.

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## iv) Make a prediction of sales for the last quarter of 2014. Give a reason for your answer.

We need to predict Q4 2014 sales.

Method 1: Use the seasonal pattern

Look at the drop from Q3 to Q4 in previous years:

- 2012: Q3 = 74 → Q4 = 68 → drop of 6
- 2013: Q3 = 76 → Q4 = 71 → drop of 5

Average drop = (6 + 5)/2 = 5.5

In 2014: Q3 = 81 → predicted Q4 = 81 - 5.5 = 75.5 ≈ 76

Method 2: Use the trend line

From earlier, we estimated the trend increases by ~2.25 per quarter.

Q3 2014 = 81 → Q4 2014 = 81 + 2.25 = 83.25

But wait — this contradicts the seasonal drop!

Actually, the trend line captures the long-term growth, while the seasonal drop is cyclical. So we should combine both:

Use the trend value for Q4 2014, then subtract the typical seasonal drop.

Alternatively, since we see a consistent drop from Q3 to Q4, and Q3 2014 is higher than previous Q3s, we can expect a similar drop.

Best approach: Seasonal adjustment

Predict Q4 2014 using the pattern:

- 2012 Q4 = 68 = 74 - 6
- 2013 Q4 = 71 = 76 - 5
- 2014 Q4 = ? = 81 - x

Assume x ≈ 5.5 → 75.5

Or round to nearest whole number: 76

✔ Prediction: 76 cars

> Reason: Based on the historical pattern, sales in Q4 are consistently lower than Q3 by approximately 5–6 units. Since Q3 2014 sales were 81, we predict Q4 2014 sales will be around 76, continuing this seasonal trend.

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## 📌 Final Summary:

- i) Graph plotted with 11 points (Q1 2012 to Q3 2014), connected to show fluctuating but generally rising sales.
- ii) Trend line drawn as a gentle upward slope, reflecting long-term growth.
- iii) Sales show an upward trend with seasonal dips each year (lowest in Q4).
- iv) Prediction for Q4 2014: 76 cars, based on consistent seasonal drop from Q3 to Q4.

✔ All tasks completed!