S2f - Stem and leaf diagrams - BossMaths.com - Free Printable

Here is the solution to Gamma Exercise 1.

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a) Ordered Back-to-Back Stem-and-Leaf Diagram

A back-to-back stem-and-leaf diagram displays two datasets side by side using a common stem. The stems represent the tens digit, and the leaves represent the units digit. For an *ordered* diagram, the leaves on each side must be sorted in ascending order.

First, let’s organize the data for each day:

Monday’s speeds (in mph):
24, 35, 30, 27, 38, 31, 30, 32, 34, 28
→ Sorted: 24, 27, 28, 30, 30, 31, 32, 34, 35, 38

Tuesday’s speeds (in mph):
27, 31, 33, 36, 18, 26, 33, 43, 21, 23
→ Sorted: 18, 21, 23, 26, 27, 31, 33, 33, 36, 43

Now, construct the back-to-back stem-and-leaf diagram. We’ll place Monday’s data on the left (leaves increasing from right to left), Tuesday’s data on the right (leaves increasing from left to right), with the stem in the middle.

```
Monday | Stem | Tuesday
| 1 | 8
4 7 8 | 2 | 1 3 6 7
0 0 1 2 4 5 8 | 3 | 1 3 3 6
| 4 | 3
```

✔ Explanation of the diagram:

- Stem “1”: Only Tuesday has a value in the 10s → leaf “8” → speed = 18 mph.
- Stem “2”:
- Monday: leaves 4, 7, 8 → speeds 24, 27, 28
- Tuesday: leaves 1, 3, 6, 7 → speeds 21, 23, 26, 27
- Stem “3”:
- Monday: leaves 0,0,1,2,4,5,8 → speeds 30,30,31,32,34,35,38
- Tuesday: leaves 1,3,3,6 → speeds 31,33,33,36
- Stem “4”:
- Tuesday: leaf 3 → speed 43 mph

This diagram allows easy visual comparison of the two distributions.

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**b) Which day was dry and sunny?

We are told that one day was wet and gloomy, the other dry and sunny. We need to infer which is which based on driving behavior.

General assumption: On wet and gloomy days, drivers tend to drive more cautiously — i.e., slower speeds. On dry and sunny days, drivers may feel more confident and drive faster.

Let’s compare the two datasets numerically and visually.

#### Summary Statistics:

Monday:
- Sorted: 24, 27, 28, 30, 30, 31, 32, 34, 35, 38
- Median = average of 5th and 6th values = (30 + 31)/2 = 30.5 mph
- Range = 38 - 24 = 14 mph
- No extreme low values; all speeds ≥ 24 mph

Tuesday:
- Sorted: 18, 21, 23, 26, 27, 31, 33, 33, 36, 43
- Median = (27 + 31)/2 = 29 mph
- Range = 43 - 18 = 25 mph
- Has very low speeds (18, 21, 23) and a very high speed (43)

#### Analysis:

- Tuesday has a wider spread (range = 25) compared to Monday (range = 14), indicating more variability in driving behavior.
- Tuesday includes the lowest recorded speed (18 mph) and the highest (43 mph).
- The presence of very slow speeds (e.g., 18, 21, 23) suggests cautious driving — typical of wet and gloomy conditions.
- Monday’s speeds are more clustered in the 27–38 range, with no extremely low or high outliers — suggesting more consistent, perhaps slightly faster, driving — typical of dry and sunny conditions.

#### Conclusion:

> Monday was the dry and sunny day.

Drivers were generally faster and more consistent in their speeds, without the extreme caution seen on Tuesday.

Tuesday had both very slow drivers (likely due to weather) and one very fast outlier (perhaps an exception), which fits the pattern of a wet/gloomy day where most drive slower but some still speed.

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✔ Final Answers:

a) Back-to-back stem-and-leaf diagram:

```
Monday | Stem | Tuesday
| 1 | 8
4 7 8 | 2 | 1 3 6 7
0 0 1 2 4 5 8 | 3 | 1 3 3 6
| 4 | 3
```

b) The dry and sunny day was Monday.