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AQA GCSE · Question 16.1 · Statistical Measures and Calculations
Here is an experiment which is designed to find the best trained dog out of Troy, Buddy, Bruno, Murphy and Bumble. Each of the five owners asks their dog to sit and then walks away. The time for which each dog sits is recorded. The experiment is repeated 4 more times. Here are the data for the five dogs. The dog which sits for the longest average time is declared the winner. Give a reason why each of the three dogs stated below could be declared the winner. In each answer you must state or calculate appropriate measures.
- Buddy
- Bruno
- Murphy
Here is an experiment which is designed to find the best trained dog out of Troy, Buddy, Bruno, Murphy and Bumble. Each of the five owners asks their dog to sit and then walks away. The time for which each dog sits is recorded. The experiment is repeated 4 more times. Here are the data for the five dogs. The dog which sits for the longest average time is declared the winner. Give a reason why each of the three dogs stated below could be declared the winner. In each answer you must state or calculate appropriate measures.
- Buddy
- Bruno
- Murphy
How to approach this question
The word "average" can mean different things (mean, median, mode). It can also be calculated in different ways (e.g., after removing outliers). For each dog, you need to find a statistical measure or method of calculation that makes them the winner.
1. **Calculate the mean** for all dogs. Who wins?
2. **Calculate the median** for all dogs. Who wins?
3. **Consider outliers.** Do any of the dogs have an unusually high or low score that might be skewing their mean? Calculate the mean for those dogs with the outlier removed. Who wins now?
4. Use your findings to construct an argument for each of the three dogs.
Full Answer
**Buddy:** Buddy could be the winner if the average used is the **mean**.
Mean for Buddy = (21+22+14+20+12)/5 = 89/5 = 17.8 seconds.
(Mean for Bruno = (39+20+17+12+12)/5 = 100/5 = 20. Mean for Murphy = (24+17+18+2+24)/5 = 85/5 = 17).
Wait, this makes Bruno the winner by mean. Let me re-read. "Give a reason why each of the three dogs...could be declared the winner". This implies different averages or criteria.
Let's re-calculate means:
Troy: (15+18+19+13+13)/5 = 78/5 = 15.6
Buddy: (21+22+14+20+12)/5 = 89/5 = 17.8
Bruno: (39+20+17+12+12)/5 = 100/5 = 20
Murphy: (24+17+18+2+24)/5 = 85/5 = 17
Bumble: (7+12+14+12+10)/5 = 55/5 = 11
By mean, Bruno is the winner.
Let's try the **median**. First, order the data for each dog:
Buddy: 12, 14, **20**, 21, 22. Median = 20.
Bruno: 12, 12, **17**, 20, 39. Median = 17.
Murphy: 2, 17, **18**, 24, 24. Median = 18.
By median, Buddy is the winner.
Let's try the **mean after removing outliers**.
Bruno has an outlier of 39. Without it, mean = (20+17+12+12)/4 = 61/4 = 15.25.
Murphy has an outlier of 2. Without it, mean = (24+17+18+24)/4 = 83/4 = 20.75.
Buddy has no obvious outliers.
So, if we remove outliers and take the mean, Murphy could be the winner.
So the final answer should be:
**Buddy:** Could be the winner if the median is used as the average. His median time is 20 seconds, which is higher than Bruno's (17) and Murphy's (18).
**Bruno:** Could be the winner if the mean is used as the average. His mean time is 20 seconds, which is the highest of all dogs.
**Murphy:** Could be the winner if outliers are removed before calculating the mean. Murphy has one low outlier (2). If this is removed, his mean becomes (24+17+18+24)/4 = 20.75 seconds, which is the highest.
The ambiguity of the word "average" allows for different interpretations, leading to different winners.
**Why Buddy could be the winner:**
If "average" is taken to be the **median**, we must order each dog's times:
- Buddy: 12, 14, **20**, 21, 22 → Median = 20 seconds.
- Bruno: 12, 12, **17**, 20, 39 → Median = 17 seconds.
- Murphy: 2, 17, **18**, 24, 24 → Median = 18 seconds.
Buddy's median time of 20 seconds is the highest, making him the winner by this measure. The median is a good measure here as it is not affected by extreme values.
**Why Bruno could be the winner:**
If "average" is taken to be the **mean**, we calculate the sum of times divided by 5:
- Bruno: (39 + 20 + 17 + 12 + 12) / 5 = 100 / 5 = 20 seconds.
- Buddy: (21 + 22 + 14 + 20 + 12) / 5 = 89 / 5 = 17.8 seconds.
- Murphy: (24 + 17 + 18 + 2 + 24) / 5 = 85 / 5 = 17 seconds.
Bruno's mean time of 20 seconds is the highest, making him the winner. However, his mean is inflated by one very high score (39), which might be an outlier.
**Why Murphy could be the winner:**
If we decide to remove **outliers** before calculating the mean, Murphy could win. Murphy has one very low score of 2 seconds, which seems anomalous. Bruno has a very high score of 39.
- If we remove Murphy's outlier of 2, his new mean is (24 + 17 + 18 + 24) / 4 = 83 / 4 = 20.75 seconds.
- If we remove Bruno's outlier of 39, his new mean is (20 + 17 + 12 + 12) / 4 = 61 / 4 = 15.25 seconds.
By removing outliers, Murphy's mean of 20.75 seconds becomes the highest, making him the winner.
Common mistakes
✗ Only calculating one type of average (e.g., the mean) for all three dogs.\n✗ Not showing any calculations to support the claims.\n✗ Failing to explain *why* the chosen measure makes that dog the winner.
Practice the full AQA GCSE Statistics Foundation Tier Paper 2
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