Hey, everyone! Welcome back to another Harrow School question. This time, we have an interesting 13+ Maths question that delves into averages. We’ll be tackling a problem that checks our skills in calculating the mean, median, mode, and range with a final comparison of two sets of results. Let’s dive right in and see if it’s an average challenge or above average.
Calculating the Mean
The problem gives us a list of marks obtained by students in a class: 20, 13, 9, 14, 17, 20, 15, and 18. We’re asked to calculate the mean mark. By this stage, you should be familiar with calculating averages, having possibly encountered it already in the 11+ exams as well.
Step-by-Step Calculation
- Sum of the Numbers:To find the mean, we first add all the numbers together.
- 20 + 13 = 33
- 33 + 9 = 42
- 42 + 14 = 56
- 56 + 17 = 73
- 73 + 20 = 93
- 93 + 15 = 108
- 108 + 18 = 126
- Number of Data Points:There are 8 numbers in total.[\text{Mean} = \frac{\text{Total Sum of Numbers}}{\text{Total Count}} = \frac{126}{8} = 15.75]
So, the mean score is 15.75 out of 20 marks.
Identifying Numbers to Remove Without Changing the Mean
Next, we’re asked to remove two marks without changing the mean.
Analysis
If we need to remove two marks without affecting the mean, we must ensure the total sum remains balanced:
- The current mean (15.75) suggests that two numbers that add up to twice this mean should sum to 31.5.
Given we need to work with whole numbers, let’s identify such pairs:
- Possible pairs include (15 and 16.5 or alternatively we can find a close hole match like 14 and 17 because we already have these numbers in the list.
So, removing the marks 14 and 17 should leave the mean unchanged.
Determining the Median and Mode
Median
The median is the middle number in a sorted list. Our sorted list is:
9, 13, 14, 14, 15, 17, 18, 20- With 8 numbers, the median is the average of the 4th and 5th numbers:[\text{Median} = \frac{14 + 15}{2} = 14.5]
Mode
The mode is the number that appears most frequently. In our list:
- 14 appears twice, making it the mode.
Summary of Findings
Median: 14.5
Mode: 14
Comparing Test Results
Finally, the teacher retested the same class the next week. The new results were:
- Mean: 13.5
- Median: 17
- Mode: 5
- Range: 15
Comparison:
Let’s compare the old results:
- Original Test Mean: 15.75
- New Test Mean: 13.5
Observation: The mean dropped, indicating a poorer performance on average.
- Original Test Median: 14.5
- New Test Median: 17
Observation: The median increased, suggesting that the middle performance improved.
- Original Test Mode: 14
- New Test Mode: 5
Observation: The mode dropped dramatically meaning a significant number of students scored the same lower mark.
- Original Test Range: Highest Mark (20) – Lowest Mark (9) = 20 – 9 = 11
- New Test Range: 15
Observation: The range is higher, indicating more variability in scores.
Conclusion
Based on these metrics, we can infer the following:
- The original test showed a better average performance (higher mean).
- The new test showed a stronger middle performance (higher median).
- The original test had a more common higher score (higher mode).
- The original test had less variability (smaller range).
From these points:
Excerpt from Analysis:
Given the data points, we conclude that although the new test had a higher median, it did not outperform the original test overall. Two out of three central tendency measures (mean and mode) were better in the original test, along with a smaller range, suggesting more consistent performance.
Let me know your thoughts in the comments below!
Wrapping Up
That wraps up our detailed review of this 13+ Maths question from Harrow School exams. Stay tuned for the next instalment of our series, where we’ll dive into another stimulating problem. Here’s to mastering averages and beyond!
Feel free to ask questions or share your insights in the comments section. Until next time, happy studying!