To help Tahir determine which average (mean, median, or mode) will give the best result for his report, let's calculate each one and compare them.

1. Mean (Average):

The mean is calculated by summing all the values and then dividing by the number of values.

List of scores:
8.4, 6.3, 7.5, 4.7,
6.7, 7.4, 7.8, 8.1,
6.5, 7.1, 5.9, 3.4,
5.6, 6.8, 7.6, 8.1

Number of scores: 16

First, sum all the values: $8.4 + 6.3 + 7.5 + 4.7 + 6.7 + 7.4 + 7.8 + 8.1 + 6.5 + 7.1 + 5.9 + 3.4 + 5.6 + 6.8 + 7.6 + 8.1 = 107.9$

Now, divide by the total number of scores (16): $\text{Mean} = \frac{107.9}{16} = 6.744$

2. Median:

The median is the middle value when the scores are arranged in order. If there is an even number of values, the median is the average of the two middle numbers.

First, arrange the scores in ascending order: $3.4, 4.7, 5.6, 5.9, 6.3, 6.5, 6.7, 6.8, 7.1, 7.4, 7.5, 7.6, 7.8, 8.1, 8.1, 8.4$

There are 16 values, so the median will be the average of the 8th and 9th values: $\frac{6.8 + 7.1}{2} = 6.95$

3. Mode:

The mode is the number that appears most frequently in the list. Looking at the scores, the value 8.1 appears twice, while all other values appear only once.

So, the mode is 8.1.

Comparison of Averages:

• Mean: 6.744
• Median: 6.95
• Mode: 8.1

Conclusion:

If Tahir wants the results to "look as good as possible," he should report the mode of 8.1, as it is the highest average among the three.

Would you like further details on any of these calculations, or do you have any additional questions?

Here are 5 follow-up questions you may want to explore:

1. How would the report change if outliers were removed from the data set?
2. What is the impact of rounding the values in the dataset?
3. How does the range of scores affect the average result?
4. What would happen if the number of batches increased?
5. How would we interpret these averages in a real-world context?

Tip: Always consider the audience when choosing which average to report—each type of average tells a different story!