### Part A: Which data display would you use to represent this data? Explain your reasoning.
To represent the given frequency data, a bar chart (or bar graph) would be the most appropriate data display.
Reasoning:
- The data is categorical, where each category represents the number of days per week that students eat vegetables.
- A bar chart will clearly show the frequency (number of students) for each category (number of days).
- It enables easy comparison of frequencies across the different categories, highlighting which days have higher or lower frequencies.
### Part B: What, if any, are the unusual features of these data? Check for outliers, clusters, and gaps. Justify your answer mathematically.
Unusual features include:
1. Outliers:
- The lower and upper limits for outliers are calculated as follows:
- First Quartile (Q1): 5.0
- Third Quartile (Q3): 7.25
- Interquartile Range (IQR): Q3 - Q1 = 2.25
- Lower limit: Q1 - 1.5 IQR = 1.625
- Upper limit: Q3 + 1.5 IQR = 10.625
- Any data point below 1.625 or above 10.625 is considered an outlier.
- In this data set, `1` is an outlier, as it falls below the lower limit.
2. Clusters:
- Clusters are areas where data is densely populated.
- There is a noticeable cluster around days 4, 5, 6, 7, and 8, where the frequencies are relatively higher.
3. Gaps:
- Gaps indicate regions where data points are missing or sparse.
- There are clear gaps between:
- Days 2-3 and 1-2, where there are no students eating vegetables for 2 or 3 days per week.
### Part C: What is the best measure of center for these data? Explain your reasoning.
Considering the distribution of the data, the median is the best measure of center.
Reasoning:
- The mean and median are calculated as follows:
- Mean: 6.0
- Median: 6.0
- Although the mean and median are equal in this data set (6.0), the median is more robust to the impact of outliers.
- Given the presence of the outlier (1), the median is a more appropriate measure of central tendency because it is not affected by extreme values and better represents the central location of the data in skewed distributions.
