Answer :
### 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.
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.
