Answer :
16. Julianna read 8 books. Which measure of center would she choose if she wanted to convince someone that she read a lot of books? Explain.
To determine which measure of center Julianna should choose to convey that she read a lot of books, we need to consider what each measure represents and how it might reflect on her reading compared to others:
- Mean (Average): It is calculated by adding all the values and dividing by the number of values. Julianna might choose the mean if her book count is higher than most of her peers, as this would raise the average.
- Median: This is the middle number when all the values are placed in ascending order. If Julianna has read more books than a majority of her peers, the median might be lower than her count, making her total look impressive in comparison.
- Mode: This is the number that appears the most frequently in the data set. If the most common number of books read is equal to or less than the number Julianna read, the mode wouldn't necessarily help her argue that she read a lot, unless she read exactly the mode count and it is relatively high.
Without more information about how many books the others have read, it's not possible to determine which measure would best support Julianna's claim. However, if Julianna has read significantly more books than most others, the mean might be her best choice as it would be influenced by her high book count. If most students read fewer books than Julianna, but some read much more, then the median might make her total seem more impressive since it is less influenced by the high values. The mode would be least likely to support her claim unless it is equal to her count and considered high.
17. The teacher wants to show that most of her students read similar numbers of books. Should the teacher use the range or the interquartile range to show this? Explain.
To demonstrate that most students read a similar number of books, the teacher should consider using a measure that indicates how closely the data values are grouped around the center.
- Range: It measures the spread of the entire dataset by subtracting the smallest value from the largest. However, the range is heavily influenced by outliers, that is, values that are much higher or lower than the rest of the data.
- Interquartile Range (IQR): This measures the spread of the middle 50% of the data, calculated by finding the difference between the first quartile (25th percentile) and the third quartile (75th percentile). IQR is not affected by outliers as it focuses on the central tendency of the data.
Given the goal is to show consistency among the students' reading, the teacher would likely choose the interquartile range. The IQR would provide a better representation of the spread of the middle bulk of the data, excluding any extreme values that might skew the interpretation when using the range.
18. Elena says that if the mean, median, and mode of a data set are equal, then there are no outliers. Is Elena correct? Explain.
Elena's statement is not necessarily correct. While it is true that having a dataset where the mean, median, and mode are all equal suggests a symmetric distribution, which could imply the absence of outliers, it's not a guarantee. It is possible to have a dataset with these measures of center being equal and still include outliers.
An outlier is an extremely high or low value compared to the rest of the data. In certain cases, such values may not affect the mean, median, and mode enough to change their equality. For instance, if a dataset is mostly consistent and symmetric but includes one outlier far from the rest, the median (middle value) and mode (most frequent value) might still reflect the central tendency of the bulk of the data, and if the outlier is not excessively distant, the mean might not be shifted too far from the median and mode. This would result in a scenario where the mean, median, and mode are nearly or exactly equal, but an outlier is still present. Therefore, one cannot infer the absence of outliers solely based on the equality of mean, median, and mode.
