Question
Based on the shape of the probability distribution below, which statistics should be used to describe its center and spread?
A histogram is shown.A histogram has 5 bars that sit on a horizontal axis, are the same width, and touch at the edges. The second bar is taller than the first. The third bar is the tallest. The fourth bar is similar in height to the second and the fifth bar is similar in height to the first.
(1 point)
Responses
mean and interquartile range
mean and interquartile range
mean and standard deviation
mean and standard deviation
median and interquartile range
median and interquartile range
median and standard deviation
A histogram is shown.A histogram has 5 bars that sit on a horizontal axis, are the same width, and touch at the edges. The second bar is taller than the first. The third bar is the tallest. The fourth bar is similar in height to the second and the fifth bar is similar in height to the first.
(1 point)
Responses
mean and interquartile range
mean and interquartile range
mean and standard deviation
mean and standard deviation
median and interquartile range
median and interquartile range
median and standard deviation
Answers
GPT-4o mini
To determine which statistics should be used to describe the center and spread of the probability distribution based on the histogram described, we need to consider the shape of the distribution.
The description indicates that the histogram has a peak (the third bar is the tallest) with bars that are similar in height around it (the second and fourth bars are similar to each other, and the fifth bar is similar in height to the first). This suggests a single peak and that the distribution may be symmetric or possibly slightly skewed.
1. **Mean**: This is typically used for symmetric distributions without outliers. However, for skewed distributions, the mean can be affected by extreme values.
2. **Median**: This is a better measure of central tendency in skewed distributions or when there are outliers, as it is less affected by extreme values.
3. **Standard Deviation**: This is used for measuring spread when the distribution is symmetric and not heavily skewed. It can be affected by outliers.
4. **Interquartile Range (IQR)**: This is a better measure of spread for skewed distributions, as it focuses on the middle 50% of the data and is not influenced by outliers.
Given that the histogram appears to have a distinct peak (which suggests potential symmetry but could also indicate the possibility of some level of skewness), a conservative approach would be to use the **median** for the center and the **interquartile range (IQR)** for the spread. This combination provides a robust measure that is appropriate in the presence of any skewness or outliers.
Therefore, the best choice based on the description of the histogram is:
**median and interquartile range**.
The description indicates that the histogram has a peak (the third bar is the tallest) with bars that are similar in height around it (the second and fourth bars are similar to each other, and the fifth bar is similar in height to the first). This suggests a single peak and that the distribution may be symmetric or possibly slightly skewed.
1. **Mean**: This is typically used for symmetric distributions without outliers. However, for skewed distributions, the mean can be affected by extreme values.
2. **Median**: This is a better measure of central tendency in skewed distributions or when there are outliers, as it is less affected by extreme values.
3. **Standard Deviation**: This is used for measuring spread when the distribution is symmetric and not heavily skewed. It can be affected by outliers.
4. **Interquartile Range (IQR)**: This is a better measure of spread for skewed distributions, as it focuses on the middle 50% of the data and is not influenced by outliers.
Given that the histogram appears to have a distinct peak (which suggests potential symmetry but could also indicate the possibility of some level of skewness), a conservative approach would be to use the **median** for the center and the **interquartile range (IQR)** for the spread. This combination provides a robust measure that is appropriate in the presence of any skewness or outliers.
Therefore, the best choice based on the description of the histogram is:
**median and interquartile range**.