Question

The distribution of test scores for two random samples of students is shown in the stem-and-leaf plots.

Answer 1

**Group A:**

Mean: 2756061122.42, Median: 23556, Range: 13456778889, IQR: 13456778888.5

**Group B:**

Mean: 93826311185.42, Median: 71123567888, Range: 913456778789, IQR: 913456778789

**Comparison:**

Group B has higher mean and median, indicating higher average scores. Group A has larger variability (range and IQR), suggesting more diverse scores.

Let's start by calculating the mean, median, range, and interquartile range for each data set.

**Group A:**

- Mean:
`\((58 + 47 + 38 + 237 + 23556 + 1123567888 + 02455577 + 2233344579 + 13456778889 + 128 + 000 + 00)/(12)\)`

To find the mean, sum up all the values and divide by the number of values.

- Median: Since the data is sorted in ascending order, the median is the middle value. In this case, it's the 6th value (23556).

- Range: The range is the difference between the maximum and minimum values. In this case, it's 13456778889 - 0.

- Interquartile Range (IQR): First, find the first quartile (Q1) and third quartile (Q3). Then, calculate IQR as Q3 - Q1.

**Group B:**

- Mean:
`\((58 + 547 + 638 + 6237 + 723556 + 71123567888 + 802455577 + 82233344579 + 913456778889 + 128 + 1000 + 100)/(12)\)`

- Median: Similar to Group A, find the middle value, which is the 6th value (71123567888).

- Range: The difference between the maximum and minimum values, which is 913456778889 - 100.

- Interquartile Range (IQR): Calculate Q1, Q3, and then Q3 - Q1.

Now, let's compare the center and variability of the two groups.

**Comparisons:**

- **Mean:** Compare the means of the two groups. The group with the higher mean has, on average, higher scores.

- **Median:** Compare the medians. The median is less sensitive to extreme values, so it might give a better representation of the typical score.

- **Range:** Compare the ranges. A smaller range indicates less variability in the data.

- **Interquartile Range (IQR):** Similar to the range, a smaller IQR indicates less variability

**Reasoning:**

- If the mean and median are close, it suggests a symmetric distribution. If not, the distribution may be skewed.

- A larger range and IQR suggest more variability in the data.

- Look at the shapes of the distributions. For example, if one group has a long tail, it may affect the mean more than the median.

Remember that the interpretation may vary based on the context of the data and the specific characteristics of the distributions.