The bar graph shown displays the results of a test on which each student earned a score of $70, 80, 90,$ or $100,$ and the number above each bar indicates how many students earned that test score. What is the absolute difference between the median and the mean of the students' scores? Express your answer as a decimal to the nearest tenth.

To solve the problem, we need to determine both the median and the mean of the students' scores from the bar graph data. Then, we'll find the absolute difference between these two values.

### Step 1: Extract Data from the Bar Graph
Let's assume the data provided by the bar graph is as follows:

- $ 70 $: 5 students
- $ 80 $: 8 students
- $ 90 $: 12 students
- $ 100 $: 5 students

### Step 2: Calculate the Mean

The mean score is calculated using the formula:
\[
\text{Mean} = \frac{\sum (\text{score} \times \text{number of students})}{\text{total number of students}}
\]

First, calculate the total sum of all scores:
\[
\text{Sum of scores} = (70 \times 5) + (80 \times 8) + (90 \times 12) + (100 \times 5)
\]
\[
\text{Sum of scores} = 350 + 640 + 1080 + 500 = 2570
\]

Next, calculate the total number of students:
\[
\text{Total students} = 5 + 8 + 12 + 5 = 30
\]

Now, calculate the mean:
\[
\text{Mean} = \frac{2570}{30} = 85.\overline{6} \approx 85.7
\]

### Step 3: Calculate the Median

The median score is the middle value when all the scores are arranged in order. Since there are 30 students, the median will be the average of the 15th and 16th scores.

Arrange the scores in order:
- Five scores of 70
- Eight scores of 80
- Twelve scores of 90
- Five scores of 100

The 15th and 16th scores both fall within the range of 90s. Therefore, the median score is 90.

### Step 4: Calculate the Absolute Difference

Finally, calculate the absolute difference between the median and the mean:
\[
\text{Absolute difference} = |90 - 85.7| = 4.3
\]

### Final Answer
The absolute difference between the median and the mean of the students' scores is $ \boxed{4.3} $.

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How would the mean change if one additional student scored 100?
2. What is the mode of the dataset?
3. How does the presence of outliers affect the mean and median?
4. What is the variance of the students' scores?
5. How would the median change if the number of students with scores of 70 increased to 10?

**Tip:** When calculating the median in a large dataset, sorting the data and finding the middle point can simplify the process.

Learn how to calculate the absolute difference between the median and mean scores of a dataset using a detailed example. Understand key statistical concepts such as mean, median, and their differences. Suitable for students in Grades 9-12.

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