https://file.aiquickdraw.com/mathbot/file/1725744059290txidvuct.jpg

From this cumulative frequency curve, estimate the median, upper and lower quartiles, and interquartile range.

To estimate the median, upper quartile (Q3), lower quartile (Q1), and interquartile range (IQR) from the cumulative frequency curve, we can follow these steps:

### 1. **Locate the Total Frequency:**
   The maximum value on the y-axis represents the total frequency, which appears to be around 50 (based on the scale of the graph).

### 2. **Estimate the Median:**
   The median corresponds to the cumulative frequency at 50%, i.e., half the total frequency. Since the total frequency is 50, the median will be at the cumulative frequency of 25.

   - Find the point where the curve crosses the line at \( \text{Cumulative Frequency} = 25 \).
   - From this point, trace vertically down to the x-axis to estimate the corresponding time.
   
   Based on the image, the median appears to be approximately **30 hours**.

### 3. **Estimate the Lower Quartile (Q1):**
   The lower quartile is the point at 25% of the total frequency. Since the total frequency is 50, Q1 will be at the cumulative frequency of 12.5.

   - Locate \( \text{Cumulative Frequency} = 12.5 \) on the y-axis and trace horizontally to the curve.
   - From this point, trace vertically down to the x-axis.
   
   The lower quartile is approximately **20 hours**.

### 4. **Estimate the Upper Quartile (Q3):**
   The upper quartile is the point at 75% of the total frequency. Since the total frequency is 50, Q3 will be at the cumulative frequency of 37.5.

   - Locate \( \text{Cumulative Frequency} = 37.5 \) on the y-axis and trace horizontally to the curve.
   - From this point, trace vertically down to the x-axis.
   
   The upper quartile is approximately **42 hours**.

### 5. **Calculate the Interquartile Range (IQR):**
   The interquartile range is the difference between the upper and lower quartiles:

   \[
   IQR = Q3 - Q1 = 42 \, \text{hours} - 20 \, \text{hours} = 22 \, \text{hours}
   \]

### Summary:
- **Median**: 30 hours
- **Lower Quartile (Q1)**: 20 hours
- **Upper Quartile (Q3)**: 42 hours
- **Interquartile Range (IQR)**: 22 hours

Would you like more details or have any questions about this process?

Here are five related questions to explore further:
1. How is the cumulative frequency curve used to estimate percentiles?
2. How would you calculate the mean from a cumulative frequency graph?
3. Can cumulative frequency graphs be used for non-numeric data?
4. What is the difference between cumulative frequency and regular frequency distributions?
5. How does the shape of a cumulative frequency graph reflect data distribution?

**Tip**: The interquartile range (IQR) is useful in identifying outliers in a dataset. Values outside of 1.5 times the IQR from Q1 or Q3 are considered potential outliers.

Learn how to estimate the median, upper quartile (Q3), lower quartile (Q1), and interquartile range (IQR) from a cumulative frequency curve. This step-by-step solution shows how to locate the median at 30 hours, the lower quartile at 20 hours, the upper quartile at 42 hours, and the IQR at 22 hours. Suitable for Grades 9-12 students.

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