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To estimate the mean height of the sunflowers, we will use the midpoints of each height class, multiply them by the corresponding frequency, and then divide the sum of these products by the total frequency. Here's the process:

1. **Determine the midpoints** of each height class:
   - For the class \( 90 < h \leq 100 \), the midpoint is \( \frac{90 + 100}{2} = 95 \).
   - For \( 100 < h \leq 110 \), the midpoint is \( \frac{100 + 110}{2} = 105 \).
   - For \( 110 < h \leq 120 \), the midpoint is \( \frac{110 + 120}{2} = 115 \).
   - For \( 120 < h \leq 130 \), the midpoint is \( \frac{120 + 130}{2} = 125 \).
   - For \( 130 < h \leq 140 \), the midpoint is \( \frac{130 + 140}{2} = 135 \).

2. **Multiply each midpoint by its frequency**:
   - \( 95 \times 8 = 760 \)
   - \( 105 \times 12 = 1260 \)
   - \( 115 \times 15 = 1725 \)
   - \( 125 \times 10 = 1250 \)
   - \( 135 \times 3 = 405 \)

3. **Sum the products**:
   \[
   760 + 1260 + 1725 + 1250 + 405 = 5400
   \]

4. **Find the total frequency** (sum of the frequencies):
   \[
   8 + 12 + 15 + 10 + 3 = 48
   \]

5. **Estimate the mean** by dividing the sum of the products by the total frequency:
   \[
   \text{Mean height} = \frac{5400}{48} = 112.5 \, \text{cm}
   \]

Thus, the estimated mean height of the sunflowers is **112.5 cm**.

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How would the mean height change if the frequency for the tallest group increased?
2. How can you calculate the median height from a grouped frequency table like this?
3. What is the range of the sunflower heights based on the given data?
4. How would you estimate the mode of the sunflower heights?
5. Can you explain why the midpoint of each class is used in this estimation?

**Tip**: Always double-check the frequencies and midpoints when estimating from grouped data to avoid errors in calculation.

Work out an estimate for the mean height of the sunflowers. The table shows information about the heights, in cm, of 48 sunflowers in a garden centre.

Learn how to estimate the mean height of sunflowers using a grouped frequency table. This problem involves calculating midpoints for each class and finding the mean using the formula for grouped data. Suitable for students in Grades 8-10.

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