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To calculate the mean daily energy output from the histogram, we need to estimate the midpoints of each interval of the energy output (kWh), multiply by the frequency for that interval, and then divide by the total frequency. However, because the histogram shows frequency density instead of raw frequency, we must first calculate the frequency for each bar.

The bars in the histogram show ranges of energy output (kWh) on the x-axis, and frequency density on the y-axis.

### Steps to solve:
1. **Identify the intervals** of energy output (kWh).
2. **Calculate the width** of each interval.
3. **Multiply the frequency density by the width** of each interval to obtain the frequency for that interval.
4. **Estimate the midpoint** of each interval.
5. **Multiply the midpoint by the frequency** to get the contribution to the total energy output for that interval.
6. **Sum the total energy output and total frequency**.
7. **Divide the total energy output by the total frequency** to find the mean.

Let's go step by step:

- **1st bar:** Interval: 0-2 kWh, Frequency density: 1. Width = 2. Frequency = \( 1 \times 2 = 2 \). Midpoint = 1.
- **2nd bar:** Interval: 2-4 kWh, Frequency density: 2. Width = 2. Frequency = \( 2 \times 2 = 4 \). Midpoint = 3.
- **3rd bar:** Interval: 4-6 kWh, Frequency density: 1.5. Width = 2. Frequency = \( 1.5 \times 2 = 3 \). Midpoint = 5.
- **4th bar:** Interval: 6-8 kWh, Frequency density: 2.5. Width = 2. Frequency = \( 2.5 \times 2 = 5 \). Midpoint = 7.

### Calculate total energy output:
\[
\text{Total energy} = (1 \times 2) + (3 \times 4) + (5 \times 3) + (7 \times 5) = 2 + 12 + 15 + 35 = 64
\]

### Calculate total frequency:
\[
\text{Total frequency} = 2 + 4 + 3 + 5 = 14
\]

### Calculate the mean energy output:
\[
\text{Mean} = \frac{64}{14} = 4.57 \text{ kWh (to 1 decimal place)}
\]

The mean daily energy output is approximately **4.6 kWh**.

Would you like more details or have any questions?

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Here are 5 follow-up questions:
1. What does frequency density represent in a histogram?
2. How would you adjust your approach if the intervals had different widths?
3. What is the importance of calculating the midpoint of each interval?
4. Can you think of other methods to estimate the mean from a histogram?
5. How would the method change if this was a cumulative frequency graph?

**Tip:** Always ensure that you carefully check the scales on both the x-axis and y-axis when working with histograms!

The histogram below shows information about the daily energy output of a solar panel for a number of days. Calculate an estimate for the mean daily energy output. If your answer is a decimal, give it to 1 decimal place.

Calculate the estimated mean daily energy output of a solar panel using a histogram. This problem covers key statistical concepts, such as frequency density, midpoints, and the formula for mean. Suitable for students in Grades 9-11.

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