Estimating Mean from a Histogram
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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.
Give your answer to $1$ d.p.
This question includes visual content: A frequency density histogram graph. The x-axis is labeled 'Energy output (kWh)' and ranges from 0 to 9 with major grid lines every 1 unit. The y-axis is labeled 'Frequency density' and ranges from 0 to 5 with major grid lines every 1 unit. There are four bars: 1) From x=0 to x=3 with height y=1. 2) From x=3 to x=5 with height y=2. 3) From x=5 to x=6 with height y=4. 4) From x=6 to x=8 with height y=2.
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In this problem, we are given a histogram showing the daily energy output of a solar panel. We need to calculate an estimate for the mean daily energy output, rounded to one decimal place.
Calculating Mean from a Histogram
To find the mean, we first need to determine the frequency for each interval. Remember that in a histogram, the area of each bar represents the frequency. The formula is frequency equals frequency density multiplied by class width.
Let's build a frequency table by looking at the bars in our histogram.
Step 1: Frequency Table
| Energy (kWh) | Width | Freq. Density | Frequency |
|---|---|---|---|
| 0 - 3 | 3 | 1.0 | 3 |
| 3 - 5 | 2 | 2.0 | 4 |
| 5 - 6 | 1 | 4.0 | 4 |
| 6 - 8 | 2 | 2.0 | 4 |
For the first bar from zero to three, the width is three and density is one, giving a frequency of three. The second bar from three to five has a width of two and density of two, so the frequency is four. The third bar from five to six has a width of one and density of four, giving a frequency of four. Finally, from six to eight, the width is two and density is two, so the frequency is four.
Since we only have intervals, we use the midpoint of each class to estimate the total energy output. Let's calculate the midpoints and then multiply them by their respective frequencies.
Step 2: Midpoints and $f \times x$
| Energy ($x$) | Midpoint ($m$) | Freq. ($f$) | $f \times m$ |
|---|---|---|---|
| 0 - 3 | 1.5 | 3 | 4.5 |
| 3 - 5 | 4.0 | 4 | 16.0 |
| 5 - 6 | 5.5 | 4 | 22.0 |
| 6 - 8 | 7.0 | 4 | 28.0 |
The midpoint is the average of the lower and upper bounds. For example, zero plus three divided by two is one point five. Multiplying midpoint by frequency gives us the estimated total for each category.
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