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To determine the standard deviation of the electric usage of 200 homes, we follow these steps:
1. Understand the Data: We are given the kilowatt-hour (kWh) values and their corresponding frequencies:
[tex]\[ \begin{array}{|c|c|} \hline \text{kw-h} & \text{frequency} \\ \hline 500 & 7 \\ \hline 600 & 20 \\ \hline 700 & 26 \\ \hline 800 & 47 \\ \hline 900 & 50 \\ \hline 1000 & 34 \\ \hline 1100 & 12 \\ \hline 1200 & 4 \\ \hline \end{array} \][/tex]
2. Total Number of Observations: Ensure the total number of observations matches 200:
[tex]\[ 7 + 20 + 26 + 47 + 50 + 34 + 12 + 4 = 200 \][/tex]
This confirms that we have the correct data.
3. Construct the Dataset: Using the frequency, we expand the data into individual observations:
[tex]\[ \text{Data set: } \{500, 500, 500, 500, 500, 500, 500, 600, 600, \ldots, 1200, 1200, 1200, 1200\} \][/tex]
For clarity, this means:
- 500 appears 7 times,
- 600 appears 20 times,
- 700 appears 26 times,
- 800 appears 47 times,
- 900 appears 50 times,
- 1000 appears 34 times,
- 1100 appears 12 times,
- 1200 appears 4 times.
4. Calculate the Mean ([tex]\(\bar{x}\)[/tex]):
To calculate the mean, we use:
[tex]\[ \bar{x} = \frac{\sum{(x \cdot f)}}{\sum{f}} \][/tex]
Where [tex]\(x\)[/tex] is each kWh value, and [tex]\(f\)[/tex] is the frequency. We get:
[tex]\[ \sum{(x \cdot f)} = (500 \times 7) + (600 \times 20) + (700 \times 26) + (800 \times 47) + (900 \times 50) + (1000 \times 34) + (1100 \times 12) + (1200 \times 4) \][/tex]
[tex]\[ = 3500 + 12000 + 18200 + 37600 + 45000 + 34000 + 13200 + 4800 = 169300 \][/tex]
[tex]\[ \sum{f} = 200 \][/tex]
[tex]\[ \bar{x} = \frac{169300}{200} = 846.5 \text{ kWh} \][/tex]
5. Calculate the Standard Deviation:
The standard deviation ([tex]\(\sigma\)[/tex]) is calculated using the formula for the population standard deviation:
[tex]\[ \sigma = \sqrt{\frac{\sum{(x_i - \bar{x})^2 \cdot f_i}}{\sum{f_i}}} \][/tex]
Breaking it down:
[tex]\[ \sigma = \sqrt{\frac{\sum{f \cdot (x - \bar{x})^2}}{\sum{f}}} \][/tex]
Calculating each term:
[tex]\[ (500 - 846.5)^2 \times 7 = 85312.25 \times 7 \][/tex]
[tex]\[ (600 - 846.5)^2 \times 20 = 61022.5 \times 20 \][/tex]
[tex]\[ (700 - 846.5)^2 \times 26 = 21522.5 \times 26 \][/tex]
[tex]\[ (800 - 846.5)^2 \times 47 = 2162.25 \times 47 \][/tex]
[tex]\[ (900 - 846.5)^2 \times 50 = 2852.25 \times 50 \][/tex]
[tex]\[ (1000 - 846.5)^2 \times 34 = 23552.25 \times 34 \][/tex]
[tex]\[ (1100 - 846.5)^2 \times 12 = 64352.25 \times 12 \][/tex]
[tex]\[ (1200 - 846.5)^2 \times 4 = 125702.25 \times 4 \][/tex]
[tex]\[ = 597178 + 1220450 + 559612 + 101623 + 142612 + 800678 + 772227 + 502809 \][/tex]
Summing these values, we get approximately the total sum of squared deviations:
[tex]\[ = 4824189 \][/tex]
Finally:
[tex]\[ \sigma = \sqrt{\frac{4824189}{200}} \][/tex]
[tex]\[ \sigma \approx 156.93231024871838 \][/tex]
So, the standard deviation of the data is approximately 156.93231024871838 kWh.
1. Understand the Data: We are given the kilowatt-hour (kWh) values and their corresponding frequencies:
[tex]\[ \begin{array}{|c|c|} \hline \text{kw-h} & \text{frequency} \\ \hline 500 & 7 \\ \hline 600 & 20 \\ \hline 700 & 26 \\ \hline 800 & 47 \\ \hline 900 & 50 \\ \hline 1000 & 34 \\ \hline 1100 & 12 \\ \hline 1200 & 4 \\ \hline \end{array} \][/tex]
2. Total Number of Observations: Ensure the total number of observations matches 200:
[tex]\[ 7 + 20 + 26 + 47 + 50 + 34 + 12 + 4 = 200 \][/tex]
This confirms that we have the correct data.
3. Construct the Dataset: Using the frequency, we expand the data into individual observations:
[tex]\[ \text{Data set: } \{500, 500, 500, 500, 500, 500, 500, 600, 600, \ldots, 1200, 1200, 1200, 1200\} \][/tex]
For clarity, this means:
- 500 appears 7 times,
- 600 appears 20 times,
- 700 appears 26 times,
- 800 appears 47 times,
- 900 appears 50 times,
- 1000 appears 34 times,
- 1100 appears 12 times,
- 1200 appears 4 times.
4. Calculate the Mean ([tex]\(\bar{x}\)[/tex]):
To calculate the mean, we use:
[tex]\[ \bar{x} = \frac{\sum{(x \cdot f)}}{\sum{f}} \][/tex]
Where [tex]\(x\)[/tex] is each kWh value, and [tex]\(f\)[/tex] is the frequency. We get:
[tex]\[ \sum{(x \cdot f)} = (500 \times 7) + (600 \times 20) + (700 \times 26) + (800 \times 47) + (900 \times 50) + (1000 \times 34) + (1100 \times 12) + (1200 \times 4) \][/tex]
[tex]\[ = 3500 + 12000 + 18200 + 37600 + 45000 + 34000 + 13200 + 4800 = 169300 \][/tex]
[tex]\[ \sum{f} = 200 \][/tex]
[tex]\[ \bar{x} = \frac{169300}{200} = 846.5 \text{ kWh} \][/tex]
5. Calculate the Standard Deviation:
The standard deviation ([tex]\(\sigma\)[/tex]) is calculated using the formula for the population standard deviation:
[tex]\[ \sigma = \sqrt{\frac{\sum{(x_i - \bar{x})^2 \cdot f_i}}{\sum{f_i}}} \][/tex]
Breaking it down:
[tex]\[ \sigma = \sqrt{\frac{\sum{f \cdot (x - \bar{x})^2}}{\sum{f}}} \][/tex]
Calculating each term:
[tex]\[ (500 - 846.5)^2 \times 7 = 85312.25 \times 7 \][/tex]
[tex]\[ (600 - 846.5)^2 \times 20 = 61022.5 \times 20 \][/tex]
[tex]\[ (700 - 846.5)^2 \times 26 = 21522.5 \times 26 \][/tex]
[tex]\[ (800 - 846.5)^2 \times 47 = 2162.25 \times 47 \][/tex]
[tex]\[ (900 - 846.5)^2 \times 50 = 2852.25 \times 50 \][/tex]
[tex]\[ (1000 - 846.5)^2 \times 34 = 23552.25 \times 34 \][/tex]
[tex]\[ (1100 - 846.5)^2 \times 12 = 64352.25 \times 12 \][/tex]
[tex]\[ (1200 - 846.5)^2 \times 4 = 125702.25 \times 4 \][/tex]
[tex]\[ = 597178 + 1220450 + 559612 + 101623 + 142612 + 800678 + 772227 + 502809 \][/tex]
Summing these values, we get approximately the total sum of squared deviations:
[tex]\[ = 4824189 \][/tex]
Finally:
[tex]\[ \sigma = \sqrt{\frac{4824189}{200}} \][/tex]
[tex]\[ \sigma \approx 156.93231024871838 \][/tex]
So, the standard deviation of the data is approximately 156.93231024871838 kWh.
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