To complete this assignment, I'll work through each part using the provided weights of the football team. Here are the weights listed again for reference:
218, 234, 210, 261, 265, 175, 191, 308, 253, 295, 253, 220, 234, 232, 189, 208, 190, 202, 159, 304, 275, 305, 308, 295
Let's calculate each part step by step:
Part 1: Calculate the Mean of the Weights
The mean is calculated by summing all the data points and dividing by the number of observations.
Calculation:
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Sum of Weights: \[ 218 + 234 + 210 + 261 + 265 + 175 + 191 + 308 + 253 + 295 + 253 + 220 + 234 + 232 + 189 + 208 + 190 + 202 + 159 + 304 + 275 + 305 + 308 + 295 = 6340 \]
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Number of Observations (n): 24
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Mean: \[ \text{Mean} = \frac{\text{Sum of Weights}}{n} = \frac{6340}{24} \approx 264.17 \]
Part 2: What is the Median of the Weights?
The median is the middle value when the data set is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle values.
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Sorted Weights: 159, 175, 189, 190, 191, 202, 208, 210, 218, 220, 232, 234, 234, 253, 253, 261, 265, 275, 295, 295, 304, 305, 308, 308
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Finding the Median:
- Since there are 24 values (an even number), the median is the average of the 12th and 13th values.
- 12th value = 234, 13th value = 253
\[ \text{Median} = \frac{234 + 253}{2} = \frac{487}{2} = 243.5 \]
Part 3: Identify the Mode of the Weights
The mode is the value that appears most frequently in the data set.
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Frequency Count:
- 159: 1
- 175: 1
- 189: 1
- 190: 1
- 191: 1
- 202: 1
- 208: 1
- 210: 1
- 218: 1
- 220: 1
- 232: 1
- 234: 2
- 253: 2
- 261: 1
- 265: 1
- 275: 1
- 295: 2
- 304: 1
- 305: 1
- 308: 2
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Modes:
- 234, 253, 295, and 308 (all appear 2 times)
Part 4: Calculate the Range of the Weights
The range is calculated as the difference between the maximum and minimum values.
- Minimum Weight: 159
- Maximum Weight: 308
Range Calculation: \[ \text{Range} = \text{Maximum} - \text{Minimum} = 308 - 159 = 149 \]
Part 5: Calculate the Variance of the Data
Variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \] where \(x_i\) represents each value, \(\mu\) is the mean, and \(n\) is the number of observations.
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Mean (μ): \(264.17\)
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Calculating Sum of Squared Deviations: \[ \sum (x_i - \mu)^2 = (218 - 264.17)^2 + (234 - 264.17)^2 + ... + (295 - 264.17)^2 \] \[ \approx 2083.77 + 910.88 + 2903.77 + ... + 931.77 = 18464.33 \]
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Variance Calculation: \[ \text{Variance} = \frac{18464.33}{24} \approx 769.35 \]
Part 6: Calculate the Standard Deviation of the Data
Standard deviation is simply the square root of the variance.
\[ \sigma \approx \sqrt{769.35} \approx 27.7 \]
Conclusion
Here is the summary of calculations:
- Mean: Approximately 264.17
- Median: 243.5
- Mode(s): 234, 253, 295, 308
- Range: 149
- Variance: Approximately 769.35
- Standard Deviation: Approximately 27.7
Make sure to properly format these calculations in your word processing document for submission. If you have any questions or need further assistance, feel free to ask!