Let's start with the data given:
Weights (in pounds): \[ 218, 234, 210, 261, 265, 175, 191, 308, 253, 295, 253, 220, 234, 232, 189, 208, 190, 202, 159, 304, 275, 305, 308, 295 \]
Part 1: Calculate the mean of the weights.
To find the mean, sum all weights and divide by the number of weights.
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Sum of weights: \[ 218 + 234 + 210 + 261 + 265 + 175 + 191 + 308 + 253 + 295 + 253 + 220 + 234 + 232 + 189 + 208 + 190 + 202 + 159 + 304 + 275 + 305 + 308 + 295 = 6095 \]
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Number of weights: There are 24 weights.
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Mean: \[ \text{Mean} = \frac{\text{Sum of weights}}{\text{Number of weights}} = \frac{6095}{24} \approx 254.79 \]
Part 2: What is the median of the weights?
To find the median, arrange the weights in ascending order and find the middle value:
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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 weights (an even number), the median is the average of the 12th and 13th values in the sorted list.
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.
- From the sorted weights:
- 234 appears 2 times
- 253 appears 2 times
Since both 234 and 253 appear twice and are the most frequent, the data is bimodal.
Thus, the modes are 234 and 253.
Part 4: Calculate the range of the weights.
The range is the difference between the maximum and minimum values in the dataset.
- Maximum weight: 308
- Minimum weight: 159
\[ \text{Range} = \text{Maximum} - \text{Minimum} = 308 - 159 = 149 \]
Part 5: Calculate the variance of the data.
Variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \] where \(x_i\) are the data points, \(\mu\) is the mean, and \(N\) is the number of data points.
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Find the deviations from the mean: \[ \mu \approx 254.79 \]
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Calculate each squared deviation: \[ (218-254.79)^2, (234-254.79)^2, (210-254.79)^2, \ldots, (295-254.79)^2 \] Calculating each deviation:
- \( (218 - 254.79)^2 \approx 1310.5841 \)
- \( (234 - 254.79)^2 \approx 430.9441 \)
- \( (210 - 254.79)^2 \approx 1986.4641 \)
- \( (261 - 254.79)^2 \approx 38.0641 \)
- \( (265 - 254.79)^2 \approx 102.4641 \)
- \( (175 - 254.79)^2 \approx 6296.9441 \)
- \( (191 - 254.79)^2 \approx 4041.0641 \)
- \( (308 - 254.79)^2 \approx 2850.6241 \)
- \( (253 - 254.79)^2 \approx 3.2041 \)
- \( (295 - 254.79)^2 \approx 1603.4641 \)
- \( (253 - 254.79)^2 \approx 3.2041 \)
- \( (220 - 254.79)^2 \approx 1197.0641 \)
- \( (234 - 254.79)^2 \approx 430.9441 \)
- \( (232 - 254.79)^2 \approx 525.0641 \)
- \( (189 - 254.79)^2 \approx 4396.4641 \)
- \( (208 - 254.79)^2 \approx 2209.0641 \)
- \( (190 - 254.79)^2 \approx 4134.6241 \)
- \( (202 - 254.79)^2 \approx 2793.6241 \)
- \( (159 - 254.79)^2 \approx 9226.9441 \)
- \( (304 - 254.79)^2 \approx 2453.1841 \)
- \( (275 - 254.79)^2 \approx 408.0641 \)
- \( (305 - 254.79)^2 \approx 2500.0641 \)
- \( (308 - 254.79)^2 \approx 2850.6241 \)
- \( (295 - 254.79)^2 \approx 1603.4641 \)
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Total of squared deviations: \[ \text{Total} \approx 19581.25 \]
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Variance: \[ \sigma^2 = \frac{19581.25}{24} \approx 815.094 \]
Part 6: Calculate the standard deviation of the data.
Standard deviation is the square root of the variance.
\[ \sigma \approx \sqrt{815.094} \approx 28.57 \]
Finally, here are all calculated values for reference:
- Mean: \( \approx 254.79 \)
- Median: \( 243.5 \)
- Mode: \( 234, 253 \)
- Range: \( 149 \)
- Variance: \( \approx 815.094 \)
- Standard Deviation: \( \approx 28.57 \)