Sure! Below are the calculations for mean, median, mode, variance, and standard deviation, as well as the sum of the weights provided.
Given Data:
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
Mean = (Sum of all data points) / (Number of data points)
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Sum of Weights: \[ 218 + 234 + 210 + 261 + 265 + 175 + 191 + 308 + 253 + 295 + 253 + 220 + 234 + 232 + 189 + 208 + 190 + 202 + 159 + 304 + 275 + 305 + 308 + 295 = 5844 \]
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Number of Data Points: 24
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Mean Calculation: \[ \text{Mean} = \frac{5844}{24} = 243.5 \]
Part 2: What is the Median?
To find the median, we need to order the data from least to greatest and find the middle value.
Ordered 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.
Number of Data Points: 24 (even), so the median is the average of the 12th and 13th values.
Median Calculation: \[ \text{12th value} = 234, \text{13th value} = 253 \] \[ \text{Median} = \frac{234 + 253}{2} = \frac{487}{2} = 243.5 \]
Part 3: Identify the Mode
The mode is the number that appears most frequently in the data set.
From the data:
- 234: 2 times
- 253: 2 times
- All other weights appear only once.
Modes: 234 and 253 (Bimodal).
Part 4: Calculate the Range
Range = (Maximum value) - (Minimum value)
- Maximum = 308
- Minimum = 159
\[ \text{Range} = 308 - 159 = 149 \]
Part 5: Calculate the Variance
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Mean = 243.5 (from Part 1).
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Calculate the squared deviations from the mean: \[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \] Where \(x_i\) are the weights and \(N\) is the number of weights.
Calculating each value squared deviation: \[ (218 - 243.5)^2 = 650.25 \ (234 - 243.5)^2 = 90.25 \ (210 - 243.5)^2 = 1130.25 \ (261 - 243.5)^2 = 306.25 \ (265 - 243.5)^2 = 462.25 \ (175 - 243.5)^2 = 4628.25 \ (191 - 243.5)^2 = 2760.25 \ (308 - 243.5)^2 = 4190.25 \ (253 - 243.5)^2 = 90.25 \ (295 - 243.5)^2 = 2656.25 \ (253 - 243.5)^2 = 90.25 \ (220 - 243.5)^2 = 552.25 \ (234 - 243.5)^2 = 90.25 \ (232 - 243.5)^2 = 132.25 \ (189 - 243.5)^2 = 3020.25 \ (208 - 243.5)^2 = 1240.25 \ (190 - 243.5)^2 = 2880.25 \ (202 - 243.5)^2 = 1690.25 \ (159 - 243.5)^2 = 7116.25 \ (304 - 243.5)^2 = 3660.25 \ (275 - 243.5)^2 = 992.25 \ (305 - 243.5)^2 = 3840.25 \ (308 - 243.5)^2 = 4190.25 \ (295 - 243.5)^2 = 2656.25 \]
Total of squared deviations: \[ 650.25 + 90.25 + 1130.25 + 306.25 + 462.25 + 4628.25 + 2760.25 + 4190.25 + 90.25 + 2656.25 + 90.25 + 552.25 + 90.25 + 132.25 + 3020.25 + 1240.25 + 2880.25 + 1690.25 + 7116.25 + 3660.25 + 992.25 + 3840.25 + 4190.25 + 2656.25 = 30783 \]
Now, divide by the number of data points: \[ \text{Variance} = \frac{30783}{24} \approx 1282.625 \]
Part 6: Calculate the Standard Deviation
Standard deviation is the square root of the variance. \[ \text{Standard Deviation} = \sqrt{1282.625} \approx 35.8 \]
Summary of Results:
- Sum of Weights: 5844
- Mean: 243.5
- Median: 243.5
- Mode: 234, 253
- Range: 149
- Variance: 1282.63
- Standard Deviation: 35.8
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