To complete this assignment, we need to calculate various statistical measures based on the provided weights of a football team's starting lineup. Let’s proceed step by step.
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
To find the mean, we sum all the weights and divide by the total number of weights.
\[ \text{Mean} = \frac{\text{Sum of weights}}{\text{Number of weights}} \]
Step 1: Calculate the Sum of Weights
\[ 218 + 234 + 210 + 261 + 265 + 175 + 191 + 308 + 253 + 295 + 253 + 220 + 234 + 232 + 189 + 208 + 190 + 202 + 159 + 304 + 275 + 305 + 308 + 295 = 6100 \]
Step 2: Calculate the Number of Weights
Count the number of weight entries:
- 24 weights
Step 3: Calculate the Mean
\[ \text{Mean} = \frac{6100}{24} \approx 254.17 \]
Part 2: Calculate the Median
To find the median, we need to sort the weights and find the middle value(s).
Step 1: Sort the Weights
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
Step 2: Find the Middle Value(s)
Since there are 24 data points (even number), the median is the average of the 12th and 13th values.
\[ \text{Median} = \frac{253 + 253}{2} = 253 \]
Part 3: Identify the Mode
The mode is the number that appears most frequently.
From the sorted weights:
- 253 appears 2 times
- 234 appears 2 times
- 295 appears 2 times
- 308 appears 2 times
- All others appear 1 time
Thus, the modes are: \[ \text{Mode} = 253, 234, 295, 308 \]
Part 4: Calculate the Range
The range is calculated by subtracting the minimum value from the maximum value.
Step 1: Identify Minimum and Maximum Values
- Minimum = 159
- Maximum = 308
Step 2: Calculate the Range
\[ \text{Range} = 308 - 159 = 149 \]
Part 5: Calculate the Variance
Variance measures how far each number in the set is from the mean and hence from every other number in the set.
Step 1: Calculate Each Deviation from the Mean
\[ \text{deviation} = (x_i - \text{Mean})^2 \]
Where \( x_i \) is each weight.
Step 2: Calculate Squared Deviations
- (218 - 254.17)² = 1296.49
- (234 - 254.17)² = 409.73
- (210 - 254.17)² = 1923.58
- (261 - 254.17)² = 46.78
- (265 - 254.17)² = 117.65
- (175 - 254.17)² = 6248.65
- (191 - 254.17)² = 3970.53
- (308 - 254.17)² = 2919.43
- (253 - 254.17)² = 1.36
- (295 - 254.17)² = 1656.11
- (253 - 254.17)² = 1.36
- (220 - 254.17)² = 1166.68
- (234 - 254.17)² = 409.73
- (232 - 254.17)² = 484.35
- (189 - 254.17)² = 4201.79
- (208 - 254.17)² = 2113.77
- (190 - 254.17)² = 4089.76
- (202 - 254.17)² = 2706.91
- (159 - 254.17)² = 9015.78
- (304 - 254.17)² = 2467.71
- (275 - 254.17)² = 432.57
- (305 - 254.17)² = 2600.07
- (308 - 254.17)² = 2919.43
- (295 - 254.17)² = 1656.11
Step 3: Sum of Squared Deviations
\[ \text{Sum of Squared Deviations} = 1296.49 + 409.73 + 1923.58 + 46.78 + 117.65 + 6248.65 + 3970.53 + 2919.43 + 1.36 + 1656.11 + 1.36 + 1166.68 + 409.73 + 484.35 + 4201.79 + 2113.77 + 4089.76 + 2706.91 + 9015.78 + 2467.71 + 432.57 + 2600.07 + 2919.43 + 1656.11 \approx 25307.57 \]
Step 4: Calculate Variance
\[ \text{Variance} = \frac{\text{Sum of Squared Deviations}}{N - 1} = \frac{25307.57}{23} \approx 1104.2 \]
Part 6: Calculate the Standard Deviation
The standard deviation is the square root of the variance.
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1104.2} \approx 33.23 \]
Summary of Results
- Mean: 254.17
- Median: 253
- Mode: 253, 234, 295, 308
- Range: 149
- Variance: 1104.2
- Standard Deviation: 33.23
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