Question

This assignment will assess your ability to calculate the mean, median, mode, variance, and standard deviation of a set of data. The weights of the starting lineup of a football team are as follows (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 Open a new word processing document and complete the following. Be sure to show all work. Part 1: Calculate the mean of the weights. Part 2: What is the median of the weights? Part 3: Identify the mode of the weights. Part 4: Calculate the range of the weights. Part 5: Calculate the variance of the data. Part 6: Calculate the standard deviation of the data.

Answer 1

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
1. Mean: **254.17**
2. Median: **253**
3. Mode: **253, 234, 295, 308**
4. Range: **149**
5. Variance: **1104.2**
6. Standard Deviation: **33.23**

Feel free to format this information into a word processing document as required for your assignment.