Question

1. teams in the national football league are divided into two conferences, the american football conference (afc) and the national football conference (nfc). the table below shows the margin of victory in each conference’s championship game for the 1990 - 2004 seasons. (make a box - and - whisker plot for each conference.)

afc championship margins of victory
48, 3, 19, 17, 4, 4, 14, 3, 13, 19, 13, 7, 17, 10, 14
mean median mode range standard deviation
min q1 median q3 max
nfc championship margins of victory
2, 31, 10, 17, 10, 11, 17, 13, 3, 5, 41, 5, 17, 11, 17
mean median mode range standard deviation
min q1 median q3 max

Explanation:

Step1: Organize AFC data

First, sort the AFC data: \(3, 3, 4, 4, 7, 10, 13, 13, 14, 14, 17, 17, 19, 19, 48\) (15 values).

Step2: Calculate AFC Mean

Sum the values: \(3 + 3 + 4 + 4 + 7 + 10 + 13 + 13 + 14 + 14 + 17 + 17 + 19 + 19 + 48 = 213\).
Mean: \(\frac{213}{15} = 14.2\).

Step3: Find AFC Median

Middle value (8th term) of sorted data: \(13\).

Step4: Determine AFC Mode

Most frequent values: \(3, 4, 13, 14, 17, 19\) (each appears twice).

Step5: Compute AFC Range

Max - Min: \(48 - 3 = 45\).

Step6: Calculate AFC Standard Deviation

1. Find deviations: \(x_i - \bar{x}\) for each \(x_i\).
2. Square deviations: \((x_i - \bar{x})^2\).
3. Sum squared deviations: \( (3 - 14.2)^2 + (3 - 14.2)^2 + \dots + (48 - 14.2)^2 = 1332.4\).
4. Variance: \(\frac{1332.4}{15} \approx 88.83\).
5. Standard Deviation: \(\sqrt{88.83} \approx 9.43\).

Step7: Identify AFC Five - Number Summary

Min: \(3\), Q1 (25th percentile, 4th term): \(4\), Median: \(13\), Q3 (75th percentile, 12th term): \(17\), Max: \(48\).

Step8: Organize NFC data

Sort NFC data: \(2, 3, 5, 5, 10, 10, 11, 11, 13, 17, 17, 17, 17, 31, 41\) (15 values).

Step9: Calculate NFC Mean

Sum the values: \(2 + 3 + 5 + 5 + 10 + 10 + 11 + 11 + 13 + 17 + 17 + 17 + 17 + 31 + 41 = 209\).
Mean: \(\frac{209}{15} \approx 13.93\).

Step10: Find NFC Median

Middle value (8th term) of sorted data: \(11\).

Step11: Determine NFC Mode

Most frequent value: \(17\) (appears four times).

Step12: Compute NFC Range

Max - Min: \(41 - 2 = 39\).

Step13: Calculate NFC Standard Deviation

1. Find deviations: \(x_i - \bar{x}\) for each \(x_i\).
2. Square deviations: \((x_i - \bar{x})^2\).
3. Sum squared deviations: \( (2 - 13.93)^2 + (3 - 13.93)^2 + \dots + (41 - 13.93)^2 = 1066.93\).
4. Variance: \(\frac{1066.93}{15} \approx 71.13\).
5. Standard Deviation: \(\sqrt{71.13} \approx 8.43\).

Step14: Identify NFC Five - Number Summary

Min: \(2\), Q1 (4th term): \(5\), Median: \(11\), Q3 (12th term): \(17\), Max: \(41\).

AFC Summary:

Mean	Median	Mode	Range	Standard Deviation	Min	Q1	Median	Q3	Max

NFC Summary:

Mean	Median	Mode	Range	Standard Deviation	Min	Q1	Median	Q3	Max

Answer:

AFC:

• Mean: \(14.2\)
• Median: \(13\)
• Mode: \(3, 4, 13, 14, 17, 19\)
• Range: \(45\)
• Standard Deviation: \(\approx 9.43\)
• Five - Number Summary: Min \(3\), Q1 \(4\), Median \(13\), Q3 \(17\), Max \(48\)

NFC:

• Mean: \(\approx 13.93\)
• Median: \(11\)
• Mode: \(17\)
• Range: \(39\)
• Standard Deviation: \(\approx 8.43\)
• Five - Number Summary: Min \(2\), Q1 \(5\), Median \(11\), Q3 \(17\), Max \(41\)