Find the arithmetic mean, standard deviation, and coefficient of variation from the following data. Also, find standard deviation of the expenditure at both cities, and determine w... Find the arithmetic mean, standard deviation, and coefficient of variation from the following data. Also, find standard deviation of the expenditure at both cities, and determine which team is more consistent in its game based on goals scored.
Understand the Problem
The question is asking for calculations related to statistical measures, including the arithmetic mean, standard deviation, and coefficient of variation from various datasets presented in a tabular format. It also involves determining the consistency of two teams based on their match scores.
Answer
Team A is more consistent in their game.
Answer for screen readers
Team A is more consistent in their game.
Steps to Solve
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Calculate the Arithmetic Mean for Team A and Team B
To find the mean, use the formula: $$ \text{Mean} = \frac{\sum (x \cdot f)}{N} $$ where $x$ is the number of goals, $f$ is the frequency (number of matches), and $N$ is the total number of matches played.
For Team A:
- Goals: 0, 1, 2
- Matches: 15, 10, 7
- Total Matches, $N_a = 15 + 10 + 7 = 32$
Calculate the mean: $$ \text{Mean}_A = \frac{(0 \cdot 15) + (1 \cdot 10) + (2 \cdot 7)}{32} = \frac{0 + 10 + 14}{32} = \frac{24}{32} = 0.75 $$
For Team B:
- Matches: 20, 10, 5
- Total Matches, $N_b = 20 + 10 + 5 = 35$
Calculate the mean: $$ \text{Mean}_B = \frac{(0 \cdot 20) + (1 \cdot 10) + (2 \cdot 5)}{35} = \frac{0 + 10 + 10}{35} = \frac{20}{35} \approx 0.57 $$
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Calculate the Variance for Team A and Team B
The variance is calculated using: $$ \sigma^2 = \frac{\sum (f \cdot (x - \text{Mean})^2)}{N} $$
For Team A:
- Calculate $(x - \text{Mean}_A)^2$:
- For 0 goals: $(0 - 0.75)^2 = 0.5625$
- For 1 goal: $(1 - 0.75)^2 = 0.0625$
- For 2 goals: $(2 - 0.75)^2 = 1.5625$
Calculate variance: $$ \sigma^2_A = \frac{15 \cdot 0.5625 + 10 \cdot 0.0625 + 7 \cdot 1.5625}{32} = \frac{8.4375 + 0.625 + 10.9375}{32} = \frac{10.625}{32} \approx 0.332 $$
For Team B:
- Calculate $(x - \text{Mean}_B)^2$:
- For 0 goals: $(0 - 0.57)^2 \approx 0.3249$
- For 1 goal: $(1 - 0.57)^2 \approx 0.1849$
- For 2 goals: $(2 - 0.57)^2 \approx 2.0569$
Calculate variance: $$ \sigma^2_B = \frac{20 \cdot 0.3249 + 10 \cdot 0.1849 + 5 \cdot 2.0569}{35} = \frac{6.498 + 1.849 + 10.2845}{35} = \frac{18.6315}{35} \approx 0.532 $$
- Calculate $(x - \text{Mean}_A)^2$:
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Calculate Standard Deviation for Team A and Team B
The standard deviation is the square root of the variance: $$ \sigma_A = \sqrt{\sigma^2_A} \quad \text{and} \quad \sigma_B = \sqrt{\sigma^2_B} $$
For Team A: $$ \sigma_A \approx \sqrt{0.332} \approx 0.576 $$
For Team B: $$ \sigma_B \approx \sqrt{0.532} \approx 0.729 $$
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Calculate Coefficient of Variation
Coefficient of Variation (CV) can be calculated using: $$ CV = \frac{\sigma}{\text{Mean}} \times 100% $$
For Team A: $$ CV_A = \frac{0.576}{0.75} \times 100% \approx 76.8% $$
For Team B: $$ CV_B = \frac{0.729}{0.57} \times 100% \approx 128.2% $$
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Determine Consistency of Teams
The team with the lower Coefficient of Variation is more consistent. Compare $CV_A$ and $CV_B$:
- Team A: 76.8%
- Team B: 128.2%
Therefore, Team A is more consistent.
Team A is more consistent in their game.
More Information
The consistency is determined by the coefficient of variation, a statistical measure of the relative variability. A lower CV indicates more consistency in performance across the matches.
Tips
- Forgetting to sum the frequency appropriately when calculating the mean.
- Incorrectly squaring the deviations when calculating variance.
- Not taking the square root of variance to find the standard deviation.
- Confusing absolute measures of variability with relative measures (like CV).
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