The mean GPA is 2.35.
The variance is 1.2115.
The standard deviation is approximately 1.104.
Probability Distribution
Here's the probability distribution table:
GPA Number of Students Probability
4.0 (A) 3 0.15
3.0 (B) 7 0.35
2.0 (C) 5 0.25
1.0 (D) 4 0.20
0.0 (F) 1 0.05
Mean
To calculate the mean (average GPA), we need to multiply each GPA by its probability and sum the products:
Mean = Σ (GPA * Probability)
Mean = (4.0 * 0.15) + (3.0 * 0.35) + (2.0 * 0.25) + (1.0 * 0.20) + (0.0 * 0.05)
Mean = 0.6 + 1.05 + 0.5 + 0.2 + 0
Mean = 2.35
Therefore, the mean GPA is 2.35.
Variance
To calculate the variance, we need to follow these steps:
Calculate the squared deviation from the mean for each GPA.
Multiply each squared deviation by its probability.
Sum the products obtained in step 2.
Variance = Σ [(GPA - Mean)^2 * Probability]
Variance = [(4.0 - 2.35)^2 * 0.15] + [(3.0 - 2.35)^2 * 0.35] + [(2.0 - 2.35)^2 * 0.25] + [(1.0 - 2.35)^2 * 0.20] + [(0.0 - 2.35)^2 * 0.05]
Variance = (2.7025 * 0.15) + (0.4225 * 0.35) + (0.1225 * 0.25) + (1.7689 * 0.20) + (5.4756 * 0.05)
Variance = 0.4054 + 0.1479 + 0.0306 + 0.3538 + 0.2738
Variance = 1.2115
Therefore, the variance is 1.2115.
Standard Deviation
Finally, to find the standard deviation, we need to take the square root of the variance:
Standard Deviation = √Variance
Standard Deviation = √1.2115
Standard Deviation ≈ 1.104
Therefore, the standard deviation is approximately 1.104.