x P(x)
0 0.3
1 0.15
2 0.2
3 0.35
Find the standard deviation of this probability distribution. Give your answer to at least 2 decimal places
To find the standard deviation of a probability distribution, we will follow these steps:
Step 1: Calculate the expected value (mean):
- Multiply each value of x by its corresponding probability (P(x))
- Sum up the results
Expected value (mean) = (0 * 0.3) + (1 * 0.15) + (2 * 0.2) + (3 * 0.35) = 0 + 0.15 + 0.4 + 1.05 = 1.60
Step 2: Calculate the variance:
- Subtract the expected value from each value of x and square the result
- Multiply each squared difference by its corresponding probability (P(x))
- Sum up the results
Variance = [(0 - 1.60)^2 * 0.3] + [(1 - 1.60)^2 * 0.15] + [(2 - 1.60)^2 * 0.2] + [(3 - 1.60)^2 * 0.35]
= [(-1.60)^2 * 0.3] + [(-0.60)^2 * 0.15] + [0.40^2 * 0.2] + [1.40^2 * 0.35]
= [2.56 * 0.3] + [0.36 * 0.15] + [0.16 * 0.2] + [1.96 * 0.35]
= 0.768 + 0.054 + 0.032 + 0.686
= 1.54
Step 3: Calculate the standard deviation:
- Take the square root of the variance
Standard deviation = √(1.54) ≈ 1.24
Therefore, the standard deviation of this probability distribution is approximately 1.24, rounded to 2 decimal places.
To find the standard deviation of a probability distribution, you can follow these steps:
1. Calculate the expected value (mean) of the distribution.
2. Calculate the variance of the distribution.
3. Take the square root of the variance to find the standard deviation.
Step 1: Calculate the Mean
The expected value (mean) of a probability distribution can be found by multiplying each value of "x" by its corresponding probability and summing them up.
Mean = (0 × 0.3) + (1 × 0.15) + (2 × 0.2) + (3 × 0.35)
Mean = 0 + 0.15 + 0.4 + 1.05
Mean = 1.6
Step 2: Calculate the Variance
The variance of a probability distribution is the average squared difference between each value and the mean. You can calculate it using the formula:
Variance = (Σ (x - mean)² * P(x))
To calculate variance, we need to subtract the mean from each value of "x", square the result, multiply it by the corresponding probability, and sum them up.
Variance = ((0 - 1.6)² * 0.3) + ((1 - 1.6)² * 0.15) + ((2 - 1.6)² * 0.2) + ((3 - 1.6)² * 0.35)
Variance = (-1.6² * 0.3) + (-0.6² * 0.15) + (0.4² * 0.2) + (1.4² * 0.35)
Variance = (2.56 * 0.3) + (0.36 * 0.15) + (0.16 * 0.2) + (1.96 * 0.35)
Variance = 0.768 + 0.054 + 0.032 + 0.686
Variance = 1.54
Step 3: Calculate the Standard Deviation
The standard deviation is simply the square root of the variance.
Standard Deviation = √(Variance)
Standard Deviation = √(1.54)
Standard Deviation ≈ 1.24 (rounded to 2 decimal places)
So, the standard deviation of this probability distribution is approximately 1.24.
You want me to find the mean
then the difference of each from the mean
then square that
then add them all and divide by 3 or 4 depending on your class' definition
then take the square root.
No way - You do it.