Consider the following data:
x 5
6
7
8
9
P(X=x)
0.2
0.1
0.1
0.2
0.4
Step 3 of 5 : Find the standard deviation. Round your answer to one decimal place.
To find the standard deviation, we first need to calculate the mean.
Mean (µ) = Σ [x * P(X=x)]
µ = (5*0.2) + (6*0.1) + (7*0.1) + (8*0.2) + (9*0.4)
µ = 1 + 0.6 + 0.7 + 1.6 + 3.6
µ = 7.5
Next, we calculate the variance:
Variance = Σ [(x-µ)^2 * P(X=x)]
Variance = [(5-7.5)^2 * 0.2] + [(6-7.5)^2 * 0.1] + [(7-7.5)^2 * 0.1] + [(8-7.5)^2 * 0.2] + [(9-7.5)^2 * 0.4]
Variance = [(-2.5)^2 * 0.2] + [(-1.5)^2 * 0.1] + [(-0.5)^2 * 0.1] + [(0.5)^2 * 0.2] + [(1.5)^2 * 0.4]
Variance = (6.25 * 0.2) + (2.25 * 0.1) + (0.25 * 0.1) + (0.25 * 0.2) + (2.25 * 0.4)
Variance = 1.25 + 0.225 + 0.025 + 0.05 + 0.9
Variance = 2.45
Finally, we calculate the standard deviation:
Standard Deviation = √Variance
Standard Deviation = √2.45
Standard Deviation ≈ 1.57
Therefore, the standard deviation of the given data is approximately 1.6 when rounded to one decimal place.