Calculating Standard Deviation of Recoveries among Patients

To calculate the standard deviation of number recoveries among 48 patients when the probability of recovering is 0.75, we need to follow the below steps:

Step 1: Determine the mean of the distribution.
The mean of a binomial distribution is calculated as:
μ = n * p
Where,
n = number of trials
p = probability of success
In this case, n = 48 and p = 0.75.
Therefore, μ = 48 * 0.75 = 36.

Step 2: Calculate the variance of the distribution.
The variance of a binomial distribution is calculated as:
σ^2 = n * p * (1 - p)
In this case, n = 48 and p = 0.75.
Therefore,
σ^2 = 48 * 0.75 * (1 - 0.75) = 9

Step 3: Calculate the standard deviation of the distribution.
The standard deviation of a binomial distribution is calculated as the square root of the variance. Therefore,
σ = √(9) = 3.

Explanation

Standard deviation is a measure of how spread out the data is from the mean. In this case, we are calculating the standard deviation of the number of recoveries among 48 patients when the probability of recovering is 0.75. This means that out of 48 patients, we expect 36 to recover (mean).

To calculate the standard deviation, we first find the mean by multiplying the number of trials (48) by the probability of success (0.75). Next, we calculate the variance using the formula n * p * (1 - p). Finally, we calculate the standard deviation by taking the square root of the variance.

The standard deviation tells us that the number of recoveries among 48 patients is expected to be spread out by about 3 patients from the mean of 36. This means that we can expect to see anywhere from 33 to 39 patients recover (mean ± 1 standard deviation).

N=48,p=.75,q=.25
48*.75*.25=9
standard deviations=√9=3