# Finding the Standard Deviation from Probability

• Samwise_geegee
In summary: From there, the rest is just algebra and integration.In summary, the question asks for the standard deviation (σ) of a random variable X, given that P(X≤500)=.5 and P(X>650)=.0227. The solution requires assuming a form of probability distribution and setting up the necessary integrals to solve for σ.
Samwise_geegee

## Homework Statement

For a certain random variable X, P(X≤500)=.5 and P(X>650)=.0227, find σ.

## Homework Equations

μ=expected value=mean

Variance=∫(X-μ)2fx(X)dx evaluated from -∞ to ∞

σ=√Variance

## The Attempt at a Solution

I'm not sure what the relationships between the standard deviation and the probabilities given are.
My only guess is that P(X≤500)=.5 also fits the definition for the median of this function but I'm not sure where the median fits into the standard deviation either. Any help is appreciated!

Samwise_geegee said:

## Homework Statement

For a certain random variable X, P(X≤500)=.5 and P(X>650)=.0227, find σ.

## Homework Equations

μ=expected value=mean

Variance=∫(X-μ)2fx(X)dx evaluated from -∞ to ∞

σ=√Variance

## The Attempt at a Solution

I'm not sure what the relationships between the standard deviation and the probabilities given are.
My only guess is that P(X≤500)=.5 also fits the definition for the median of this function but I'm not sure where the median fits into the standard deviation either. Any help is appreciated!

The question as written does not allow for a unique solution. You need to assume a form of probability distribution, such as Normal or Poisson or Gamma or ... . I suggest you try it for the case of normally-distributed X.

So you have a 50% chance of X being less than or equal to 500, and a 2.27% chance of it being greater than 650. That means there's a 47.73% chance for 500 < X <= 650. Now, knowing this you should be able to divide your integral into three separate integrals, as you now know the PDF values for each region.

## 1. What is the formula for finding the standard deviation from probability?

The formula for finding the standard deviation from probability is the square root of the variance, which is calculated by subtracting the mean of the probability distribution from each value and squaring the result, then dividing the sum of those squared values by the total number of values.

## 2. How is the standard deviation related to probability?

The standard deviation measures the spread or variability of a probability distribution. It tells us how much the values in a distribution differ from the mean, and therefore gives us an idea of how likely it is for a particular value to occur.

## 3. Can the standard deviation be negative?

No, the standard deviation cannot be negative. It is always a positive value because it is the square root of the variance, which is calculated by squaring the differences between values and the mean.

## 4. What does a high or low standard deviation indicate about a probability distribution?

A high standard deviation indicates that the values in the distribution are spread out from the mean, while a low standard deviation indicates that the values are closer to the mean. This can tell us how much uncertainty there is in the data and how likely it is for a particular value to occur.

## 5. How is the standard deviation used in statistical analysis?

The standard deviation is used in statistical analysis to measure the variability of a data set, to compare different data sets, and to make predictions about the likelihood of future events. It is an important tool for understanding and interpreting data, and is often used in conjunction with other statistical measures.

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