# Mean and standard deviation problem

• snoggerT
In summary: Since you are asked about the mean and standard deviation of a transformed r.v., I'm going to assume you have been exposed to this concept.So, in summary, the mean and standard deviation of a transformed random variable x are -11 and 4 respectively.
snoggerT
The mean and standard deviation of a random variable x are -11 and 4 respectively. Find the mean and standard deviation of the given random variables:

1) y=x+7

2) v=8x

3) w=8x+7

2. Homework Equations : E(x) = u, E(ax+b) = aE(x)+b

## The Attempt at a Solution

I've gotten the standard deviations for these problems, though I'm still not sure why they are what they are. I can't figure out how to find the mean though. Can somebody please explain all of this to me? thanks.

If you know E(X) = -11, what do your relevant equaitons tell you about E(X + 7)?

LCKurtz said:
If you know E(X) = -11, what do your relevant equaitons tell you about E(X + 7)?

- I get it. Can you explain the standard deviation portion for me? for instance, when v=8x, the standard deviation is 8*4, but I'm not sure why. Is x the standard deviation in the equation?

When you're dealing with the mean of a transformed random variable X, you can use expectation, E(...), to find the mean of the transformed variable. When you're dealing with the standard deviation, you need something else, Var(...), or variance of a random variable. As you probably know, the variance is the square of the standard deviation, or equivalently, the standard deviation is the square root of the variance.

Since you are asked about the mean and standard deviation of a transformed r.v., I'm going to assume you have been exposed to this concept.

In my book on mathematical statistics, there is a theorem that says:
Let X be a random variable and let a and b be constants. Define Y = aX + b. Then
Var(Y) = a2Var(X)​

Elsewhere in my text Var(X) is defined as E( (X - mu)2 ), which turns out to be equal to E(X2) - mu2.

In the problem, V = 8X, what would be Var(V)? Further, what would be the standard deviation of V?

## 1. What is the difference between mean and standard deviation?

The mean, also known as average, is a measure of central tendency that represents the sum of all values in a data set divided by the number of values. It gives an overall idea of the data set. On the other hand, the standard deviation is a measure of spread or variability in the data set, indicating how much the data points deviate from the mean. It provides information about the consistency or variability of the data set.

## 2. How do I calculate the mean and standard deviation?

To calculate the mean, add all the values in the data set and divide by the number of values. For example, if the data set is 5, 7, 9, 10, the mean would be (5+7+9+10)/4 = 7.75. To calculate the standard deviation, first calculate the mean. Then, for each value in the data set, subtract the mean from the value, square the result, and add all the squared values. Divide this sum by the number of values and take the square root of the result. This will give you the standard deviation.

## 3. Why is standard deviation important?

Standard deviation is important because it provides information about the variability in a data set. A small standard deviation indicates that the data points are close to the mean, while a large standard deviation indicates that the data points are spread out from the mean. This information is useful in understanding the consistency or variability of the data set and making comparisons between different data sets.

## 4. How is standard deviation used in data analysis?

Standard deviation is used in data analysis to measure the spread or variability of a data set. It is often used in conjunction with the mean to summarize and describe a data set. It is also used to identify outliers, which are data points that are significantly different from the rest of the data. Additionally, standard deviation is used in statistical tests to determine the significance of differences between groups or variables.

## 5. Can standard deviation be negative?

No, standard deviation cannot be negative. Since it is a measure of variability, it is always a positive value. If the standard deviation is close to zero, it means that the data points are close to the mean, while a larger standard deviation indicates that the data points are more spread out.

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