# Mean and standard deviation for linear combinations

• engineer23
In summary, the given problem involves finding the mean and standard deviation for a data set in X, with a given mean of 100 and standard deviation of 10. To find the mean and standard deviation for a new data set in Y, represented by 2(Yi-5)/10 + 7, you need to know the relationship between X and Y. If the expression 2(Yi - 5)/10 + 7 represents Xi, then the mean ybar can be found by multiplying the mean xbar by 5 and subtracting 30, and the standard deviation Sy can be found by multiplying the standard deviation Sx by 5.
engineer23

## Homework Statement

Data set in X with mean xbar = 100 and standard deviation Sx = 10

Find ybar and Sy for 2(Yi-5)/10 + 7

## The Attempt at a Solution

All the problems I have seen are in the form yi = axi + b in which case the mean ybar = a(xbar) + b and Sy = aSx

What does Yi represent exactly? How does it relate to xi?

I don't see how you can find ybar and Sy unless you know the relationship between X and Y.

You show an expression 2(Yi - 5)/10 + 7 = (Yi - 5)/5 + 7. Could this be Xi?

If that's the case, then you can solve for Yi.

Xi = (Yi - 5)/5 + 7
==> Xi - 7 = (Yi - 5)/5
==> 5(Xi - 7) = (Yi - 5)
==> 5(Xi - 7) + 5 = Yi
or
Yi = 5Xi - 30

## 1. What is a linear combination?

A linear combination is a mathematical operation in which two or more variables are multiplied by constants and then added together. It is commonly used in statistics to calculate the mean and standard deviation of a set of data.

## 2. How is the mean calculated for a linear combination?

The mean for a linear combination is calculated by multiplying each variable by its corresponding constant, adding all the products together, and then dividing by the total number of values in the data set.

## 3. How is the standard deviation calculated for a linear combination?

The standard deviation for a linear combination is calculated by first finding the variance of the data set, which is the average of the squared differences from the mean. Then, the square root of the variance is taken to find the standard deviation.

## 4. Can the mean and standard deviation be calculated for any type of data set?

Yes, the mean and standard deviation can be calculated for any type of data set, as long as the data follows a linear relationship and there are no extreme outliers that could skew the results.

## 5. How are the mean and standard deviation used in data analysis?

The mean and standard deviation are important measures of central tendency and variability in a data set. They can be used to summarize and compare different data sets, identify patterns and trends, and make predictions based on the data.

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