Finding Linear Transform for <x>=5, SD=2 to Reach <y>=20, SD=4

In summary, to find the linear transform for a given set of data, you need to calculate the mean and standard deviation for both the input and output variables. Then, use the formula y = mx + b to determine the slope and y-intercept of the linear transformation. Changing the mean and standard deviation will directly affect the slope and y-intercept of the transform, and the transform can be used to predict values outside of the given data set, but the accuracy may vary.
  • #1
Ezekiel20
2
0
let <x> =5 and Standard Dev =2. Which linear tranform y=ax+b results in <y>=20 and standard dev=4?
 
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  • #2
You asked the same question in UK Physics forum, and I answered it there. (a=2, b=10)
 
  • #3


To find the linear transform that will result in <y>=20 and standard dev=4, we can use the formula y = ax + b, where a is the slope and b is the y-intercept.

First, we need to find the value of a. We know that the standard deviation is directly related to the slope (a) by the formula SD(y) = a * SD(x). Since we want the standard deviation of y to be 4, we can set up the equation as 4 = a * 2. Solving for a, we get a = 2.

Next, we need to find the value of b. We can use the fact that the mean remains the same after a linear transformation, so we can set up the equation 5 = 2 * 5 + b, since the mean of x is 5. Solving for b, we get b = -5.

Therefore, the linear transform y = 2x - 5 will result in <y>=20 and standard dev=4. This can also be verified by plugging in the values of <x>=5 and SD=2 into the formula y = 2x - 5, which will give us y = 10 - 5 = 5, which is the mean of y, and SD(y) = 2 * 2 = 4, which is the desired standard deviation.
 

1. How do you find the linear transform for a given set of data?

To find the linear transform for a given set of data, you need to first calculate the mean and standard deviation for both the input and output variables. Then, use the formula y = mx + b to determine the slope (m) and y-intercept (b) of the linear transformation. Once you have these values, you can plug them into the equation to transform any input value into an output value.

2. How do you calculate the mean and standard deviation for a set of data?

To calculate the mean of a set of data, add all of the values together and divide by the total number of values. To calculate the standard deviation, you first need to find the variance by subtracting each data point from the mean, squaring the differences, and then finding the average of these values. The standard deviation is then the square root of the variance.

3. What does =5 and =20 mean in the context of this problem?

In this context, =5 means that the mean of the input variable is 5, and =20 means that the mean of the output variable is 20. These values are used to determine the linear transform for the given data set.

4. How does changing the mean and standard deviation affect the linear transform?

The mean and standard deviation of the input and output variables will directly affect the slope and y-intercept of the linear transformation. A larger mean or standard deviation will result in a steeper slope, while a smaller mean or standard deviation will result in a flatter slope. The y-intercept will also shift based on these values.

5. Can the linear transform be used to predict values outside of the given data set?

Yes, the linear transform can be used to predict values outside of the given data set. However, the accuracy of these predictions may vary depending on the distribution of the data and the number of data points. It is important to note that the linear transform is an estimation and may not always be exact.

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