Variance of a weighted population

In summary, the conversation discussed the computation of the population mean and variance using the given equation. The equation for variance was defined as the expected value of (x-E{x})^2, and an alternative equation was also mentioned. It was suggested to compute the expected value of x^2 and subtract the square of the previously calculated mean to find the variance.
  • #1
rogo0034
37
0

Homework Statement


KEnHA.png



Homework Equations





The Attempt at a Solution



Was able to get the population mean of 5.3 with 4*.2 + 5*.4 + 6*.3 + 7*.1
But for some reason I'm drawing blank for the variance (which i guess is .81)
 
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  • #2
rogo0034 said:

Homework Statement


KEnHA.png

Homework Equations


The Attempt at a Solution



Was able to get the population mean of 5.3 with 4*.2 + 5*.4 + 6*.3 + 7*.1
But for some reason I'm drawing blank for the variance (which i guess is .81)

Variance is defined as:
[tex]var\{x\}=E\{(x-E\{x\})^2\}[/tex]
So if you wanted to directly apply the equation above, you would compute it the same way as mean, except instead of doing a weighted sum of x, you would do a weighted sum of (x-E{x})^2. So for each x, before doing the weighted sum, you would subtract off the mean you found already and square it.

But there is a popular reworking of the equation:
[tex]var\{x\}=E\{x^2-2xE\{x\}+E^2\{x\}\}=E\{x^2\}-E^2\{x\}[/tex]

So in this case, you would compute the expected value of x^2 (the same way you did the mean except use x^2 instead of x). Then, subtract the square of the expectation you already found.
 
  • #3
Got it! thanks, that was easier
 

1. What is the definition of variance of a weighted population?

The variance of a weighted population is a measure of the spread or variability of the data points in a population, taking into account the weights assigned to each data point. It is calculated by taking the sum of the squared differences between each data point and the weighted mean, divided by the sum of the weights.

2. How is the variance of a weighted population different from the variance of an unweighted population?

The variance of an unweighted population only considers the raw data values, while the variance of a weighted population takes into account the weights assigned to each data point. This means that data points with higher weights will have a greater impact on the overall variance.

3. What is the purpose of calculating the variance of a weighted population?

The variance of a weighted population is useful in situations where the data points have varying levels of importance or significance. By taking into account the weights, we can get a more accurate measure of the variability of the data and make more informed decisions.

4. Can the variance of a weighted population be negative?

No, the variance of a weighted population cannot be negative. This is because the squared differences between each data point and the weighted mean will always result in a positive value, and when divided by the sum of the weights, it will always result in a positive variance.

5. How can the variance of a weighted population be used in statistical analysis?

The variance of a weighted population can be used in various statistical analyses such as hypothesis testing and confidence interval calculations. It can also be used to compare the variability between different populations with different weights assigned to their data points.

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