Answer to Question #143519 in Statistics and Probability for Vladimyr Lubin

Question

Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results.
Listed below are the measured radiation absorption rates​ (in W/kg) corresponding to various cell phone models. If one of each model is measured for radiation and the results are used to find the measures of​ variation, are the results typical of the population of cell phones that are in​ use?
0.74,0.72,1.38,0.67,1.09,0.76,1.08,0.96,0.74,0.98,0.87

Answer 1

The range of a set of data is the difference between the highest and lowest values in the set.

$$range=1.38-0.67=0.71W/kg$$

The variance (σ²) is a measure of how far each value in the data set is from the mean. Here is how it is defined:

1. Subtract the mean from each value in the data. This gives you a measure of the distance of each value from the mean.
2. Square each of these distances (so that they are all positive values), and add all of the squares together.
3. Divide the sum of the squares by the number of values in the data set.

$mean=($ 0.74+0.72+1.38+0.67+1.09+0.76+1.08+0.96+0.74+0.98+0.87)/11=0.908

$variance(=((0.74-0.908)^2+(0.72-0.908)^2+(1.38-0.908)^2+(0.67-0.908)^2+(1.09-0.908)^2+(0.76-0.908)^2+(1.08-0.908)^2+(0.96-0.908)^2+(0.74-0.908)^2+(0.98-0.908)^2+(0.87-0.908)^2)/11=0.04228W^2/kg^2$

The standard deviation (σ) is simply the (positive) square root of the variance.

$\sigma= \sqrt{\smash[b]{variance}}=\sqrt{\smash[b]{0.04228}}=0.206W/kg$

If one of each model is measured for radiation and the results are used to find the measures of​ variation, are the results typical of the population of cell phones that are in​ use? -NO

The results for the measures of variation is not typical of the population of cell phones that are in use because some models of cell phones will have a longer market share than others. Measures from different models should be weighted according to their size in population