For the distribution of Exercise 1, find the variance, the...

For the distribution of Exercise 1, find the variance, the standard deviation, the expected absolute deviation, and the range. Also, find these same values under the assumption that the largest value of $X$ is 12 rather than 6 . In light of your results, comment on the relative merit of the different measures of dispersion.

Key Concepts

Variance

Variance is a numerical measure of how far individual data points in a distribution are spread out from the mean. It is calculated by averaging the squared deviations of each data point from the mean, and it provides a sense of the overall dispersion within the data set.

Standard Deviation

The standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the original data. It indicates the typical distance of the data points from the mean, making it more interpretable compared to the variance.

Expected Absolute Deviation

The expected absolute deviation, often known as the mean absolute deviation, is the average of the absolute differences between each data point and the mean. Unlike variance, it does not square the deviations, making it less sensitive to extreme values or outliers.

Range

The range is the simplest measure of dispersion, defined as the difference between the maximum and minimum values in the data set. It provides a quick sense of the spread of the data but can be highly influenced by any extreme values present.

Measures of Dispersion

Measures of dispersion are statistical tools that assess the variability or spread within a data set. They are crucial for understanding the degree of variability in relation to the mean and help in comparing the consistency of different distributions, while also highlighting their sensitivity to outliers.

Transcript

00:01 In this problem, we're looking at the number of words per pages on a dictionary.

00:09 And we want to know the standard deviation among pages and see if this standard deviation, this variance can help us guess if we could estimate the number of words in the whole book.

00:26 So we have a sample of 1, 2, 3, 4, 5.

00:31 Six, seven, eight, nine, ten.

00:36 We have a sample of ten pages.

00:39 So that's pages and that's the number of word per pages.

00:44 So in order to get the standard deviation, first we must get the mean.

00:48 Now this you should probably have done in the previous chapter of the book.

00:53 So i'm going to go very quick over it.

00:55 So the mean is one over n and being the number of pages so 10 times the sum of every value so it would be in this case one over 10 times 51 plus 63 plus blah blah plus 53 plus 79 so you sum all the number of pages in your sample and you divide by 10 and you will get 53 .3.

01:42 So i can you can do this with your calculator or a spreadsheet software like excel or whatever google's google sheet can easily get this and come from your result.

01:54 Now i'm due to get the standard deviation the formula is a bit more complicated but not too hard.

02:02 So you have one divided by n minus one times the sun the sum of every data point minus the average square and you take this square root of all of this.

02:24 So i'm going to expand a little bit to show you how it should look, but i'm going to let you do most of the calculation yourself because it simply plug in and use your calculator.

02:39 So you're going to have 1 over 9 because 10 minus 1 is 9 times...

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