Difference between standard deviation and kurtosis

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Standard deviation and kurtosis measure different aspects of data distribution. Standard deviation indicates the spread of data points around the mean, while kurtosis assesses the shape of the distribution, particularly its tails and peak compared to a normal distribution. Kurtosis is calculated as the fourth moment about the mean divided by the square of the variance, often adjusted by subtracting three. A positive kurtosis indicates a sharper peak and longer tails, while a negative kurtosis suggests a flatter peak. Understanding these distinctions clarifies their unique roles in statistical analysis.
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I'm new to studying statistics but it seems to me like standard deviation and kurtosis measure the same thing.
The higher the standard deviation, the more spread out the data is, while the lower the kurtosis the more spread out the data is.

I'm sure I'm wrong about this so can someone help me out with what I'm missing?
 
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Kurtosis measures something quite distinct from the variance. The kurtosis is the fourth moment about the mean divided by the square of the variance, less three. (Not everyone subtracts 3 from the result). Regardless of the variance, all normal distributions have a kurtosis of 0 (or 3 depending on how you define kurtosis).

One way to look at kurtosis is that it measures the width of a random distribution compared to that of a normal distribution with the same mean and variance as the distribution in question. If the distribution is symmetric about the mean, a distribution with a positive kurtosis will have a sharper peak and much longer tails than does the normal distribution. The opposite is the case for distributions with a negative kurtosis.
 
Ok i understand now. I think I was looking at figures incorrectly. Thank you.
 
The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...

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