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What is the difference between measures of center and measures of variability?
To put it simple, try to capture the nonrandomness of a random variable or population. , on the other hand, try to capture how this process varies.The intuitive notion is quite straightforward so I will give some examples of each.
This answer , has a good explanation of measures of center.
The most common measures of variability are basically of two types:

Range, interquartile range and similar . The range is simply ##"Largest value  lowest value"## of the population, pay attention that the range doesn't make sense in case of a distribution, as you already know their lowest and maximum value, it is only use in empirical data to have a sense of how far apart your data points are spread. but it doesn't take into account how densely packed the data or distribution is . There is interquartile range , Interdecile range , and others measures are that similar to the range.

Variance and standard deviation The variance measures how spread out is the data, it does however take into account 2 things: The farther from the mean the point is, the larger it contributes to the variance and it takes into account every point and the probability of it happening. Its formula is: ##E(XE(X))^2##, the expected value ponder the probability of each event happening and the square "punishes" how father apart the value is from the mean. If you are more into the "practical side of statistics" the sample variance may be easier for you to understand the inner workings of the formula, the sample variance is: ##1/(n1)*sum_(i=1)^n(y_i(sum_(k=1)^ny_k)/n)##. The standard deviation is simply the square root of the variance.
There are other measures of variance , some of them used in specific context and/or in more advanced analysis