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The sample mean gives a somewhat crude ("one-number summary") measure of the aggregate size of the data in a data set. The variance on the other hand, measures how "spread out" are the data on either side of their mean.
Find the mean, variance and standard deviation of this data set: -1,-1,0,1,2,3,3
Here are 4 scatterplots comparing data sets each of size n = 100 and all having mean = 5. The standard deviations are 1, 5, 10, and 100 respectively. Note that the higher the standard deviation, the more "spread out" are the data.
Let X be a random variable. Recall that the expected value, E(X), plays the role for the (theoretical) probability distribution of X that the sample mean plays for a relative frequency distribution. Similarly, there is a quantity called the variance associated with a probability distribution.
Calculate the variance of the Binomial distribution with n = 5, p = 1/2.
The probability that an observation of a random variable will lie between
E(X) - c and E(X) + c is at least
A probability distribution has a mean of 10 and a standard deviation of 2. Use the Chebyshev inequality to estimate the probability that an observation will lie between 2 and 18 (inclusive).
This scatter plot is of 1000 observations of a probability distribution with mean 4 and standard deviation 2. According to chebyshev's inequality, the probability that an observation will lie between 0 and 8 should be at least 0.75, i.e., at least 75% of the observations should lie in that range. In fact, all but about 20 + 10 + 2 + 2 = 34, i.e, about 97% of the data, lie in that range.