The Sample Variance and Standard Deviation
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.
- Definition
- Motivation of the definition
- Numerical Example
Find the mean, variance and standard deviation of this data set: -1,-1,0,1,2,3,3
- Calculation on the TI-83
- Comparisons of histograms
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.
- Variations for population and tabular data
The Variance and Standard Deviation of a Random Variable
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.
- Definition
- Numerical Example
Calculate the variance of the Binomial distribution with n = 5, p = 1/2.
Chebyshev's Inequality
- Statement
The probability that an observation of a random variable will lie between E(X) - c and E(X) + c is at least
- Example
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).
- Data example
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.