Statistics Chapter 3.2 Measures of Dispersion
Dispersion
degree to which data are spread out
range (R)
difference between largest and smallest data value
using chebychevs inequality when asked to find a %
just used 1-1/k^2 (100) to find particular number
sample standard deviation ( s ) Sx
of a variable is the square root of the sum of squared deviations about the sample mean divided by n-1, where n is the sample size
using chebychev inequality: when given 2 particular variable:
reverse the equation k - μ/ σ then plug resulting value into 1-1/k^2 (100%) to find percentage
the larger the standard deviation the more ______
the *larger* the standard deviation, the *more dispersion* the distribution has
using chebychevs inequality when asked to find actual times or anything besides a percentage:
used mean and standard deviations to find answer (1- 1/k^2) lie between (μ - kσ and μ + kσ)
For chebychevs inequality, what percentage of observations lie within (1/2^2)(100%) that is *2 standard deviation*?
*at least 75% lie within k=2 standard deviation of the mean*
For chebychevs inequality, what percentage of observations lie within (1/2^3)(100%) that is *3 standard deviation*?
*at least 88.9% lie within k=3 standard deviation of the mean*
Standard deviation is based on?
*deviation about the mean*
Chebyshev's Inequality
-used for any shape distribution -for any data at least (1- 1/k^2)(100%) of the observations lie within k standard deviations of the mean -where k is any number greater than 1 (1- 1/k^2) lie between (μ - kσ and μ + kσ)
standard deviation
-uses all the data values in the computation - numerical value used to indicate how widely individuals in a group vary -measure of how spread out numbers are
population standard deviation ( σ ) on calculator (σX)
-variable is the square root of the sum of squared individuals about the population mean divided by the number of observations in the population, N -That is the square root of the mean of the squared deviation sabout the population mean
3 numerical measures that describe dispersion aka spread measures of dispersion
1.range 2.standard deviation 3.variance
empirical rule
used only with *bell-shaped* distributions -68% of the data will lie within 1 standard deviation of the mean (μ - 1σ and μ +1σ) (34%/34%) -95% of the data will lie within 2 standard deviations of the mean (μ - 2σ and μ +2σ) (13.5%, 13.5%) -99.7% of the data will lie within 3 standard deviations of the mean (μ - 3σ and μ +3σ) (2.35%/ 2.35%) very small part will be one half 0.15% and other half 0.15%
variance
variance of a variable is the square of the standard deviation the *population variance* is σ^2 the *sample variance* is s^2 the units of measure in variance are squared values. so if the value is measured in dollars then the variance is measured in dollars squared
when do you use the empirical rule
when the shape of the distribution is *bell-shaped*
when will you use chebychev?
when the shape of the distribution is *unknown*
