Stats 3.2 measures of dispersion

There are two solutions to this

We could find the mean of the absolute values of the deviations about the mean Or We could find the mean of the squared deviations because squaring a nonzero number always results in a positive number.

The remaining notes are handwritten

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Numerical measures for describing the dispersion or spread

1. The range 2. Standard deviation 3. Variance Section 3.4 talks about interquartile range (IQR), another measure of dispersion.

This result follows from the fact that observations greater than the mean are offset by observations less than the mean.

Because this sum is 0, we can't use the average deviation about the mean as a measure of spread.

Central tendency describe the typical value of a variable. Dispersion is the degree to which the data are spread out.

Central tendency isn't sufficient in describing a distribution.

For a population, the deviation about the mean for the kth observation is xi-u.

For a sample, the deviation about the mean for the kth observation is xi-x.

See computational formula on page 132

Page 133 sample of sd formulas on notebook paper

The range, R, of a variable is the difference between the largest and the smallest data value. That is

Range = R = largest data value - smallest data value.

Measures of dispersion are meant to describe how spread out data are. They describe how far each observation is from the typical value.

Standard deviation is based on the deviation about the mean.

The population standard deviation of a variable is the square root of the sum of squared deviations 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 deviations about the population mean.

The range is affected by extreme values in the data set, so the range isn't resistant.

The range is computed using only 2 values in the data set (the largest and the smallest). Standard deviation uses all the data values in the computations.

First approach leads to a measure of dispersion called the mean absolute deviation (MAD) (problem 42)

The second approach leads to variance. The problem with variance is that squaring the deviations about the mean leads to squared units of measure which is difficult to interpret.

Recall |a| = a if a < or equal to 0, and |a| = -a if a < 0 so |3| = 3 and |-3| = 3.

The sum of all deviations about the mean must equal zero.

The further an observation is from the mean, the larger the absolute value of deviation.

The sum of all deviations about the mean must equal zero. That is Summation(xi-u)= 0 and Summation(xi-x)=0

The population standard deviation is symbolically represented by (see notes) lowercase Greek sigma.

Where x1, x2, ...., xN are the N observations in the population and u is the population mean. See formula on page 132