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 MAD = mean absolute deviation

  = average of absolute deviations from the median

the bigger MAD, the further is your typical x_i from the median ® more spread out

data 1 = {1, 3, 5} Q2 = 3

data 2 = {2, 3, 4} Q2 = 3

Certainly data 1 is more spread out than data 2.

Note: MAD is not resistant because data in the 'tails' will affect the average value. (I suppose you could look at median absolute deviation: that would be a resistant measure of spread.)

This is a very popular measure of spread. You have to be careful using it if your data are skewed because it is sensitive to outliers.

Start with a list {x₁,…x_n}

Construct the mean

Note: both s and s² will be large if the observations are widely spread out from the mean.

Example:

data 1 = {1, 3, 5}

data 2 = {2, 3, 4}

• Sometimes when we compute

or s² or other numerical values in statistics we will make mistakes because the calculator (say) can only remember a fixed number of digits after the decimal point. This is called round off error.

• For our course, don't worry about this

• Some people get caught up in trying to be too precise

• For example, say I want the mean of {0, 1, 1}. It is .66 or .667 or .66667 etc.

• Don't get too carried away because the original didn't have any decimal points. Let's not get confused with 'false' precision.

 Computing Formula For Variance

(takes fewer additions/multiplications to compute

Maris = {8, 13, 14, 16, 23, 26, 28, 33, 39, 61}

 Units of Measurement

 Note that is in the same units as all the {x₁,…,x_n} but s² is not (it is in units squared) also s is in the same units as x_i.

What happens if you make a linear transformation of a list of data.

Start with {x₁, x₂…, x_n} Transform each x_i to y_i = a+bx_i Where a and b are constant numbers.

so à scaled by a factor b²

also s_y = bs_x à scaled by a factor of proportionality b

Try it out with a simple data set {x₁, x₂, x₃} = (-1, 0, 1)