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Range is a measure of spread that measures the difference between the maximum and minimum values of the data set.
The interquartile range, or IQR, of a data set is a measure of spread that measures the difference between Q_3 and Q_1, the upper and lower quartiles.
IQR=Q_3-Q_1
The following applet shows how to find the IQR of different data sets.
The mean absolute deviation (MAD) is a measure of the spread of a data set that measures how much the data elements differ from the mean. The mean absolute deviation is the average distance between each data value and the mean.
MAD = |x_1-x|+|x_2-x|+⋯+|x_n-x|/n
The variance is a measure of spread of a set of data that measures how much the data elements deviate from the mean. Mathematically, the variance is the average of the squares of the difference between each data value x_i and the mean x. (x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n The variance is the square of the standard deviation σ, so it is usually denoted as σ ^2. σ = sqrt((x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n) ⇕ σ ^2 = (x_1 - x)^2 + (x_2 - x)^2 + ⋯ + (x_n - x)^2/n The applet below calculates the variance in the data set on the number line. Points can be moved to change the data.
For each data value, x - x can now be calculated and added to a table. This shows how much each data point varies from the mean.
x | x - x |
---|---|
1 | 1 - 5 = - 4 |
5 | 5 - 5= 0 |
3 | 3 - 5 = - 2 |
4 | 4 - 5 = - 1 |
5 | 5 - 5 = 0 |
12 | 12 - 5 = 7 |
Square the deviations, and add them to a new column in the table.
x | x - x | (x - x)^2 |
---|---|---|
1 | - 4 | (- 4)^2 = 16 |
5 | 0 | 0^2 = 0 |
3 | -2 | (- 2)^2 = 4 |
4 | - 1 | (- 1)^2 = 1 |
5 | 0 | 0^2 = 0 |
12 | 7 | 7^2 = 49 |
Finally, take the square root of the just found quotient to get the standard deviation. Here, the fraction is used instead of the quotient, to avoid rounding errors. σ = sqrt(70/6) ≈ 3.4 Thus, a randomly chosen value from this data set is expected to deviate roughly 3.4 units from the mean.