Variability

The mean tells us where a histogram balances. But in almost every histogram we have seen, the values spread out on both sides of the mean. How far from the mean can they be? To answer this question, we will develop a measure of variability about the mean.

Variance

The mean squared deviation

• while it does not give us an idea of spread, it is not the same scale as the original variables as its units are the square of the original

Standard Deviation

The SD of a list is defined as the root mean square of deviations from average.

Measures how far the numbers on the list are from their average

Working with SD

#calculating average

np.mean(nba13.column(‘Height’))

79.06

About how far off are the player’s heights from the average? This is a measure by the SD of the heights, which Is about 3.45 inches

np.std(nba13.column(‘Height’))

3.45

The towering center Hasheem Thabeet of the Oklahoma City Thunder was the tallest player at a height of 87 inches.

Thabeet was about 8 inches above the average height

7.934

Thats a deviation from average, and it is about 2.3 times the SD:

(87- mean_height)/ sd_height

2.29

The height of the tallest player was about 2.3 SDs above average

What we have observed is that the tallest and shortest players were both just a few SDs away from the average height. This is an example of why the SD is a useful measure of spread. No matter what the shape of the histogram, the average and the SD together tell you a lot about where the histogram is situated on the number line.

Chebychev’s Bounds

Chebychev’s result gives a lower bound, not exact answer in an approximation. For example, the percent of entries in the range “average +- 2 SDs” might be quite a but larger than 75%. But it cannot be smaller”

Standard Units

The quantity z measures standard units, the number of standard deviations above average.

example: united

we will create a new column called delay(Standard unites) by applying he function standard_units to the column f delay times. This allows us to see all the delay times in minutes as well as their corresponding values in standard units

united = united.with_column(‘Delay(Standard Units), standard_units(united.column (‘Delay’))

united