Measures of Spread

Estimators of the Spread

The Standard Deviation (SD)

	It is difficult to understand why statisticians commonly limit their enquiries to averages and do not revel in more comprehensive views. Their souls seem as dull to the charm of variety as the of the native of one of our English countries, whose retrospect of Switzerland was that, if its mountains could be thrown into its lakes, two nuisances would be gotten rid of at once.
	Sir Francis Galton, 1822-1911

Statistics Joke: Three statisticians were duck hunting. A duck took off from the tall grass and flew close to the hunters. One fired his shotgun and missed to the left; the other fired and missed to the right. The third said: "We got him!"

The standard deviation SD is defined as

SD = √ average of squared deviations from the average

SD = √

(x₁ - x)² + ... + (x_n - x)²

n
R uses a slightly different definition for the SD:

SD+ = √

(x₁ - x)² + ... + (x_n - x)²

n - 1
Most statisticians prefer SD+ for theoretical reasons; the argument goes like this:
because the true center of the dataset is unknown, you must estimate it using x, so the SD loses a little accuracy. Therefore divide by n-1 instead of n to compensate. (Dividing by n-1 instead of n makes SD⁺ a little larger than SD.)
If n is large, SD and SD+ are very close in value.
The SD describes about how far an observation tends to be away the sample average.
Example 1: What is the SD of these observations?
1. 20 10 15 15
2. Average = (20 + 10 + 15 + 15) / 4 = 60 / 4 = 15
3. Deviations from Average = 5 -5 0 0
4. Average of Squared Deviations = (5² + (-5)² + 0² + 0²) / 4 = 12.5
5. SD = √average of squared deviations = √12.5= 3.54
The SD gives a plus or minus estimate of a measurement. If we measure the width of a part 100 times and obtain a average measurement of 3.4544 cm, with an SD of 0.0037 cm, we can say that the width of the part is 3.4544 ± 0.0037 cm.
For a bell-shaped histogram, the SD is the best estimate of the spread.
Warning: Outliers can cause an inflated value of the SD. For this reason, SD and SD⁺ are not resistant statistics for estimating the spread of a histogram.
If the histogram of a dataset is bell-shaped, it can be described with only two summary statistics, the average which estimates the center, x and the SD, which estimates the spread.
The average and SD form a parsimonious summary of the data.
If a histogram is not bell-shaped, then the five statistics Q0, Q1, Q2, Q3 and Q4 may form a better summary.

MAD

You may think that the SD is overly complicated. Why do we need to square the deviations and then take the square root at the end? Why not just use the average deviation?

    AD = ((x₁ - x) + ... + (x_n - x)) / n

Reason: AD is always 0:

    AD = ((x₁ - x) + ... + (x_n - x)) / n

         = (x₁ + ... + x_n) / n - (x + ... + x) / n = x - n x / n = x - x = 0
This calculation shows that the positive deviations and the negative deviations always exactly cancel each other out.
A better possibility is the mean absolute deviation or MAD defined as

MAD = (|x₁ - x| + ... + |x_n - x|) / n
From 1914 to 1920, the statistician R. A. Fisher debated with the astronomer Sir Eddington (both from England) over whether the SD or MAD was better for measuring spread. In 1920, Fisher showed that, if the data follows a normal histogram, the MAD is 14% less accurate than the SD. The variance (SD²) is also easier to manipulate algebraically than the MAD. However, if the data have even a small number of outliers, the MAD is better.
You can use R to compute the MAD for a data vector x like this: mad(x).
Reference: YouTube Video
https://www.youtube.com/watch?v=JfGQTSL93i8

Other Measures of Spread

We already know of two simple measures of spread: the interquartile range: IQR = Q3 - Q1, and the range: Q4 - Q1. Statisticians prefer to use IQR because the range is suceptable to the influence of outliers. As a rough rule of thumb, the range ≈ 3 × SD.