| MadSci Network: General Biology |
The two most commonly used statistics for describing a distribution of
data are the mean (measure of central tendency - where the center of
the distribution is) and the standard deviation (measure of variability or
spread). In the image below are two sets of normal distributions - on the left
are three distributions with the same spread but different means, and on the right
are three distributions with the same mean but different standard deviations or spread.
The standard deviation is a measure of how spread out a random variable is. As an example, consider rolling a six sided die where the numbers 1, 2, 3, 4, 5, and 6 are all equally likely so that each has a 1/6 chance of coming up. Then the mean (also called expected value) of this random variable is (1/6)(1) + (1/6)(2) + (1/6)(3) + (1/6)(4) + (1/6)(5) + (1/6)(6) = 3 1/2.
If I had a 100 sided die, then the mean would come out to be 50 1/2. However, when I roll the 100 sided die, the numbers are likely to be quite a ways away from 50 1/2. They are likely to be much farther away than the numbers coming from a 6 sided die are from 3 1/2. One way to measure this spread is by using standard deviation.
Standard deviation is defined as follows. It is the square root of the mean of the squared deviations from the mean of the random variable. Let's break that apart one piece at a time. The deviation from the mean is the difference from the mean. So if I rolled a 5 on the six sided die, the deviation is 5 - 3 1/2 = 1 1/2. If I rolled a 1, the deviation would be 1 - 3 1 /2 = -2 1/2. That's why we square the deviations, so that it is always positive. Altogether, the mean of the square of the deviations for the six sided die is
(1/6)(1 - 3 1/2)^2 + (1/6)(2 - 3 1/2)^2 + (1/6)(3 - 3 1/2)^2 + (1/6)(4 - 3 1/2)^2 + (1/6)(5 - 3 1/2)^2 + (1/6)(6 - 3 1/2)^2 = 35/12.
The standard deviation is the square root of 35/12, or about 1.72. If you compute the standard deviation for the hundred sided die, it comes out to be 28.9, which is much larger, indicating that this random variable is more spread out.
Now, statisticians are often presented with data presumed to come from some random distribution. For instance, the data may be the numbers
3, 7, 7, 10, 10, 11
Suppose that we wish to estimate the standard deviation of the data. Then first we estimate the mean, a. In this case the estimate for the mean is (3 + 7 + 7 + 10 + 10 + 11) / 6 = 8. Then we use the formula
s^2 = [(x_1 - a)^2 + ... (x_k - a)^2 ] / (k - 1)
where k is the number of data elements. For our data
s^2 = [(3 - 8)^2 + (7 - 8)^2 + (7 - 8)^2 + (10 - 8)^2 + (10 - 8)^2 + (11 - 8)^2] / 5 = 8 4/5
This is an estimate for s^2, to find the estimate s for the standard deviation, we just take the square root, so s is about 2.97. To find out more about probability and statistics concepts, check out http://www.britannica.com (search for "probability" or "statistics") for a high level view. The site http://nilesonline.com/stats/ gives a very basic introduction to statistics intended for journalists. Finally, http://euclid.math.fsu.edu/Science/stats.html contains several links to statistical resources.
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