Measures of Spread, Variability and Standard Deviation
Different samples/populations contain different levels of variability that can be expressed with a value known as standard deviation. Standard deviation measures the spread of the numbers in a given data set from its mean value. This means that among the entire set of values, the average difference between each number is given to us by the standard deviation, s. This is an easy way for statisticians to explain how much variability there is within a set of values of random variable by organizing the data in such a way that there’s a standard for how different each individual is from the average.
Let’s take the IQ test for example. Say we have a group of 20 individuals, randomly selected from the population who take the IQ tests.
The individuals receive the following IQ scores:
97, 86, 90, 90, 95, 100, 100, 104, 102, 113, 93, 95, 88, 87, 89, 92, 98, 101, 100 , 100 If we were to arrange it in order: 86, 87, 88, 89, 90, 90, 92, 93, 95, 95, 97, 98, 100, 100, 100, 100, 101, 102, 104, 113
First, we have to take the mean, call it M, of the entire data set, which would be. M = Sum of values (X) / total number of values (n) = 1,920 / 20 = 96
With this, we get a sense of where the set of values tends to favor. Now, the differences between each score from the mean vary disproportionately. The point of using standard deviation is to take these values and make it so that the average difference is one set number.
In order to get the standard deviation, we need to calculate the variances, or how much each score deviates from the mean.
The way we do this is as follows:
We take the deviations of each score from the mean (X - M):
86-96 = -10
92-96 = -4
98- 96 = 2
100 - 96 = 4
102-96 = 6
and so on….
We then must square the deviations we get, and add them up. Squaring these values gets rid of any negative values. This way the sum of our variances won’t add up to 0, giving us zero variance in the sample (which isn't true).
We take the sum of the variances squared divided by the sample size (n-1).
Σ(X - M)2 / (n-1) =
875.999/(20-1) =
46.10526 = s2
This is called the sample squared variances. Now, we just take the square root of the sum squared variance, s2, to compute the sample standard deviation, s.
s = √s2 = √Σ(X-M)2/ (n-1) = √(46.10526) = 6.79
Summary of notations and formulas:
s2 = Σ(X - M)2
__________
n-1
s (standard deviation) = √s2 =
√Σ(X - M)2
__________
n-1
Challenge
Calculate the standard deviation of the set of scores listed:
9, 11, 8, 7 , 4, 5, 6, 3, 10
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