A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out. A standard deviation close to zero indicates that data points are close to the mean, whereas a high or low standard deviation indicates data points are respectively above or below the mean. In Image 7, the curve on top is more spread out and therefore has a higher standard deviation, while the curve below is more clustered around the mean and therefore has a lower standard deviation.
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[Image 7: High and low standard deviation curves. Source: University of North Carolina, 2012.]
To calculate the standard deviation, use the following formula:
In this formula, σ is the standard deviation, x1 is the data point we are solving for in the set, µ is the mean, and N is the total number of data points. Let’s go back to the class example, but this time look at their height. To calculate the standard deviation of the class’s heights, first calculate the mean from each individual height. In this class there are nine students with an average height of 75 inches. Now the standard deviation equation looks like this:
The first step is to subtract the mean from each data point. Then square the absolute value before adding them all together. Now divide by 9 (the total number of data points) and finally take the square root to reach the standard deviation of the data:
| Height in inches x | Mean µ | Subtract mean from each data point x - µ | Result x | Square each value x2 | Sum of Squares ∑ x | Variance | Standard deviation σ=√x |
|---|---|---|---|---|---|---|---|
| 56 | 75 | 56 – 75 | -19 | 361 | 784 | 87.1 | 9.3 |
| 65 | 65 – 75 | -10 | 100 | ||||
| 74 | 74 – 75 | -1 | 1 | ||||
| 75 | 75 – 75 | 0 | 0 | ||||
| 76 | 76 – 75 | 1 | 1 | ||||
| 77 | 77 – 75 | 2 | 4 | ||||
| 80 | 80 – 75 | 5 | 25 | ||||
| 81 | 81 – 75 | 6 | 36 | ||||
| 91 | 91 – 75 | 16 | 256 |
[Figure 2: The step-by-step process of finding the standard deviation of sample data]
This data shows that 68% of heights were 75 inches plus or minus 9.3 inches (1 standard deviation away from the mean), 95% of heights were 75’’ plus or minus 18.6’’ (2 standard deviations away from the mean), and 99.7% of heights were 75’’ plus or minus 27.9’’ (3 standard deviations away from the mean).
[2] The University of North Carolina at Chapel Hill “Density Curves and Normal Distributions” 9/12/06. Web.
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