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Standard Deviation and Variance
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Learn about standard deviation and variance, which we use to measure how tightly or loosely clustered the data is around the mean.
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
Variance & SD on Calculator (Sx vs σ)
SL 4.3
The variance σ2 of a dataset measures the spread of data around the mean.
The standard deviation σ is the square root of the variance. The advantage of the standard deviation is that is has the same units as the original data.
When you use a calculator to find standard deviation:
Enter your data into L1 using STAT > EDIT. Then, use STAT > CALC > 1-Var Stats and enter L1 as your list by clicking 2ND then 1.
You will see two values: Sx and σx. We use Sx when the data is a sample of a large population, and σx when the data represents the entire population. The difference is due to the fact that a sample will usually have a smaller variance than the population, because there are fewer elements.
Finding variance by hand
AHL 4.14
The mathematical definition of variance, which the calculator is using under the hood, is
σ2=ni=1∑kfi(xi−μ)2=ni=1∑kfixi2−μ2📖
Here, xi are all the existing values in the dataset, and fi is the frequency of each value.
The standard deviation is thus simply the square root of this:
σ=⎷ni=1∑kfi(xi−μ)2📖
Constant changes to data
SL 4.3
If we have a dataset with mean xˉ and standard deviation σ, then if we
add a constant +b to the dataset, the mean increases by b and the standard deviation does not change
scale the values by a, then both the mean and the standard deviation are scaled by a.
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