The term "sample deviation" usually refers to the sample standard deviation, which measures the amount of variation or dispersion in a set of sample data points. Here's how to calculate it:
Understanding Sample Standard Deviation
The sample standard deviation (often denoted as s) gives an idea of how spread out the data values are around the sample mean. A higher standard deviation indicates a greater spread, while a lower one means data points are clustered closer to the mean.
Steps to Calculate Sample Standard Deviation
Here's a breakdown of the steps, using the formula provided:
s= ⎷1n−1n∑x=1(xi−¯x)2
Where:
- s represents the sample standard deviation.
- n is the number of observations in the sample.
- xi represents each individual observation.
- x̄ (x-bar) represents the sample mean.
1. Calculate the Sample Mean (x̄)
- Sum all the values in your sample.
- Divide the sum by the total number of observations (n).
2. Calculate the Deviations from the Mean
- For each data point (xi), subtract the sample mean (x̄). This gives you xi - x̄.
3. Square the Deviations
- Square each of the deviations calculated in the previous step: (xi - x̄)2.
4. Sum the Squared Deviations
- Add up all the squared deviations from Step 3. This gives you ∑( xi - x̄)2.
5. Divide by (n - 1)
- Divide the sum of the squared deviations (from Step 4) by (n - 1). This is one less than the number of data points in your sample.
- This result is called the sample variance.
6. Take the Square Root
- Take the square root of the result from Step 5. This gives you the sample standard deviation (s).
Example
Let’s consider a sample data set: 2, 4, 6, 8.
-
Calculate the mean: (2+4+6+8)/4 = 20/4 = 5 (x̄ = 5).
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Calculate the deviations:
- 2 - 5 = -3
- 4 - 5 = -1
- 6 - 5 = 1
- 8 - 5 = 3
-
Square the deviations:
- (-3)2 = 9
- (-1)2 = 1
- 12 = 1
- 32 = 9
-
Sum the squared deviations: 9 + 1 + 1 + 9 = 20
-
Divide by (n-1): 20 / (4 - 1) = 20/3 = 6.67 (This is the sample variance.)
-
Take the square root: √6.67 ≈ 2.58
Therefore, the sample standard deviation for the data set 2, 4, 6, and 8 is approximately 2.58.
Key Points
- Sample standard deviation is used when working with a sample from a population, instead of the entire population.
- The division by (n-1) rather than n is known as Bessel’s correction and is necessary for the sample standard deviation to be an unbiased estimate of the population standard deviation.
- A higher standard deviation implies that data points are more spread out from the mean, whereas a lower value suggests data points are closer to the mean.
