# Formula for Standard Deviation

Standard deviation is a statistical measure that describes the amount of variability or dispersion in a set of data. It measures how spread out the data points are from the mean or average value of the data set. The standard deviation is calculated by taking the square root of the variance, which is the average of the squared differences between each data point and the mean. A larger standard deviation indicates that the data points are more spread out from the mean, while a smaller standard deviation indicates that the data points are closer to the mean. The standard deviation is commonly used in many fields, such as finance, psychology, and engineering, to analyze and compare data sets.

## Standard Deviation Formula

The formula for calculating the standard deviation is:

s = √[Σ(x – x̄)² / (n – 1)]

Where:

- s is the sample standard deviation
- x is each value in the sample
- x̄ is the sample mean
- n is the sample size
- Σ is the summation symbol, meaning to add up all the values in the brackets

In words, the formula tells us to:

- Subtract the mean from each data point.
- Square the differences.
- Add up all the squared differences.
- Divide by n-1 (sample size minus one).
- Take the square root of the result to obtain the sample standard deviation.

### Example

Let’s say we have a sample of 5 test scores: 85, 90, 92, 88, and 89.

- Calculate the sample mean: x̄ = (85 + 90 + 92 + 88 + 89) / 5 = 88.8
- Subtract the mean from each data point: (85 – 88.8), (90 – 88.8), (92 – 88.8), (88 – 88.8), (89 – 88.8) = -3.8, 1.2, 3.2, -0.8, 0.2
- Square the differences: (-3.8)², (1.2)², (3.2)², (-0.8)², (0.2)² = 14.44, 1.44, 10.24, 0.64, 0.04
- Add up all the squared differences: 14.44 + 1.44 + 10.24 + 0.64 + 0.04 = 26.8
- Divide by n-1 (sample size minus one): 26.8 / (5 – 1) = 6.7
- Take the square root of the result to obtain the sample standard deviation: √6.7 ≈ 2.59

Therefore, the sample standard deviation for the test scores is approximately 2.59.

here’s an example of a table on standard deviation:

Sample | Mean | Data Points | Differences from Mean | Squared Differences | Sum of Squared Differences | Standard Deviation |
---|---|---|---|---|---|---|

A | 5.5 | 4, 5, 6, 8 | -0.5, 0.5, 0.5, 2.5 | 0.25, 0.25, 0.25, 6.25 | 6.0 | 2.19 |

B | 10.0 | 8, 9, 11, 13 | -2.0, -1.0, 1.0, 3.0 | 4.0, 1.0, 1.0, 9.0 | 15.0 | 2.74 |

In this table, we have two samples, A and B. For each sample, we have calculated the mean, data points, differences from the mean, squared differences, and sum of squared differences. Finally, we have calculated the standard deviation for each sample using the formula:

s = √[Σ(x – x̄)² / (n – 1)]

Note that the standard deviation for sample A is 2.19 and the standard deviation for sample B is 2.74. The table shows that sample B has a larger standard deviation than sample A, indicating that the data points in sample B are more spread out from the mean than in sample A.

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