The standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a dataset. It tells you how much the individual data points differ from the mean (average) of the dataset. A low standard deviation indicates that the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Formula for Standard Deviation

For a dataset $x_1, x_2, x_3, \ldots, x_n$:

1. Compute the mean $\mu$ (for a population) or $\bar{x}$ (for a sample): $\text{Mean } \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$
2. Subtract the mean from each data point and square the result to calculate the squared deviations: $(x_i - \bar{x})^2$
3. Compute the variance:
   • For a sample: $s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}$
   • For a population: $\sigma^2 = \frac{\sum_{i=1}^n (x_i - \mu)^2}{n}$
4. Take the square root of the variance to find the standard deviation:
   • Sample standard deviation: $s = \sqrt{s^2}$
   • Population standard deviation: $\sigma = \sqrt{\sigma^2}$

Example Calculation:

For a sample dataset: $[4, 8, 6, 5, 3]$:

1. Mean: $\bar{x} = \frac{4 + 8 + 6 + 5 + 3}{5} = 5.2$
2. Deviations squared: $(4-5.2)^2, (8-5.2)^2, (6-5.2)^2, (5-5.2)^2, (3-5.2)^2$ $(4-5.2)^2 = 1.44,\ (8-5.2)^2 = 7.84,\ (6-5.2)^2 = 0.64,\ (5-5.2)^2 = 0.04,\ (3-5.2)^2 = 4.84$
3. Variance: $s^2 = \frac{1.44 + 7.84 + 0.64 + 0.04 + 4.84}{4} = 3.7$
4. Standard deviation: $s = \sqrt{3.7} \approx 1.92$

The standard deviation of this dataset is approximately $1.92$.

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Would you like more details, have questions, or want me to solve a specific example for you?

Here are 5 related questions for further learning:

1. What is the difference between population and sample standard deviation?
2. How is standard deviation used in real-world data analysis?
3. Can you calculate the standard deviation for a dataset with given values?
4. What does a standard deviation of 0 indicate about a dataset?
5. How is standard deviation related to variance?

Tip: Remember, standard deviation is always non-negative, as it measures the spread of data around the mean!