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Standard deviation is a crucial statistical measure that quantifies how much individual data points deviate from the mean. It is represented by sigma (σ) and is the square root of the variance, which is the average of squared deviations. This concept is essential for understanding data spread and is visualized through a normal distribution curve, where the empirical rule applies. The text provides a step-by-step calculation example and discusses its importance in data analysis.
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Standard deviation is a statistical measure that quantifies the degree of variation or dispersion of a set of data values from their mean
Variance is the square of the standard deviation and indicates the average squared deviation of each data point from the mean
Standard deviation and variance are both fundamental statistical measures, with variance often used in more complex methods and standard deviation providing a more direct understanding of data spread
Standard deviation is calculated using the formula σ = √[Σ(xi - μ)² / N], where Σ denotes the sum of all squared deviations, xi is an individual data point, μ is the mean of the dataset, and N is the number of data points
To calculate standard deviation, the mean is first computed, then the difference between each data point and the mean is squared, summed, and divided by the number of data points, with the square root of this result being the standard deviation
Standard deviation and variance provide insight into the consistency and variability within datasets, with standard deviation being particularly useful for normally distributed data
The normal distribution curve, also known as the bell curve, is a graph that displays how data is distributed in relation to the mean, with the x-axis representing data values in standard deviations from the mean and the y-axis representing the frequency or probability density of these values
The empirical rule, also known as the 68-95-99.7 rule, states that in a normal distribution, about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations
The normal distribution curve aids in predicting value distribution and provides a visual representation of how data is spread around the mean
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Symbol representing standard deviation
Greek letter sigma (σ)
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Formula to calculate standard deviation
σ = √[Σ(xi - μ)² / N]
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Relationship between standard deviation and variance
Standard deviation is the square root of the variance
