How to Calculate Dispersion

Dispersion Calculation:

\[ Variance = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n} \] \[ Standard\ Deviation = \sqrt{Variance} \]

Mean (Average):

Variance:

Standard Deviation:

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1. What is Dispersion?

Dispersion measures describe how spread out or varied a set of data values are. The two most common measures of dispersion are variance and standard deviation, which quantify the average distance of data points from the mean.

2. How to Calculate Dispersion

The calculator uses the following formulas:

\[ Variance = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n} \] \[ Standard\ Deviation = \sqrt{Variance} \]

Where:

\( x_i \) — Individual data values
\( \bar{x} \) — Mean (average) of all values
\( n \) — Number of data values
\( \sum \) — Summation symbol

Explanation: Variance measures the average squared deviation from the mean, while standard deviation provides a measure of spread in the original units of the data.

3. Importance of Dispersion Measures

Details: Understanding dispersion is crucial for statistical analysis as it helps determine the reliability of the mean, identify outliers, and assess the variability within a dataset. Low dispersion indicates data points are clustered closely around the mean, while high dispersion suggests greater spread.

4. Using the Calculator

Tips: Enter numerical values separated by commas (e.g., 5,8,12,15,20). The calculator will compute the mean, variance, and standard deviation of your dataset.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between variance and standard deviation?
A: Variance is the average of squared deviations from the mean, while standard deviation is the square root of variance. Standard deviation is expressed in the same units as the original data.

Q2: When should I use population variance vs sample variance?
A: Use population variance when working with the entire population. For samples, use sample variance (dividing by n-1 instead of n) to get an unbiased estimate.

Q3: What does a high standard deviation indicate?
A: A high standard deviation indicates that data points are spread out over a wider range of values, suggesting greater variability in the dataset.

Q4: Can dispersion be negative?
A: No, both variance and standard deviation are always non-negative values since they are based on squared differences.

Q5: How does dispersion relate to normal distribution?
A: In a normal distribution, about 68% of values fall within one standard deviation of the mean, and about 95% within two standard deviations.