Uncertainty (Standard Deviation) Calculator

Standard Deviation Formula:

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

Mean (Average):

Uncertainty (Standard Deviation):

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1. What is Uncertainty (Standard Deviation)?

Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much the individual data points differ from the mean value of the dataset.

2. How Does the Calculator Work?

The calculator uses the standard deviation formula:

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

Where:

\( \sigma \) — Standard deviation (uncertainty)
\( x_i \) — Individual values in the dataset
\( \bar{x} \) — Mean (average) of the values
\( n \) — Number of values in the dataset

Explanation: The formula calculates how far each value is from the mean, squares these differences, averages them (divides by n-1 for sample standard deviation), and takes the square root.

3. Importance of Uncertainty Calculation

Details: Standard deviation is crucial in statistics to understand the spread of data points. It's used in quality control, finance, research, and many scientific fields to assess variability and reliability of measurements.

4. Using the Calculator

Tips: Enter your values separated by commas (e.g., 5.2, 5.5, 5.3). You need at least 2 values to calculate standard deviation. The calculator will ignore any non-numeric values.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between population and sample standard deviation?
A: Population standard deviation divides by N, while sample standard deviation divides by N-1 (Bessel's correction). This calculator uses sample standard deviation.

Q2: When should I use standard deviation?
A: Use it when you need to quantify how spread out your data is from the mean. It's particularly useful for normally distributed data.

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, showing more variability.

Q4: Can standard deviation be negative?
A: No, standard deviation is always non-negative because it's derived from squared differences.

Q5: How many decimal places should I report?
A: Generally, report standard deviation with one more decimal place than your original measurements.