Physics
Errors in Measurement
Standard Deviation (σ)
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⚡ Quick Summary
Standard deviation (σ) tells you how spread out your measurements are. A smaller σ means your measurements are clustered closer to the average, while a larger σ means they're more spread out. It's a measure of the uncertainty in your measurements.
σ = √[ (1/N) * Σ (xi - x̄)² ]
where:
xi = individual measurement
x̄ = average of all measurements
N = number of measurements
Standard deviation (σ) quantifies the uncertainty in a set of measurements. It is calculated as the square root of the average of the squared differences between each measurement and the average measurement.
Uncertainty and True Value
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⚡ Quick Summary
After taking measurements and finding the average, we use the standard deviation to estimate how close our average is to the 'true' value. We can say there's a good chance (95%) the real value lies within a range around our average (average ± 1.96 * standard deviation). If you want to be even more sure (99%), expand the range to average ± 3 * standard deviation.
95% Confidence Interval: x̄ ± 1.96σ
Higher Confidence Interval: x̄ ± 3σ
The standard deviation (σ) provides an estimate of the uncertainty in the average value. The interval x̄ ± 1.96σ represents a 95% confidence interval for the true value. The interval x̄ ± 3σ represents a higher confidence interval.