Physics experiments often require repeated measurements of the same quantity.
Even when the true physical constant is fixed, the observed measurements vary due to instrument precision and random error.
Statistics helps scientists summarize these measurements and estimate the true value more accurately.
The speed of light in vacuum is approximately
\[ c = 299,792,458 \text{ meters per second} \]
However, real experiments measure slightly different values each time.
Scientists therefore use statistical methods to estimate the true value.
A simple model for experimental measurements is
\[ X_i = \mu + \epsilon_i \]
where
Often we assume
\[ \epsilon_i \sim N(0,\sigma^2) \]
meaning measurement errors follow a normal distribution.
The sample mean estimates the true value:
\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i \]
If measurements are unbiased, this average should be close to the true speed of light.
Measurement variability is described by the sample standard deviation:
\[ s = \sqrt{\frac{1}{n-1}\sum (x_i-\bar{x})^2} \]
A 95% confidence interval is
\[ \bar{x} \pm 1.96\frac{s}{\sqrt{n}} \]
## trial speed ## 1 1 299792753 ## 2 2 299789029 ## 3 3 299792806 ## 4 4 299792969 ## 5 5 299792260 ## 6 6 299793296 ## 7 7 299792454 ## 8 8 299790689 ## 9 9 299792336 ## 10 10 299793250
set.seed(301) true_c <- 299792458 n <- 120 sigma <- 1200 measurements <- round(rnorm(n, mean = true_c, sd = sigma)) light_data <- data.frame( trial = 1:n, speed = measurements )
Experimental physics relies heavily on statistical analysis.
By using averages, standard deviation, confidence intervals, and visualization, scientists can estimate physical constants more reliably.
This example demonstrates how statistical methods help interpret repeated measurements of the speed of light.