A p-value measures how surprising the data are if the null hypothesis is true.
It answers this question:
\[ \text{p-value} = P(\text{data at least as extreme as observed} \mid H_0 \text{ is true}) \]
Smaller p-values mean the observed result is less consistent with the null hypothesis.
We will use the built-in mtcars data set and test whether the mean miles per gallon is different from 18.
We test
\[ H_0 : \mu = 18 \]
against
\[ H_a : \mu \ne 18 \]
The one-sample t statistic is
\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \]
Large values of \(|t|\) give smaller p-values.
The solid line is the sample mean and the dashed line is the null mean.
For a two-sided t-test,
\[ p\text{-value} = P\left(|T| \ge |t_{\text{obs}}| \mid H_0\right) \]
If the p-value is smaller than the chosen significance level \(\alpha\), we reject the null hypothesis.
The p-value is the area in the tails beyond the observed test statistic.
This interactive chart shows the null distribution and the observed cutoff.
ggplot(null_df, aes(x = t)) +
geom_histogram(
aes(y = after_stat(density)),
bins = 40,
fill = "gray80",
color = "white"
) +
geom_vline(
xintercept = c(-abs(t_obs), abs(t_obs)),
color = "#8C1D40",
linetype = "dashed",
linewidth = 1
) +
annotate(
"text",
x = abs(t_obs),
y = 0.38,
label = paste0("observed |t| = ", round(abs(t_obs), 2)),
color = "#8C1D40",
hjust = -0.05
) +
labs(
title = "Null Distribution of t Statistics",
x = "t statistic under H0",
y = "Density"
) +
coord_cartesian(xlim = c(-6, 6)) +
theme_minimal(base_size = 16)
Observed t statistic: 1.962
p-value: 0.0588
Because the p-value is greater than 0.05, we fail to reject \(H_0\) at the 5% level.
P-values are one of the most common tools for making decisions in statistics.