How to Find a P-Value from a t-Score in Python?

A p-value represents the probability of obtaining results at least as extreme as observed, assuming the null hypothesis is true. A t-score measures how many standard deviations a sample mean is from the population mean. In Python, we can easily convert t-scores to p-values using SciPy's statistical functions.

What is P-value?

In statistics, the p-value is the probability of generating outcomes at least as extreme as the observed results of a statistical hypothesis test, assuming the null hypothesis is valid. A smaller p-value indicates stronger evidence against the null hypothesis. Typically, p-values below 0.05 are considered statistically significant.

What is t-score?

A t-score (also called t-statistic) measures how many standard deviations a sample mean differs from the hypothesized population mean. It's commonly used in t-tests when the population standard deviation is unknown and sample sizes are small.

Syntax for Finding P-value from t-score

The scipy.stats.t.sf() function calculates the p-value from a t-score using this syntax:

scipy.stats.t.sf(abs(t_score), df)

Where:

• t_score ? The calculated t-statistic
• df ? Degrees of freedom (typically n-1 for one-sample tests)

Left-tailed Test

For a left-tailed test, we calculate the p-value directly using the t-score. Let's find the p-value for t = -0.77 with df = 15:

import scipy.stats

# Find p-value for left-tailed test
t_score = -0.77
df = 15
p_value = scipy.stats.t.sf(abs(t_score), df)

print(f"t-score: {t_score}")
print(f"p-value: {p_value:.4f}")

t-score: -0.77
p-value: 0.2266

Since the p-value (0.2266) is greater than 0.05, we would not reject the null hypothesis at ? = 0.05 significance level.

Right-tailed Test

For a right-tailed test, we use the same formula. Here's an example with t = 1.87 and df = 24:

import scipy.stats

# Find p-value for right-tailed test
t_score = 1.87
df = 24
p_value = scipy.stats.t.sf(abs(t_score), df)

print(f"t-score: {t_score}")
print(f"p-value: {p_value:.4f}")

t-score: 1.87
p-value: 0.0369

Since the p-value (0.0369) is less than 0.05, we would reject the null hypothesis at ? = 0.05 significance level.

Two-tailed Test

For a two-tailed test, we multiply the result by 2 since we consider both tails of the distribution. Example with t = 1.24 and df = 22:

import scipy.stats

# Find p-value for two-tailed test
t_score = 1.24
df = 22
p_value = scipy.stats.t.sf(abs(t_score), df) * 2

print(f"t-score: {t_score}")
print(f"p-value: {p_value:.4f}")

t-score: 1.24
p-value: 0.2280

Since the p-value (0.2280) is greater than 0.05, we would not reject the null hypothesis at ? = 0.05 significance level.

Complete Example with All Test Types

import scipy.stats

def calculate_p_value(t_score, df, test_type="two-tailed"):
    """Calculate p-value from t-score for different test types"""
    
    if test_type == "left-tailed":
        p_value = scipy.stats.t.cdf(t_score, df)
    elif test_type == "right-tailed":
        p_value = scipy.stats.t.sf(abs(t_score), df)
    else:  # two-tailed
        p_value = scipy.stats.t.sf(abs(t_score), df) * 2
    
    return p_value

# Example calculations
tests = [
    (-0.77, 15, "left-tailed"),
    (1.87, 24, "right-tailed"),
    (1.24, 22, "two-tailed")
]

for t_score, df, test_type in tests:
    p_val = calculate_p_value(t_score, df, test_type)
    print(f"{test_type}: t={t_score}, df={df}, p-value={p_val:.4f}")

left-tailed: t=-0.77, df=15, p-value=0.2266
right-tailed: t=1.87, df=24, p-value=0.0369
two-tailed: t=1.24, df=22, p-value=0.2280

Conclusion

Converting t-scores to p-values in Python is straightforward using SciPy's t.sf() function. Remember to multiply by 2 for two-tailed tests and use the appropriate significance level to make statistical decisions about your hypothesis.