This calculator helps you compare the means between two independent groups to determine if they are significantly different. Perfect for comparing test scores between two classes, treatment effectiveness between groups, or any scenario where you need to know if two populations truly differ.
Ready to compare your groups? to see the step-by-step process in action, or upload your own data to discover if your groups are truly different.
Two-Sample T-Test is a statistical test used to determine whether there is a significant difference between the means of two independent groups. It's particularly useful when comparing two different treatments, methods, or groups to each other.
Test Statistic:
Degrees of freedom:
For equal variances (Student's t-test):
For unequal variances (Welch's t-test):
Confidence Interval:
Standard Error (SE) for equal variances:
where pooled standard deviation:
Standard Error (SE) for unequal variances:
Where:
While both Welch's t-test and Student's t-test are used to compare means between two groups, they differ in their assumptions and applications:
| Aspect | Welch's T-Test | Student's T-Test |
|---|---|---|
| Variance Assumption | Does not assume equal variances | Assumes equal variances |
| Degrees of Freedom | Calculated using Welch–Satterthwaite equation above | |
| Robustness | More robust when variances are unequal | Less robust when variances are unequal |
| Sample Size Sensitivity | Less sensitive to unequal sample sizes | More sensitive to unequal sample sizes |
| Use Case | Preferred when variances or sample sizes are unequal | Used when variances are assumed to be equal |
Key Distinction: The primary difference lies in the assumption of equal variances. Welch's t-test does not require this assumption, making it more appropriate for comparing groups with unequal variances.
Both tests share the following assumptions:
In practice, Welch's t-test is often recommended as the default choice for comparing two means, as it maintains good control over Type I error rates and statistical power across a wider range of scenarios compared to Student's t-test.
We want to compare two teaching methods by examining test scores:
Given Data:
Hypotheses:
Null Hypothesis (): (no difference between methods)
Alternative Hypothesis (): (there is a difference between methods)
Step-by-Step Calculation:
Conclusion:
, we reject the null hypothesis. There is sufficient evidence to conclude that there is a significant difference between the two teaching methods (). We are 95% confident that the true difference in means lies between 0.54 and 9.46.
Cohen's d for two independent samples:
For unequal variances (preferred when using Welch's t-test):
Interpretation guidelines:
To determine required sample size per group (n) for desired power (1-β):
Where:
Reject if:
Where is:
Standard format for scientific reporting:
Remember to report whether Welch's or Student's t-test was used and justify the choice based on the equality of variances.
library(tidyverse)
library(car)
library(effsize)
set.seed(42)
group1 <- rnorm(30, mean = 75, sd = 8) # Method A
group2 <- rnorm(35, mean = 70, sd = 10) # Method B
# Combine data
data <- tibble(
score = c(group1, group2),
method = factor(c(rep("A", 30), rep("B", 35)))
)
# Basic summary statistics
summary_stats <- data %>%
group_by(method) %>%
summarise(
n = n(),
mean = mean(score),
sd = sd(score)
)
# Levene's test for equality of variances
car::leveneTest(score ~ method, data = data)
# Welch's t-test (default)
t_test_result <- t.test(score ~ method, data = data)
# Student's t-test (if equal variances assumed)
t_test_equal_var <- t.test(score ~ method, data = data, var.equal = TRUE)
# Effect size
cohens_d <- effsize::cohen.d(score ~ method, data = data)
# Visualization
ggplot(data, aes(x = method, y = score, fill = method)) +
geom_boxplot(alpha = 0.5) +
geom_jitter(width = 0.2, alpha = 0.5) +
theme_minimal() +
labs(title = "Comparison of Test Scores by Method")import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.stats.power import TTestIndPower
# Generate sample data
np.random.seed(42)
group1 = np.random.normal(75, 8, 30) # Method A
group2 = np.random.normal(70, 10, 35) # Method B
# Create a DataFrame for easier plotting with seaborn
import pandas as pd
df = pd.DataFrame({
'Score': np.concatenate([group1, group2]),
'Method': ['A']*30 + ['B']*35
})
# Basic summary statistics
def get_summary(data):
return {
'n': len(data),
'mean': np.mean(data),
'std': np.std(data, ddof=1),
'se': stats.sem(data)
}
summary1 = get_summary(group1)
summary2 = get_summary(group2)
# Test for equal variances
_, levene_p = stats.levene(group1, group2)
# Perform t-tests
# Welch's t-test (unequal variances)
t_stat, p_value = stats.ttest_ind(group1, group2, equal_var=False)
# Calculate Cohen's d
pooled_sd = np.sqrt((summary1['std']**2 + summary2['std']**2) / 2)
cohens_d = abs(summary1['mean'] - summary2['mean']) / pooled_sd
# Create visualization
plt.figure(figsize=(12, 5))
# Subplot 1: Boxplot
plt.subplot(1, 2, 1)
sns.boxplot(data=df, x='Method', y='Score')
plt.title('Score Distribution by Method')
# Subplot 2: Distribution
plt.subplot(1, 2, 2)
sns.histplot(data=df, x='Score', hue='Method', element="step",
stat="density", common_norm=False)
plt.title('Score Distribution Density')
plt.tight_layout()
plt.show()
# Print results
print("Summary Statistics:")
print(f"Method A: Mean = {summary1['mean']:.2f}, SD = {summary1['std']:.2f}, n = {summary1['n']}")
print(f"Method B: Mean = {summary2['mean']:.2f}, SD = {summary2['std']:.2f}, n = {summary2['n']}")
print(f"Levene's test p-value: {levene_p:.4f}")
print(f"Welch's t-test: t = {t_stat:.4f}, p = {p_value:.4f}")
print(f"Cohen's d: {cohens_d:.4f}")Consider these alternatives when assumptions are violated:
This calculator helps you compare the means between two independent groups to determine if they are significantly different. Perfect for comparing test scores between two classes, treatment effectiveness between groups, or any scenario where you need to know if two populations truly differ.
Ready to compare your groups? to see the step-by-step process in action, or upload your own data to discover if your groups are truly different.
Two-Sample T-Test is a statistical test used to determine whether there is a significant difference between the means of two independent groups. It's particularly useful when comparing two different treatments, methods, or groups to each other.
Test Statistic:
Degrees of freedom:
For equal variances (Student's t-test):
For unequal variances (Welch's t-test):
Confidence Interval:
Standard Error (SE) for equal variances:
where pooled standard deviation:
Standard Error (SE) for unequal variances:
Where:
While both Welch's t-test and Student's t-test are used to compare means between two groups, they differ in their assumptions and applications:
| Aspect | Welch's T-Test | Student's T-Test |
|---|---|---|
| Variance Assumption | Does not assume equal variances | Assumes equal variances |
| Degrees of Freedom | Calculated using Welch–Satterthwaite equation above | |
| Robustness | More robust when variances are unequal | Less robust when variances are unequal |
| Sample Size Sensitivity | Less sensitive to unequal sample sizes | More sensitive to unequal sample sizes |
| Use Case | Preferred when variances or sample sizes are unequal | Used when variances are assumed to be equal |
Key Distinction: The primary difference lies in the assumption of equal variances. Welch's t-test does not require this assumption, making it more appropriate for comparing groups with unequal variances.
Both tests share the following assumptions:
In practice, Welch's t-test is often recommended as the default choice for comparing two means, as it maintains good control over Type I error rates and statistical power across a wider range of scenarios compared to Student's t-test.
We want to compare two teaching methods by examining test scores:
Given Data:
Hypotheses:
Null Hypothesis (): (no difference between methods)
Alternative Hypothesis (): (there is a difference between methods)
Step-by-Step Calculation:
Conclusion:
, we reject the null hypothesis. There is sufficient evidence to conclude that there is a significant difference between the two teaching methods (). We are 95% confident that the true difference in means lies between 0.54 and 9.46.
Cohen's d for two independent samples:
For unequal variances (preferred when using Welch's t-test):
Interpretation guidelines:
To determine required sample size per group (n) for desired power (1-β):
Where:
Reject if:
Where is:
Standard format for scientific reporting:
Remember to report whether Welch's or Student's t-test was used and justify the choice based on the equality of variances.
library(tidyverse)
library(car)
library(effsize)
set.seed(42)
group1 <- rnorm(30, mean = 75, sd = 8) # Method A
group2 <- rnorm(35, mean = 70, sd = 10) # Method B
# Combine data
data <- tibble(
score = c(group1, group2),
method = factor(c(rep("A", 30), rep("B", 35)))
)
# Basic summary statistics
summary_stats <- data %>%
group_by(method) %>%
summarise(
n = n(),
mean = mean(score),
sd = sd(score)
)
# Levene's test for equality of variances
car::leveneTest(score ~ method, data = data)
# Welch's t-test (default)
t_test_result <- t.test(score ~ method, data = data)
# Student's t-test (if equal variances assumed)
t_test_equal_var <- t.test(score ~ method, data = data, var.equal = TRUE)
# Effect size
cohens_d <- effsize::cohen.d(score ~ method, data = data)
# Visualization
ggplot(data, aes(x = method, y = score, fill = method)) +
geom_boxplot(alpha = 0.5) +
geom_jitter(width = 0.2, alpha = 0.5) +
theme_minimal() +
labs(title = "Comparison of Test Scores by Method")import numpy as np
import scipy.stats as stats
import matplotlib.pyplot as plt
import seaborn as sns
from statsmodels.stats.power import TTestIndPower
# Generate sample data
np.random.seed(42)
group1 = np.random.normal(75, 8, 30) # Method A
group2 = np.random.normal(70, 10, 35) # Method B
# Create a DataFrame for easier plotting with seaborn
import pandas as pd
df = pd.DataFrame({
'Score': np.concatenate([group1, group2]),
'Method': ['A']*30 + ['B']*35
})
# Basic summary statistics
def get_summary(data):
return {
'n': len(data),
'mean': np.mean(data),
'std': np.std(data, ddof=1),
'se': stats.sem(data)
}
summary1 = get_summary(group1)
summary2 = get_summary(group2)
# Test for equal variances
_, levene_p = stats.levene(group1, group2)
# Perform t-tests
# Welch's t-test (unequal variances)
t_stat, p_value = stats.ttest_ind(group1, group2, equal_var=False)
# Calculate Cohen's d
pooled_sd = np.sqrt((summary1['std']**2 + summary2['std']**2) / 2)
cohens_d = abs(summary1['mean'] - summary2['mean']) / pooled_sd
# Create visualization
plt.figure(figsize=(12, 5))
# Subplot 1: Boxplot
plt.subplot(1, 2, 1)
sns.boxplot(data=df, x='Method', y='Score')
plt.title('Score Distribution by Method')
# Subplot 2: Distribution
plt.subplot(1, 2, 2)
sns.histplot(data=df, x='Score', hue='Method', element="step",
stat="density", common_norm=False)
plt.title('Score Distribution Density')
plt.tight_layout()
plt.show()
# Print results
print("Summary Statistics:")
print(f"Method A: Mean = {summary1['mean']:.2f}, SD = {summary1['std']:.2f}, n = {summary1['n']}")
print(f"Method B: Mean = {summary2['mean']:.2f}, SD = {summary2['std']:.2f}, n = {summary2['n']}")
print(f"Levene's test p-value: {levene_p:.4f}")
print(f"Welch's t-test: t = {t_stat:.4f}, p = {p_value:.4f}")
print(f"Cohen's d: {cohens_d:.4f}")Consider these alternatives when assumptions are violated: