Pooled Standard Deviation Calculator

Understanding Pooled Standard Deviation

The pooled standard deviation is a measure that combines the standard deviations of two independent samples into a single value, representing the average spread of values around the mean. It is commonly used in hypothesis testing and effect size calculations where combining data from multiple groups is necessary.

Key Components of Pooled Standard Deviation

• Sample Standard Deviations (\( s_1 \) and \( s_2 \)): The standard deviations of the two independent samples.
• Sample Sizes (\( n_1 \) and \( n_2 \)): The number of observations in each sample.
• Degrees of Freedom: The combined degrees of freedom for both samples, calculated as \( n_1 + n_2 - 2 \).

Formula for Pooled Standard Deviation

The formula for pooled standard deviation is derived from the pooled variance formula:

where:

• \( s_1 \): Standard deviation of sample 1
• \( s_2 \): Standard deviation of sample 2
• \( n_1 \): Sample size of sample 1
• \( n_2 \): Sample size of sample 2

Programmatically Calculating Pooled Standard Deviation

Below are examples for calculating the pooled standard deviation in JavaScript, Python, and R.

1. Using JavaScript (with jStat)

In JavaScript, the jStat library can be used to calculate the pooled standard deviation:

// Define inputs
const sample1 = [10, 15, 14, 18, 20];
const sample2 = [12, 17, 16, 19, 21];

// Calculate variances and sample sizes
const variance1 = jStat.variance(sample1, true); // 14.5
const variance2 = jStat.variance(sample2, true); // 12.5
const n1 = sample1.length;
const n2 = sample2.length;

// Calculate pooled variance
const pooledVariance = ((n1 - 1) * variance1 + (n2 - 1) * variance2) / (n1 + n2 - 2);

// Calculate pooled standard deviation
const pooledStdDev = Math.sqrt(pooledVariance);
console.log(`Pooled Standard Deviation: ${pooledStdDev.toFixed(5)}`);

2. Using Python

In Python, you can calculate the pooled standard deviation using basic statistical operations:

# Define inputs
sample1 = [10, 15, 14, 18, 20]
sample2 = [12, 17, 16, 19, 21]

# Calculate variances and sample sizes
variance1 = sum((x - sum(sample1) / len(sample1))**2 for x in sample1) / (len(sample1) - 1)  # 14.5
variance2 = sum((x - sum(sample2) / len(sample2))**2 for x in sample2) / (len(sample2) - 1)  # 12.5
n1 = len(sample1)
n2 = len(sample2)

# Calculate pooled variance
pooled_variance = ((n1 - 1) * variance1 + (n2 - 1) * variance2) / (n1 + n2 - 2)

# Calculate pooled standard deviation
pooled_std_dev = pooled_variance**0.5
print(f"Pooled Standard Deviation: {pooled_std_dev:.5f}")

3. Using R

In R, you can calculate the pooled standard deviation as follows:

# Define inputs
sample1 <- c(10, 15, 14, 18, 20)
sample2 <- c(12, 17, 16, 19, 21)

# Calculate variances and sample sizes
variance1 <- var(sample1)  # 14.5
variance2 <- var(sample2)  # 12.5
n1 <- length(sample1)
n2 <- length(sample2)

# Calculate pooled variance
pooled_variance <- ((n1 - 1) * variance1 + (n2 - 1) * variance2) / (n1 + n2 - 2)

# Calculate pooled standard deviation
pooled_std_dev <- sqrt(pooled_variance)
cat("Pooled Standard Deviation:", round(pooled_std_dev, 5))

Further Reading

• Standard Deviation on Wikipedia
• Explore More Calculators on the Research Scientist Pod