Z Score Calculator for 2 Population Samples in Statistics

Z-Score Formula for Two Populations:

\[ Z = \frac{x_1 - x_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Mean of Sample 1 (x₁):

Standard Deviation of Sample 1 (σ₁):

Size of Sample 1 (n₁):

Mean of Sample 2 (x₂):

Standard Deviation of Sample 2 (σ₂):

Size of Sample 2 (n₂):

Z-Score:

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1. What is the Z-Score for Two Populations?

The Z-score for two population samples is a statistical measurement that describes the difference between two sample means in terms of standard deviations. It's used to determine if the difference between two sample means is statistically significant.

2. How Does the Calculator Work?

The calculator uses the following formula:

\[ Z = \frac{x_1 - x_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Where:

\( x_1 \) — Mean of sample 1
\( x_2 \) — Mean of sample 2
\( \sigma_1 \) — Standard deviation of sample 1
\( \sigma_2 \) — Standard deviation of sample 2
\( n_1 \) — Size of sample 1
\( n_2 \) — Size of sample 2

Explanation: The numerator represents the difference between sample means, while the denominator is the standard error of the difference between means.

3. Importance of Z-Score Calculation

Details: The Z-score is crucial for hypothesis testing, particularly in Z-tests. It helps determine whether observed differences between groups are statistically significant or likely due to chance.

4. Using the Calculator

Tips: Enter the means, standard deviations, and sample sizes for both groups. All values must be valid (sample sizes > 0, standard deviations ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: When should I use a Z-test vs a T-test?
A: Use a Z-test when sample sizes are large (typically n > 30) and population standard deviations are known. For smaller samples or unknown population SDs, use a T-test.

Q2: How do I interpret the Z-score?
A: Generally, |Z| > 1.96 indicates statistical significance at α=0.05 level. Higher absolute values indicate stronger evidence against the null hypothesis.

Q3: What assumptions does this test make?
A: Assumes independent samples, normally distributed populations (or large sample sizes), and known population standard deviations.

Q4: Can I use this for proportions?
A: A similar but different formula exists for comparing two proportions. This calculator is specifically for comparing means.

Q5: What if my standard deviations are equal?
A: The formula still applies. Some tests pool variances when SDs are equal, but this calculator uses the more general formula.