Z Score Calculator For 2 Population Samples Mean

Z-Score Formula for Two Population Means:

\[ Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Sample Mean 1 (x̄₁):

Standard Deviation 1 (σ₁):

Sample Size 1 (n₁):

Sample Mean 2 (x̄₂):

Standard Deviation 2 (σ₂):

Sample Size 2 (n₂):

Z-Score:

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1. What is the Z-Score for Two Population Means?

The Z-score for two population means measures how many standard deviations apart the means of two samples are from each other. It's used in hypothesis testing to determine if there's a statistically significant difference between two population means.

2. How Does the Calculator Work?

The calculator uses the Z-score formula:

\[ Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]

Where:

\( \bar{x}_1 \) — Mean of sample 1
\( \bar{x}_2 \) — 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 measures the difference between sample means, while the denominator accounts for the standard error of this difference.

3. Importance of Z-Score Calculation

Details: The Z-score is fundamental in statistical hypothesis testing, allowing researchers to determine if observed differences between groups are likely due to chance or represent true population differences.

4. Using the Calculator

Tips: Enter all required parameters (means, standard deviations, and sample sizes) for both samples. Standard deviations must be positive, and sample sizes must be at least 1.

5. Frequently Asked Questions (FAQ)

Q1: When should I use a Z-test vs a t-test?
A: Use Z-test when population variances are known and sample sizes are large (>30). For small samples or unknown population variances, use t-test.

Q2: How do I interpret the Z-score?
A: Typically, |Z| > 1.96 suggests 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: No, for comparing proportions you need a different Z-score formula specifically designed for proportion comparisons.

Q5: What's the relationship between Z-score and p-value?
A: The Z-score can be converted to a p-value using the standard normal distribution. Higher |Z| corresponds to smaller p-values.