Z Statistic Calculator Two Sample

Two-Sample Z-Statistic Formula:

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

Sample 1 Mean (x̄₁):

Sample 1 SD (σ₁):

Sample 1 Size (n₁):

Sample 2 Mean (x̄₂):

Sample 2 SD (σ₂):

Sample 2 Size (n₂):

Z-Statistic:

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1. What is the Two-Sample Z-Statistic?

The two-sample Z-statistic is used to compare the means of two independent groups when the population standard deviations are known. It's commonly used in hypothesis testing to determine if there's a statistically significant difference between the two groups.

2. How Does the Calculator Work?

The calculator uses the two-sample Z-test 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 population 1
\( \sigma_2 \) — Standard deviation of population 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 standardizes this difference by the standard error of the difference between means.

3. When to Use This Test

Details: Use this test when comparing means from two independent samples with known population standard deviations, sample sizes are large (typically n > 30), and data is normally distributed.

4. Using the Calculator

Tips: Enter the mean, standard deviation, and sample size 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 Z-test when population standard deviations are known or sample sizes are large (n > 30). Use t-test for smaller samples with unknown population standard deviations.

Q2: What does the Z-score tell me?
A: The Z-score indicates how many standard deviations the difference between means is from zero. Higher absolute values indicate more significant differences.

Q3: What's a good sample size for this test?
A: While the test works with any sample size, larger samples (n > 30 per group) provide more reliable results through the Central Limit Theorem.

Q4: Can I use this for proportions?
A: No, for comparing proportions between two groups, use the two-proportion Z-test formula instead.

Q5: How do I interpret the Z-score?
A: Compare your Z-score to critical values from the standard normal distribution. Typically, |Z| > 1.96 indicates significance at α = 0.05 level.