Z Score Calculator For 2 Populations

Z-score Formula for Two Population Means:

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

Mean of Population 1 (x₁):

Mean of Population 2 (x₂):

Standard Deviation of Population 1 (σ₁):

Standard Deviation of Population 2 (σ₂):

Sample Size of Population 1 (n₁):

Sample Size of Population 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 populations are. It's used in hypothesis testing to determine if there's a significant difference between two population means when the population standard deviations are known.

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 population 1
\( x_2 \) — Mean of population 2
\( \sigma_1 \) — Standard deviation of population 1
\( \sigma_2 \) — Standard deviation of population 2
\( n_1 \) — Sample size of population 1
\( n_2 \) — Sample size of population 2

Explanation: The numerator measures the difference between means, while the denominator calculates the standard error of the difference.

3. Importance of Z-Score Calculation

Details: The Z-score is fundamental in statistical hypothesis testing, particularly in Z-tests. It helps determine if 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 populations. Standard deviations must be positive, and sample sizes must be at least 1.

5. Frequently Asked Questions (FAQ)

Q1: When should I use this Z-score formula?
A: Use when comparing means of two independent populations with known standard deviations and large sample sizes (typically n > 30).

Q2: What does the Z-score value indicate?
A: Higher absolute Z-scores indicate greater difference between means relative to variability. Scores beyond ±1.96 are typically significant at p < 0.05.

Q3: How is this different from a t-test?
A: Z-tests use known population standard deviations, while t-tests use sample standard deviations. Use t-tests for small samples or unknown population SDs.

Q4: Can I use this for proportions?
A: No, there's a different Z-score formula for comparing two population proportions.

Q5: What if my standard deviations are unknown?
A: You should use a two-sample t-test instead, which uses sample standard deviations.