Z Score Calculator For 2 Population Samples In R

Z-Score Equation:

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

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 Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

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

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

2. How Does the Calculator Work?

The calculator uses the Z-score equation for two population samples:

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

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 measures the difference between sample 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 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 samples. All values must be valid (sample sizes > 0, standard deviations ≥ 0).

5. Frequently Asked Questions (FAQ)

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

Q2: What does the Z-score value mean?
A: A higher absolute Z-score indicates a greater difference between the sample means relative to the expected variation.

Q3: How is this different from a t-test?
A: Z-tests are used when population standard deviations are known (or with large samples), while t-tests are used when they must be estimated from the sample.

Q4: What's a significant Z-score value?
A: Typically, |Z| > 1.96 indicates significance at α=0.05 level, but this depends on your specific hypothesis test.

Q5: Can I use this for small samples?
A: For small samples (n < 30), a t-test is generally more appropriate unless you know the population standard deviations.