1. What is Z-Score for Sample Mean?

The Z-score for sample mean measures how many standard errors a sample mean (x̄) is from the population mean (μ). It's used in hypothesis testing to determine if a sample comes from a particular population.

2. How Does the Calculator Work?

The calculator uses the Z-score formula for sample means:

Where:

• \( Z \) — Z-score (dimensionless)
• \( x \) — Sample mean
• \( \mu \) — Population mean
• \( \sigma \) — Population standard deviation
• \( n \) — Sample size

Explanation: The numerator measures how far the sample mean is from the population mean, while the denominator is the standard error of the mean (how much sample means typically vary from the population mean).

3. Importance of Z-Score Calculation

Details: Z-scores are crucial for hypothesis testing (z-tests), constructing confidence intervals, and determining how unusual a sample result is relative to the population.

4. Using the Calculator

Tips: Enter all required values. Population standard deviation must be positive and sample size must be at least 1. The result is dimensionless.

5. Frequently Asked Questions (FAQ)

Q1: When should I use this Z-score formula?
A: Use when you know the population standard deviation and have a sufficiently large sample size (typically n ≥ 30).

Q2: What does the Z-score value mean?
A: A Z-score of 0 means the sample mean equals the population mean. Positive values indicate the sample mean is above the population mean, negative values indicate it's below.

Q3: What if I don't know the population standard deviation?
A: Use t-score with sample standard deviation instead, especially for small sample sizes.

Q4: How is this different from regular Z-score?
A: The regular Z-score (for individual values) uses σ in denominator, while this version (for sample means) uses σ/√n (standard error).

Q5: What's considered a "significant" Z-score?
A: Typically |Z| > 1.96 indicates statistical significance at α=0.05 level (two-tailed test).