Z-Statistic Calculator
Use this calculator to determine the Z-statistic (also known as a Z-score) for a sample mean, given the population mean, population standard deviation, and sample size. This is a fundamental tool in hypothesis testing to assess how many standard deviations a sample mean is from the population mean.
Understanding the Z-Statistic
The Z-statistic, often referred to as a Z-score, is a standardized value that indicates how many standard deviations an element is from the mean of a population. In the context of hypothesis testing, it specifically tells us how many standard deviations a sample mean is away from the population mean.
Why is the Z-Statistic Important?
The Z-statistic is crucial for performing Z-tests, which are a type of hypothesis test used to determine if there is a statistically significant difference between a sample mean and a known or hypothesized population mean. It's particularly useful when the population standard deviation is known and the sample size is large (typically n > 30), or when the population is known to be normally distributed.
The Z-Statistic Formula
The formula for calculating the Z-statistic for a sample mean is:
Z = (X̄ - μ) / (σ / √n)
- X̄ (Sample Mean): The mean of your observed sample data.
- μ (Population Mean): The mean of the entire population, or the hypothesized population mean under the null hypothesis.
- σ (Population Standard Deviation): The standard deviation of the entire population.
- n (Sample Size): The number of observations in your sample.
- σ / √n (Standard Error of the Mean): This term represents the standard deviation of the sampling distribution of the sample means.
Interpreting Your Z-Statistic
Once calculated, the Z-statistic can be compared to critical values from a standard normal distribution table (Z-table) to determine the p-value. The p-value helps you decide whether to reject or fail to reject the null hypothesis. A larger absolute Z-statistic indicates that the sample mean is further away from the population mean, making it less likely that the observed difference occurred by random chance.
- A Z-statistic close to 0 suggests that the sample mean is very close to the population mean.
- A large positive Z-statistic indicates the sample mean is significantly higher than the population mean.
- A large negative Z-statistic indicates the sample mean is significantly lower than the population mean.
Common critical values for a two-tailed test at a 0.05 significance level are ±1.96. If your calculated Z-statistic falls outside this range (e.g., Z > 1.96 or Z < -1.96), you would typically reject the null hypothesis, concluding that there is a statistically significant difference.
Example Calculation
Let's say a school claims the average IQ of its students is 100 with a population standard deviation of 15. You take a random sample of 40 students and find their average IQ is 105.
- Sample Mean (X̄) = 105
- Population Mean (μ) = 100
- Population Standard Deviation (σ) = 15
- Sample Size (n) = 40
Using the formula:
Standard Error = 15 / √40 ≈ 15 / 6.324 ≈ 2.372
Z = (105 - 100) / 2.372 = 5 / 2.372 ≈ 2.107
In this example, the calculated Z-statistic is approximately 2.107. Since 2.107 is greater than 1.96 (the critical value for a 0.05 significance level, two-tailed test), you would reject the null hypothesis and conclude that the sample's average IQ is significantly different from the claimed population average of 100.