Step-by-Step Guide to Calculating Z-Scores

Calculating Z-Scores

Z-scores, also known as standard scores, measure how many standard deviations a data point is from the mean of the data set. This guide will walk you through calculating Z-scores with detailed explanations and examples.

What is a Z-Score?

A Z-score indicates the relative position of a data point within a data set. It is calculated by subtracting the mean from the data point and dividing the result by the standard deviation.

Z-Score Formula

The formula for calculating a Z-score is:

Z = (X - μ) / σ

Where:

• X is the value of the data point,
• μ is the mean of the data set,
• σ is the standard deviation of the data set.

Steps to Calculate Z-Scores

1. Calculate the Mean (μ): Sum all data points and divide by the number of points.
2. Calculate the Standard Deviation (σ): Use the formula for standard deviation.
3. Apply the Z-Score Formula: Subtract the mean from each data point and divide by the standard deviation.

Example: Calculating Z-Scores

Given the data set: 10, 12, 15, 18, 20:

Calculate the Mean:

μ = (10 + 12 + 15 + 18 + 20) / 5 = 15

Calculate the Standard Deviation:

σ = sqrt(((10 - 15)^2 + (12 - 15)^2 + (15 - 15)^2 + (18 - 15)^2 + (20 - 15)^2) / 5)

σ ≈ 3.42

Calculate Z-Scores:

• Z₁₀ = (10 – 15) / 3.42 ≈ -1.46
• Z₁₂ = (12 – 15) / 3.42 ≈ -0.88
• Z₁₅ = (15 – 15) / 3.42 = 0
• Z₁₈ = (18 – 15) / 3.42 ≈ 0.88
• Z₂₀ = (20 – 15) / 3.42 ≈ 1.46

Interpretation of Z-Scores

• Positive Z-Score: The data point is above the mean.
• Negative Z-Score: The data point is below the mean.
• Zero Z-Score: The data point is exactly at the mean.

Practical Applications

• Statistics: Comparing data points from different distributions.
• Finance: Assessing investment risks.
• Education: Analyzing test scores.

Conclusion

Calculating Z-scores is essential for understanding the relative position of data points within a data set. By following the steps outlined, you can accurately compute Z-scores and interpret their meaning.