# How to calculate z

**Introduction**

Whether you’re a student, researcher, or data analyst, understanding how to calculate a z-score is an essential skill. Also known as the standard score or z-value, the z-score measures the relative position of a data point within a dataset, transforming raw scores into standardized values which can be easily compared and analyzed. In this article, we will walk you through the process of calculating a z-score step by step.

**What is a Z-Score?**

A z-score is a measure of how many standard deviations an observation or data point is from the mean (average) of the dataset. This score is useful for determining how exceptional or unusual a value is when compared to other data points in a distribution. High absolute z-score values indicate that a data point lies far from the mean, while low absolute values indicate that it’s close to the mean.

**Calculating Z-Score: A Step-By-Step Guide**

To calculate the z-score for any given data point within a dataset, you’ll need basic statistical information such as the mean and standard deviation of your dataset. Follow these steps:

**1. Calculate the Mean (μ):** You’ll need to find the average value for your entire dataset first. To do this, sum up all

data points and divide the total by the number of data points in your dataset.

Formula: μ = Σx / N

**2. Calculate Standard Deviation (σ):** To find the standard deviation, calculate the squared difference between each data point and your calculated mean. Next, calculate the average of these squared differences and take its square root.

Formula: σ = √(Σ(x – μ)^2 / N)

**3. Calculate Z-Score**: Subtract the mean from your desired data point, then divide this result by your calculated standard deviation.

Formula: z = (x – μ) / σ

Example

To better understand the process, let’s use an example dataset: {15, 20, 25, 30, 35}

**1. Calculate the Mean (μ):**

μ = (15 + 20 + 25 +30 + 35) / 5

μ = 125 / 5

μ = 25

**2. Calculate Standard Deviation (σ):**

σ = √((15-25)^2 + (20-25)^2 + (25-25)^2 (30-25)^2 + (35-25)^2) / 5

σ = √((-10)^2 + (-5)^2 + (0)^2 +(5)^2 +(10)^2) /5

σ = √(100+25+0+25+100)/5

σ = √(250/5)

σ = √50 ≈ 7.07

**3. Calculate Z-Score for a data point, e.g., x = 30:**

z = (30 – 25) / 7.07

z ≈ 0.71

**Conclusion**

The z-score of approximately .71 implies that our data point of interest is roughly .71 standard deviations above the mean value of the dataset. By learning to calculate z-scores, you can effectively compare data points and analyze large sets of data efficiently and confidently.