Z-score Calculator

Calculate standard scores and analyze normal distribution data

Calculation Mode

Input Values

Data Value (x)*

The individual data point or observation

Mean (μ)*

The average of all values in the dataset

Standard Deviation (σ)*

Measure of variability in the dataset (must be positive)

Show p-values and percentiles

Z-score Results

0.4138

Z-score

62.000

Data Value (x)

58.750

Mean (μ)

7.854

Std Dev (σ)

66.05%

Percentile

66.0% of values are below this point

0.660490

P(Z ≤ z)

Probability of getting this value or lower

0.339510

P(Z > z)

Probability of getting a higher value

Formula Used

z = (x - μ) / σ = (62.000 - 58.750) / 7.854 = 0.4138

Interpretation

Typical value (0.41 standard deviations from mean). This is within the normal range of variation. The data point is above the mean.

Example: Test Scores

Given Data

Test scores: 50, 53, 62, 70

Student's score: 62 points

Question: What is the z-score?

Solution

Mean (μ): (50+53+62+70)/4 = 58.75

Std Dev (σ): 7.854

Z-score: (62 - 58.75) / 7.854 = 0.41

Result

The score of 62 is 0.41 standard deviations above the mean, which is in the 66th percentile.

Z-score Interpretation

z = 0

Exactly at the mean

±1

|z| ≈ 1

~68% of data within ±1σ

±2

|z| ≈ 2

~95% of data within ±2σ

±3

|z| ≈ 3

~99.7% of data within ±3σ

Common Z-score Values

90th percentile:z = 1.28

95th percentile:z = 1.645

97.5th percentile:z = 1.96

99th percentile:z = 2.33

99.5th percentile:z = 2.576

99.9th percentile:z = 3.09

Understanding Z-scores

What is a Z-score?

A Z-score (also called standard score) measures how many standard deviations a data point is from the mean. It allows you to compare values from different normal distributions by standardizing them.

Key Properties

•Mean = 0: The average z-score is always 0
•Std Dev = 1: Standard deviation of z-scores is 1
•Positive/Negative: Above/below the mean

Formula & Calculation

Z-score Formula

z = (x - μ) / σ

z = z-score (standard score)

x = individual data value

μ = population mean

σ = population standard deviation

Reverse Calculations

Find x: x = z × σ + μ

Find μ: μ = x - z × σ

Find σ: σ = (x - μ) / z

Applications

•Education: Standardizing test scores across different exams
•Quality Control: Six Sigma methodology for process improvement
•Research: Identifying outliers and unusual observations
•Finance: Risk assessment and portfolio analysis

Reading Z-score Tables

Z-score tables provide the area under the standard normal curve to the left of a given z-value, which represents the cumulative probability or percentile.

Steps to use Z-table:

1. Find your z-value to one decimal in the left column
2. Find the second decimal in the top row
3. The intersection gives you the cumulative probability
4. Multiply by 100 to get the percentile