Whiz Tools

Z-Score Calculator

Introduction

The z-score (or standard score) is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. The z-score is a crucial tool in statistics, allowing for the standardization of different datasets and the identification of outliers.

Formula

The z-score is calculated using the following formula:

z=x−μσz = \frac{x - \mu}{\sigma}

Where:

  • zz = z-score
  • xx = individual data point
  • μ\mu = mean of the dataset
  • σ\sigma = standard deviation of the dataset

This formula calculates the number of standard deviations a data point is from the mean.

Calculation

To compute the z-score of a data point:

  1. Calculate the Mean (μ\mu):

    Sum all data points and divide by the number of data points.

    μ=∑i=1nxin\mu = \frac{\sum_{i=1}^{n} x_i}{n}

  2. Calculate the Standard Deviation (σ\sigma):

    • Variance (σ2\sigma^2):

      σ2=∑i=1n(xi−μ)2n\sigma^2 = \frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}

    • Standard Deviation:

      σ=σ2\sigma = \sqrt{\sigma^2}

  3. Compute the Z-Score:

    Substitute the values into the z-score formula.

    z=x−μσz = \frac{x - \mu}{\sigma}

Edge Cases

  • Zero Standard Deviation (σ=0\sigma = 0):

    When all data points are identical, the standard deviation is zero, making the z-score undefined because you cannot divide by zero. In this case, the concept of a z-score does not apply.

  • Data Point Equal to Mean (x=μx = \mu):

    If the data point equals the mean, the z-score is zero, indicating it is exactly average.

  • Non-Numeric Inputs:

    Ensure all inputs are numeric. Non-numeric inputs will result in calculation errors.

Cumulative Probability

The cumulative probability associated with a z-score represents the probability that a random variable from a standard normal distribution will be less than or equal to the given value. It's the area under the normal distribution curve to the left of the specified z-score.

Mathematically, the cumulative probability PP is calculated using the cumulative distribution function (CDF) of the standard normal distribution:

P(Z≤z)=Φ(z)=12π∫−∞ze−t2/2 dtP(Z \leq z) = \Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2} \, dt

Where:

  • Φ(z)\Phi(z) = CDF of the standard normal distribution at zz

The cumulative probability is essential in statistics for determining the likelihood of a value occurring within a certain range. It is widely used in fields like quality control, finance, and social sciences.

SVG Diagram

Below is an SVG diagram illustrating the standard normal distribution curve and the z-score:

μ x z

Standard Normal Distribution

Figure: Standard Normal Distribution Curve with Z-Score Shaded

This diagram shows the normal distribution curve with the mean μ\mu at the center. The shaded area represents the cumulative probability up to the data point xx, corresponding to the z-score.

Use Cases

Applications

  • Standardization Across Different Scales:

    Z-scores allow comparison between data from different scales by standardizing the datasets.

  • Outlier Detection:

    Identifying data points that are significantly distant from the mean (e.g., z-scores less than -3 or greater than 3).

  • Statistical Testing:

    Used in hypothesis testing, including z-tests, to determine if a sample mean significantly differs from a known population mean.

  • Quality Control:

    In manufacturing, z-scores help monitor processes to ensure outputs remain within acceptable limits.

  • Finance and Investment:

    Assessing stock performance by comparing returns relative to the average market performance.

Alternatives

  • T-Score:

    Similar to the z-score but used when the sample size is small and the population standard deviation is unknown.

  • Percentile Rank:

    Indicates the percentage of scores in its frequency distribution that are equal to or lower than it.

  • Standard Deviation Units:

    Using raw standard deviation values without standardizing as z-scores.

History

The concept of the z-score stems from the work on the normal distribution by Carl Friedrich Gauss in the early 19th century. The standard normal distribution, fundamental to z-scores, was further developed by statisticians such as Abraham de Moivre and Pierre-Simon Laplace. The use of z-scores became widespread with the advancement of statistical methods in the 20th century, particularly in psychological testing and quality control.

Examples

Excel

## Calculate z-score in Excel
## Assuming data point in cell A2, mean in cell B2, standard deviation in cell C2
=(A2 - B2) / C2

R

## Calculate z-score in R
calculate_z_score <- function(x, mean, sd) {
  if (sd == 0) {
    stop("Standard deviation cannot be zero.")
  }
  z <- (x - mean) / sd
  return(z)
}

## Example usage:
x <- 85
mu <- 75
sigma <- 5
z_score <- calculate_z_score(x, mu, sigma)
print(paste("Z-score:", z_score))

MATLAB

% Calculate z-score in MATLAB
function z = calculate_z_score(x, mu, sigma)
    if sigma == 0
        error('Standard deviation cannot be zero.');
    end
    z = (x - mu) / sigma;
end

% Example usage:
x = 90;
mu = 80;
sigma = 8;
z = calculate_z_score(x, mu, sigma);
fprintf('Z-score: %.2f\n', z);

JavaScript

// Calculate z-score in JavaScript
function calculateZScore(x, mu, sigma) {
  if (sigma === 0) {
    throw new Error('Standard deviation cannot be zero.');
  }
  return (x - mu) / sigma;
}

// Example usage:
const x = 100;
const mu = 85;
const sigma = 7;
try {
  const z = calculateZScore(x, mu, sigma);
  console.log(`Z-score: ${z.toFixed(2)}`);
} catch (error) {
  console.error(error.message);
}

Python

## Calculate z-score in Python
def calculate_z_score(x, mu, sigma):
    if sigma == 0:
        raise ValueError("Standard deviation cannot be zero.")
    return (x - mu) / sigma

## Example usage:
x = 95
mu = 88
sigma = 4
try:
    z = calculate_z_score(x, mu, sigma)
    print("Z-score:", round(z, 2))
except ValueError as e:
    print(e)

Java

// Calculate z-score in Java
public class ZScoreCalculator {
    public static double calculateZScore(double x, double mu, double sigma) {
        if (sigma == 0) {
            throw new IllegalArgumentException("Standard deviation cannot be zero.");
        }
        return (x - mu) / sigma;
    }

    public static void main(String[] args) {
        double x = 110;
        double mu = 100;
        double sigma = 5;

        try {
            double z = calculateZScore(x, mu, sigma);
            System.out.printf("Z-score: %.2f%n", z);
        } catch (IllegalArgumentException e) {
            System.err.println(e.getMessage());
        }
    }
}

C/C++

// Calculate z-score in C++
#include <iostream>
#include <stdexcept>

double calculate_z_score(double x, double mu, double sigma) {
    if (sigma == 0) {
        throw std::invalid_argument("Standard deviation cannot be zero.");
    }
    return (x - mu) / sigma;
}

int main() {
    double x = 130;
    double mu = 120;
    double sigma = 10;

    try {
        double z = calculate_z_score(x, mu, sigma);
        std::cout << "Z-score: " << z << std::endl;
    } catch (const std::exception &e) {
        std::cerr << e.what() << std::endl;
    }

    return 0;
}

Ruby

## Calculate z-score in Ruby
def calculate_z_score(x, mu, sigma)
  raise ArgumentError, "Standard deviation cannot be zero." if sigma == 0
  (x - mu) / sigma
end

## Example usage:
x = 105
mu = 100
sigma = 5
begin
  z = calculate_z_score(x, mu, sigma)
  puts "Z-score: #{z.round(2)}"
rescue ArgumentError => e
  puts e.message
end

PHP

<?php
// Calculate z-score in PHP
function calculate_z_score($x, $mu, $sigma) {
  if ($sigma == 0) {
    throw new Exception("Standard deviation cannot be zero.");
  }
  return ($x - $mu) / $sigma;
}

// Example usage:
$x = 115;
$mu = 110;
$sigma = 5;

try {
  $z = calculate_z_score($x, $mu, $sigma);
  echo "Z-score: " . round($z, 2);
} catch (Exception $e) {
  echo $e->getMessage();
}
?>

Rust

// Calculate z-score in Rust
fn calculate_z_score(x: f64, mu: f64, sigma: f64) -> Result<f64, String> {
    if sigma == 0.0 {
        return Err("Standard deviation cannot be zero.".to_string());
    }
    Ok((x - mu) / sigma)
}

fn main() {
    let x = 125.0;
    let mu = 115.0;
    let sigma = 5.0;

    match calculate_z_score(x, mu, sigma) {
        Ok(z) => println!("Z-score: {:.2}", z),
        Err(e) => println!("{}", e),
    }
}

C#

// Calculate z-score in C#
using System;

public class ZScoreCalculator
{
    public static double CalculateZScore(double x, double mu, double sigma)
    {
        if (sigma == 0)
            throw new ArgumentException("Standard deviation cannot be zero.");
        return (x - mu) / sigma;
    }

    public static void Main()
    {
        double x = 135;
        double mu = 125;
        double sigma = 5;

        try
        {
            double z = CalculateZScore(x, mu, sigma);
            Console.WriteLine($"Z-score: {z:F2}");
        }
        catch (ArgumentException e)
        {
            Console.WriteLine(e.Message);
        }
    }
}

Go

// Calculate z-score in Go
package main

import (
    "errors"
    "fmt"
)

func calculateZScore(x, mu, sigma float64) (float64, error) {
    if sigma == 0 {
        return 0, errors.New("standard deviation cannot be zero")
    }
    return (x - mu) / sigma, nil
}

func main() {
    x := 140.0
    mu := 130.0
    sigma := 5.0

    z, err := calculateZScore(x, mu, sigma)
    if err != nil {
        fmt.Println(err)
    } else {
        fmt.Printf("Z-score: %.2f\n", z)
    }
}

Swift

// Calculate z-score in Swift
func calculateZScore(x: Double, mu: Double, sigma: Double) throws -> Double {
    if sigma == 0 {
        throw NSError(domain: "Standard deviation cannot be zero.", code: 1, userInfo: nil)
    }
    return (x - mu) / sigma
}

// Example usage:
let x = 120.0
let mu = 110.0
let sigma = 5.0

do {
    let z = try calculateZScore(x: x, mu: mu, sigma: sigma)
    print("Z-score: \(String(format: "%.2f", z))")
} catch let error as NSError {
    print(error.domain)
}

References

  1. Standard Score - Wikipedia

    https://en.wikipedia.org/wiki/Standard_score

  2. Understanding Z-Scores - Statistics Solutions

    https://www.statisticssolutions.com/understanding-z-scores/

  3. Normal Distribution and Z-Scores - Khan Academy

    https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/a/z-scores-review

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