How to calculate z-scores in R

statistics

z-score

Learn calculate z-scores in r with clear examples and explanations.

Published

March 26, 2026

Introduction

A z-score is a statistical measure that describes how far a data point is from the mean of a dataset, expressed in terms of standard deviations. Z-scores are essential for standardizing data across different scales, identifying outliers, and comparing values from different distributions. In this tutorial, we’ll learn how to calculate z-scores in R using both manual calculations and built-in functions.

Understanding Z-Scores

The z-score formula is: z = (X - μ) / σ

Where: - X is the individual data point - μ (mu) is the mean of the dataset
- σ (sigma) is the standard deviation of the dataset

Interpreting z-scores: - A z-score of 0 means the data point equals the mean - Positive z-scores indicate values above the mean - Negative z-scores indicate values below the mean - The magnitude shows how many standard deviations away from the mean

For example, if test scores have a mean of 70 and standard deviation of 10, a score of 85 has a z-score of (85-70)/10 = 1.5, meaning it’s 1.5 standard deviations above average.

Setting Up Our Data

First, let’s load the necessary packages and examine our dataset. We’ll use the Palmer penguins data to demonstrate z-score calculations.

library(tidyverse)
library(palmerpenguins)

Let’s explore the penguins dataset and select two numeric variables for our z-score analysis.

penguins |> 
  glimpse()

We can see various measurements for different penguin species. Let’s focus on bill depth and body mass for our z-score calculations.

Preparing the Data

We need to clean our data by removing missing values and selecting our variables of interest.

df <- penguins |>
  drop_na() |>
  select(bill_depth_mm, body_mass_g)

Now let’s examine the basic statistics of our selected variables.

df |> 
  head()

df |> 
  summary()

This shows us the distribution of our variables. Notice the different scales - bill depth is measured in millimeters (around 15-20) while body mass is in grams (around 3000-6000).

Manual Z-Score Calculation

Let’s calculate z-scores manually using the formula. This helps us understand exactly what’s happening mathematically.

df <- df |>
  mutate(
    bill_depth_zscore_manual = (bill_depth_mm - mean(bill_depth_mm)) / sd(bill_depth_mm),
    body_mass_zscore_manual = (body_mass_g - mean(body_mass_g)) / sd(body_mass_g)
  )

df |> 
  head()

Now our data includes the original measurements and their corresponding z-scores. Notice how the z-scores are on a similar scale regardless of the original units.

Using R’s Built-in Scale Function

R provides a convenient scale() function that calculates z-scores automatically. Let’s use it and compare with our manual calculation.

df <- df |>
  mutate(
    bill_depth_zscore = c(scale(bill_depth_mm)),
    body_mass_zscore = c(scale(body_mass_g))
  )

The c() function converts the matrix output of scale() into a vector that works well with mutate().

Let’s verify our calculations by examining the summary statistics of our z-scores.

df |>
  select(contains("zscore")) |>
  summary()

Notice that z-scores have a mean of approximately 0 and standard deviation of 1. This is the key property of standardized scores.

Verifying Our Calculations

Let’s confirm that both methods produce identical results.

all.equal(df$bill_depth_zscore_manual, df$bill_depth_zscore)

This should return TRUE, confirming our manual calculation matches R’s built-in function.

Visualizing Z-Score Transformations

Let’s create visualizations to understand how z-scores transform our data. First, let’s see the relationship between original values and their z-scores.

df |>
  ggplot(aes(x = bill_depth_mm, y = bill_depth_zscore)) +
  geom_point(alpha = 0.7) +
  labs(
    title = "Original Bill Depth vs Z-Score",
    x = "Bill Depth (mm)",
    y = "Z-Score"
  )

Scatter plot showing the linear relationship between original penguin bill depth in millimeters and its calculated z-score in R, illustrating how standardization rescales raw measurements to a mean of zero and unit standard deviation.

This plot shows the linear relationship between original values and z-scores. The transformation preserves the relative relationships while changing the scale.

Let’s also verify that both calculation methods produce identical results visually.

df |>
  ggplot(aes(x = bill_depth_zscore_manual, y = bill_depth_zscore)) +
  geom_point(alpha = 0.7) +
  geom_abline(intercept = 0, slope = 1, color = "red", linetype = "dashed") +
  labs(
    title = "Manual vs Built-in Z-Score Calculation",
    x = "Manual Z-Score",
    y = "Built-in Z-Score"
  )

The points should fall exactly on the diagonal line, confirming both methods are equivalent.

Comparing Distributions

One powerful use of z-scores is comparing values across different distributions. Let’s create a comparison plot.

df |>
  select(bill_depth_zscore, body_mass_zscore) |>
  pivot_longer(everything(), names_to = "variable", values_to = "zscore") |>
  ggplot(aes(x = zscore, fill = variable)) +
  geom_histogram(alpha = 0.7, bins = 30) +
  facet_wrap(~variable) +
  labs(
    title = "Distribution of Z-Scores",
    x = "Z-Score",
    y = "Count"
  )

Faceted histogram of z-scores for penguin bill depth and body mass calculated in R, showing both distributions centered near zero with standard deviation one, demonstrating how the scale() function standardizes variables with very different original units onto a common axis.

Both distributions are now on the same standardized scale, making them directly comparable despite their very different original units.

Summary

Z-scores are powerful tools for standardizing data and making comparisons across different scales. In R, you can calculate them manually using the formula (x - mean(x)) / sd(x) or use the convenient scale() function. Z-scores transform any distribution to have a mean of 0 and standard deviation of 1, while preserving the original relationships between data points. This makes them invaluable for data analysis, outlier detection, and comparing measurements across different variables or datasets.

--- title: "How to calculate z-scores in R" description: "Learn calculate z-scores in r with clear examples and explanations." date: 2026-03-26 categories: ['statistics', 'z-score'] format: html: code-fold: false code-tools: true --- ## Introduction A z-score is a statistical measure that describes how far a data point is from the mean of a dataset, expressed in terms of standard deviations. Z-scores are essential for standardizing data across different scales, identifying outliers, and comparing values from different distributions. In this tutorial, we'll learn how to calculate z-scores in R using both manual calculations and built-in functions. ## Understanding Z-Scores The z-score formula is: **z = (X - μ) / σ** Where: - X is the individual data point - μ (mu) is the mean of the dataset - σ (sigma) is the standard deviation of the dataset **Interpreting z-scores:** - A z-score of 0 means the data point equals the mean - Positive z-scores indicate values above the mean - Negative z-scores indicate values below the mean - The magnitude shows how many standard deviations away from the mean For example, if test scores have a mean of 70 and standard deviation of 10, a score of 85 has a z-score of (85-70)/10 = 1.5, meaning it's 1.5 standard deviations above average. ## Setting Up Our Data First, let's load the necessary packages and examine our dataset. We'll use the Palmer penguins data to demonstrate z-score calculations. ```r library(tidyverse) library(palmerpenguins) ``` Let's explore the penguins dataset and select two numeric variables for our z-score analysis. ```r penguins |> glimpse() ``` We can see various measurements for different penguin species. Let's focus on bill depth and body mass for our z-score calculations. ## Preparing the Data We need to clean our data by removing missing values and selecting our variables of interest. ```r df <- penguins |> drop_na() |> select(bill_depth_mm, body_mass_g) ``` Now let's examine the basic statistics of our selected variables. ```r df |> head() ``` ```r df |> summary() ``` This shows us the distribution of our variables. Notice the different scales - bill depth is measured in millimeters (around 15-20) while body mass is in grams (around 3000-6000). ## Manual Z-Score Calculation Let's calculate z-scores manually using the formula. This helps us understand exactly what's happening mathematically. ```r df <- df |> mutate( bill_depth_zscore_manual = (bill_depth_mm - mean(bill_depth_mm)) / sd(bill_depth_mm), body_mass_zscore_manual = (body_mass_g - mean(body_mass_g)) / sd(body_mass_g) ) ``` ```r df |> head() ``` Now our data includes the original measurements and their corresponding z-scores. Notice how the z-scores are on a similar scale regardless of the original units. ## Using R's Built-in Scale Function R provides a convenient `scale()` function that calculates z-scores automatically. Let's use it and compare with our manual calculation. ```r df <- df |> mutate( bill_depth_zscore = c(scale(bill_depth_mm)), body_mass_zscore = c(scale(body_mass_g)) ) ``` The `c()` function converts the matrix output of `scale()` into a vector that works well with `mutate()`. Let's verify our calculations by examining the summary statistics of our z-scores. ```r df |> select(contains("zscore")) |> summary() ``` Notice that z-scores have a mean of approximately 0 and standard deviation of 1. This is the key property of standardized scores. ## Verifying Our Calculations Let's confirm that both methods produce identical results. ```r all.equal(df$bill_depth_zscore_manual, df$bill_depth_zscore) ``` This should return `TRUE`, confirming our manual calculation matches R's built-in function. ## Visualizing Z-Score Transformations Let's create visualizations to understand how z-scores transform our data. First, let's see the relationship between original values and their z-scores. ```r df |> ggplot(aes(x = bill_depth_mm, y = bill_depth_zscore)) + geom_point(alpha = 0.7) + labs( title = "Original Bill Depth vs Z-Score", x = "Bill Depth (mm)", y = "Z-Score" ) ``` ![Scatter plot showing the linear relationship between original penguin bill depth in millimeters and its calculated z-score in R, illustrating how standardization rescales raw measurements to a mean of zero and unit standard deviation.](/images/statistics/calculate-z-scores-in-r-scatter-original-vs-zscore-ggplot.png) This plot shows the linear relationship between original values and z-scores. The transformation preserves the relative relationships while changing the scale. Let's also verify that both calculation methods produce identical results visually. ```r df |> ggplot(aes(x = bill_depth_zscore_manual, y = bill_depth_zscore)) + geom_point(alpha = 0.7) + geom_abline(intercept = 0, slope = 1, color = "red", linetype = "dashed") + labs( title = "Manual vs Built-in Z-Score Calculation", x = "Manual Z-Score", y = "Built-in Z-Score" ) ``` The points should fall exactly on the diagonal line, confirming both methods are equivalent. ## Comparing Distributions One powerful use of z-scores is comparing values across different distributions. Let's create a comparison plot. ```r df |> select(bill_depth_zscore, body_mass_zscore) |> pivot_longer(everything(), names_to = "variable", values_to = "zscore") |> ggplot(aes(x = zscore, fill = variable)) + geom_histogram(alpha = 0.7, bins = 30) + facet_wrap(~variable) + labs( title = "Distribution of Z-Scores", x = "Z-Score", y = "Count" ) ``` ![Faceted histogram of z-scores for penguin bill depth and body mass calculated in R, showing both distributions centered near zero with standard deviation one, demonstrating how the scale() function standardizes variables with very different original units onto a common axis.](/images/statistics/calculate-z-scores-in-r-distribution-histogram-ggplot.png) Both distributions are now on the same standardized scale, making them directly comparable despite their very different original units. ## Summary Z-scores are powerful tools for standardizing data and making comparisons across different scales. In R, you can calculate them manually using the formula `(x - mean(x)) / sd(x)` or use the convenient `scale()` function. Z-scores transform any distribution to have a mean of 0 and standard deviation of 1, while preserving the original relationships between data points. This makes them invaluable for data analysis, outlier detection, and comparing measurements across different variables or datasets.