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Range, Variance, Standard Deviation

Created:September 2, 2024
Last Updated:March 31, 2025

This Range, Variance, and Standard Deviation Calculator helps you analyze the spread and variability of your data distribution. It calculates the range (difference between maximum and minimum values), variance (average squared deviation from the mean), and standard deviation (square root of variance), helping you understand how dispersed your data points are from their average. The calculator also provides a boxplot to visualize the distribution of your data and an error bar plot to show the mean with standard deviation.

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Range

Definition

The range is the difference between the largest and smallest values in a dataset, measuring the total spread of values.

Formula

Range=xmax−xmin\text{Range} = x_{max} - x_{min}

Example

For the numbers: 2,4,7,8,112, 4, 7, 8, 11

Range=11−2=9\text{Range} = 11 - 2 = 9

Key Points

  • Simple to calculate but sensitive to outliers
  • Only uses two values, ignoring all values in between

Variance

Definition

The variance measures how far a set of numbers are spread out from their mean. It's calculated as the average squared difference from the mean.

Formula

Sample Variance:

s2=∑i=1n(xi−xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}

Population Variance:

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

Example

For the numbers: 2,4,4,62, 4, 4, 6

Mean = 4

s2=(2−4)2+(4−4)2+(4−4)2+(6−4)24−1=4+0+0+43=83≈2.67\begin{align*} s^2 &= \frac{(2-4)^2 + (4-4)^2 + (4-4)^2 + (6-4)^2}{4-1} \\ &= \frac{4 + 0 + 0 + 4}{3} = \frac{8}{3} \approx 2.67 \end{align*}

Key Points

  • Uses squared differences, making it sensitive to outliers
  • Units are squared (e.g., if data is in meters, variance is in meters squared)

Standard Deviation

Definition

The standard deviation is the square root of the variance, providing a measure of spread in the same units as the original data.

Formula

Sample Standard Deviation:

s=∑i=1n(xi−xˉ)2n−1s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}

Population Standard Deviation:

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

Example

Using the previous variance example:

s=2.67≈1.63s = \sqrt{2.67} \approx 1.63

Key Points

  • In same units as original data, making it more interpretable than variance
  • Approximately 68% of data falls within one standard deviation of the mean in a normal distribution

Creating Frequency Tables in R

Here is an example of how to calculate the range, variance, and standard deviation of a dataset in R using the tidyverse package.

R
library(tidyverse)

tips <- read.csv("https://raw.githubusercontent.com/plotly/datasets/master/tips.csv")

spread_measures <- tips |> 
  summarise(
    mean = mean(tip),
    min = min(tip),
    max = max(tip),
    range = max - min,
    IQR = IQR(tip),
    sample_var = var(tip),
    sample_sd = sd(tip),
    pop_var = var(tip) * (n() - 1) / n(),
    pop_sd = sqrt(pop_var)
  )

# boxplot
ggplot(tips, aes(x = "", y = tip)) +
  geom_boxplot(fill = "steelblue") +
  labs(
    title = "Boxplot of Tips",
    y = "Tip Amount"
  ) +
  theme_minimal()

# error bar plot
ggplot(spread_measures, aes(x = "", y = mean, ymin = mean - sample_sd, ymax = mean + sample_sd)) +
  geom_errorbar(width = 0.2, color = "red") +
  geom_point(aes(y = mean), color = "red", size = 3) +
  labs(
    title = "Mean and Standard Deviation of Tips",
    y = "Tip Amount"
  ) +
  theme_minimal()

Related Calculators

Coefficient of Variation (CV) Calculator

Percentile, Quartile Calculator

Normal Distribution Calculator

Box Plot

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