How to Calculate Population Standard Deviation with N and X

Population standard deviation is a measure of the dispersion of data points in a population. It quantifies how much the individual data points deviate from the population mean. This guide explains how to calculate it using sample size n and data points x.

What is Population Standard Deviation?

The population standard deviation (σ) is a statistical measure that describes the amount of variation or dispersion in a set of values. Unlike sample standard deviation, which estimates the population standard deviation from a sample, the population standard deviation is calculated from the entire population.

Standard deviation is widely used in statistics, finance, and quality control to understand data distribution and make informed decisions. A lower standard deviation indicates that data points tend to be closer to the mean, while a higher standard deviation indicates greater dispersion.

Formula

The formula for population standard deviation is:

σ = √(Σ(xᵢ - μ)² / N)

Where:

σ = population standard deviation
xᵢ = each individual data point
μ = population mean
N = total number of data points in the population

This formula calculates the square root of the average of the squared differences from the mean. The population mean μ is calculated as:

μ = Σxᵢ / N

Step-by-Step Calculation

Collect all data points in the population (x₁, x₂, ..., xₙ).
Calculate the population mean (μ) using the formula above.
For each data point, subtract the mean and square the result: (xᵢ - μ)².
Sum all the squared differences: Σ(xᵢ - μ)².
Divide the sum by the total number of data points (N).
Take the square root of the result to get the population standard deviation (σ).

Example Calculation

Let's calculate the population standard deviation for the following data set: 4, 7, 13, 16.

Calculate the mean: (4 + 7 + 13 + 16) / 4 = 40 / 4 = 10.
Calculate each squared difference:
- (4 - 10)² = 36
- (7 - 10)² = 9
- (13 - 10)² = 9
- (16 - 10)² = 36
Sum of squared differences: 36 + 9 + 9 + 36 = 90.
Divide by N (4): 90 / 4 = 22.5.
Take the square root: √22.5 ≈ 4.743.

The population standard deviation for this data set is approximately 4.743.

Interpretation

The population standard deviation provides several insights:

It measures the average distance of each data point from the mean.
A smaller standard deviation indicates that data points are closer to the mean.
A larger standard deviation indicates greater dispersion in the data.
It helps in comparing the variability of different populations.

Note: Population standard deviation is different from sample standard deviation, which divides by (n-1) instead of N. This guide focuses on the population version.

FAQ

What is the difference between population and sample standard deviation?: The main difference is in the denominator of the formula. Population standard deviation divides by N (total population size), while sample standard deviation divides by (n-1) to correct for bias in small samples.
When should I use population standard deviation?: Use population standard deviation when you have data for the entire population, not just a sample. This is common in small populations or when you can measure every member.
How is standard deviation different from variance?: Variance is the square of standard deviation. Standard deviation is in the same units as the original data, while variance is in squared units. Both measure dispersion but on different scales.
Can standard deviation be negative?: No, standard deviation is always non-negative because it's calculated as the square root of a squared value. The result is always zero or positive.
What does a high standard deviation mean?: A high standard deviation indicates that data points are spread out over a wider range of values. This suggests greater variability or inconsistency in the data.