Standard Deviation Calculator

Calculate standard deviation, variance, mean, and margin of error for a given set of data points.

Data Input

Enter Data Points

Separate numbers by commas, spaces, or new lines.

Calculation Mode

Population

Sample

Quick Tip

Use Population if your data includes every member of the group. Use Sample if you are analyzing a subset to represent a larger group.

Standard Deviation (σ)

4.8989794856

Count (N)

Sum (Σx)

144

Mean (μ)

Variance (σ²)

Margin of Error (Confidence Interval)

Level	Margin of Error
68.3%, σx̄	18 ±1.732 (±9.62%)
90%, 1.645σx̄	18 ±2.849 (±15.83%)
95%, 1.96σx̄	18 ±3.395 (±18.86%)
99%, 2.576σx̄	18 ±4.462 (±24.79%)
99.9%, 3.291σx̄	18 ±5.700 (±31.67%)
99.99%, 3.891σx̄	18 ±6.739 (±37.44%)
99.999%, 4.417σx̄	18 ±7.650 (±42.50%)
99.9999%, 4.892σx̄	18 ±8.473 (±47.07%)

SEM (σx̄) = 1.7320508076

Frequency Table

Value	Frequency
10	1 (12.5%)
12	1 (12.5%)
16	2 (25.0%)
21	1 (12.5%)
23	3 (37.5%)

Master Statistical Data Analysis with MyApexCalc

Whether you are analyzing test scores in a classroom, assessing risk in a financial portfolio, evaluating quality control metrics in manufacturing, or processing scientific research observations, understanding how data spreads out is critical. While the average (mean) gives you a central starting point, it doesn't tell you if your numbers are closely grouped or wildly scattered. Our free online Standard Deviation Calculator takes the pain out of statistics, serving as an instant tool for determining standard deviation, variance, and mean with a single click.

Understanding the Mathematics: How to Find the Standard Deviation

Standard deviation measures the average distance of each data point in a set from the mean of that set. To find the standard deviation of your data, the calculation engine runs through several steps.

First, we calculate the arithmetic mean (μ or x̄) of the dataset. Next, we find the squared difference of each individual data point (x_i) from that mean. Finally, we sum those squared differences and calculate the average. The exact equation for standard deviation depends entirely on whether your data represents an entire population or just a sample size:

1. Sample Standard Deviation (s)

Use this Standard Deviation formula if your dataset is a representative subset of a larger group. It utilizes Bessel's correction (n - 1) in the denominator to correct for bias:

s = √[ Σ(x_i - x̄)² / (n - 1) ]

2. Population Standard Deviation (σ)

Use this calculation if your dataset represents every single member of the group you are evaluating (such as every student in a single school class):

σ = √[ Σ(x_i - μ)² / n ]

x_i: individual value

x̄ / μ: arithmetic mean

n: total data points

Σ: summation symbol

Step-by-Step Practical Example

Let's analyze a simple sample dataset of five numbers: 2, 4, 4, 4, 5.5.

Step 1: Find the Mean

Mean (x̄) = (2 + 4 + 4 + 4 + 5.5) / 5 = 19.5 / 5 = 3.9

Step 2: Subtract Mean and Square Result

(2 - 3.9)² = 3.61
(4 - 3.9)² = 0.01 (x3)
(5.5 - 3.9)² = 2.56

Step 3: Sum the Squares

Σ = 3.61 + 0.01 + 0.01 + 0.01 + 2.56 = 6.2

Step 4: Final Result

Variance (s²) = 6.2 / (5 - 1) = 1.55

Standard Deviation (s) = √1.55 ≈ 1.245

Why Choose MyApexCalc?

Side-by-Side Results
View both Sample and Population calculations instantly so you never have to worry about selecting the wrong option beforehand.
Flexible Input Formats
Simply paste your raw numbers separated by commas, spaces, or line breaks to process your data immediately.