Standard Deviation Calculator
Standard Deviation Calculator
Calculate standard deviation, variance, mean, and margin of error for a given set of data points.
Separate numbers by commas, spaces, or new lines.
Standard Deviation (σ)
4.8989794856
Count (N)
8
Sum (Σx)
144
Mean (μ)
18
Variance (σ²)
24
| 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
| Value | Frequency |
|---|---|
| 10 | 1 (12.5%) |
| 12 | 1 (12.5%) |
| 16 | 2 (25.0%) |
| 21 | 1 (12.5%) |
| 23 | 3 (37.5%) |
Worked Examples
A teacher analyzes scores for a class of 5 students: 85, 90, 88, 92, 85. They treat this as a Population.
The calculator finds a mean of 88 and a standard deviation of 2.83. This low deviation indicates the students performed consistently close to each other.
A factory tests a Sample of 6 bolts for length (mm): 10.1, 9.9, 10.2, 10.0, 9.8, 10.0.
The tool calculates a mean of 10.0 and a sample standard deviation of 0.141. This helps the engineer determine if the machinery is within the required 0.15mm tolerance level.
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 (xi) 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:
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):
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 Solve Statistics with MyApexCalc?
Simultaneous Sample & Population Results
View both calculations side-by-side on your screen so you never have to worry about selecting the wrong option beforehand.
Complete Statistical Breakdown
Get your dataset's mean, variance, count (n), sum, minimum, maximum, and range along with your standard deviation results.
Flexible Input Formats
Simply paste your raw numbers separated by commas, spaces, or line breaks to process your data immediately.