Standard Deviation Calculator
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Standard Deviation Calculator
Calculate the dispersion of a dataset from its mean.
What is Standard Deviation?
Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (average) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values. It is a key indicator of how representative the mean is of the data.
Why is Standard Deviation Important?
Standard deviation is widely used across many fields, including finance, science, engineering, education, and social sciences.
- Finance: To measure the risk or volatility of an investment. A higher standard deviation for a stock's returns suggests greater risk.
- Science/Research: To understand the reliability and variability of experimental results. It helps in determining if observed differences are statistically significant.
- Quality Control: To monitor the consistency of manufactured products. Low standard deviation means higher quality and consistency.
- Social Sciences: To analyze the spread of survey responses or demographic data.
How to Calculate Standard Deviation
There are two common types of standard deviation: population standard deviation (σ) and sample standard deviation (s). The calculator below computes the sample standard deviation, which is typically used when your data is a sample from a larger population.
Formula for Sample Standard Deviation (s):
The formula for sample standard deviation is:
s = sqrt( Σ(xi - x̄)² / (n - 1) )
Where:
s is the sample standard deviation.Σ (Sigma) means "sum of".xi represents each individual data point in the dataset.x̄ (x-bar) is the mean (average) of the dataset.n is the number of data points in the sample.(n - 1) is used for sample standard deviation to provide a less biased estimate of the population standard deviation. This is known as Bessel's correction.
Steps to Calculate Standard Deviation:
- Calculate the Mean (x̄): Sum all the data points and divide by the number of data points (n).
- Calculate Deviations: For each data point (xi), subtract the mean (x̄) to find the deviation:
(xi - x̄).
- Square the Deviations: Square each of the deviations calculated in the previous step:
(xi - x̄)².
- Sum the Squared Deviations: Add up all the squared deviations:
Σ(xi - x̄)².
- Calculate the Variance: Divide the sum of squared deviations by
(n - 1) for sample standard deviation. This value Σ(xi - x̄)² / (n - 1) is called the sample variance.
- Take the Square Root: Calculate the square root of the variance to get the standard deviation (s).
Example Calculation:
Let's calculate the standard deviation for the dataset: 10, 12, 15, 11, 13
- n = 5 (number of data points)
- Mean (x̄) = (10 + 12 + 15 + 11 + 13) / 5 = 61 / 5 = 12.2
- Deviations (xi – x̄):
- 10 – 12.2 = -2.2
- 12 – 12.2 = -0.2
- 15 – 12.2 = 2.8
- 11 – 12.2 = -1.2
- 13 – 12.2 = 0.8
- Squared Deviations (xi – x̄)²:
- (-2.2)² = 4.84
- (-0.2)² = 0.04
- (2.8)² = 7.84
- (-1.2)² = 1.44
- (0.8)² = 0.64
- Sum of Squared Deviations = 4.84 + 0.04 + 7.84 + 1.44 + 0.64 = 14.8
- Variance (s²) = 14.8 / (5 – 1) = 14.8 / 4 = 3.7
- Standard Deviation (s) = sqrt(3.7) ≈ 1.924
This means the data points in this sample tend to be approximately 1.924 units away from the mean of 12.2.
function calculateStandardDeviation() {
var dataInput = document.getElementById("dataValues").value;
var errorMessageDiv = document.getElementById("errorMessage");
var resultDiv = document.getElementById("standardDeviationResult");
errorMessageDiv.innerText = ""; // Clear previous errors
resultDiv.innerText = "–"; // Reset result
if (dataInput.trim() === "") {
errorMessageDiv.innerText = "Please enter at least two data values.";
return;
}
var dataArray = dataInput.split(',')
.map(function(item) {
return parseFloat(item.trim());
})
.filter(function(value) {
return !isNaN(value);
});
if (dataArray.length < 2) {
errorMessageDiv.innerText = "Please enter at least two valid numbers.";
return;
}
var n = dataArray.length;
var sum = 0;
for (var i = 0; i < n; i++) {
sum += dataArray[i];
}
var mean = sum / n;
var sumSquaredDeviations = 0;
for (var i = 0; i < n; i++) {
var deviation = dataArray[i] – mean;
sumSquaredDeviations += deviation * deviation;
}
// Calculate sample variance (divide by n-1)
var variance = sumSquaredDeviations / (n – 1);
// Calculate standard deviation (square root of variance)
var standardDeviation = Math.sqrt(variance);
resultDiv.innerText = standardDeviation.toFixed(4); // Display with 4 decimal places
}