Standard Deviation Calculator

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standard deviation calculator
Sample Standard DeviationPopulation Standard Deviation
Results:
function calculateSD(){var rawData=document.getElementById('data_input').value;var type=document.getElementById('data_type').value;var showSteps=document.getElementById('show_steps').checked;var numbers=rawData.split(/[,\s\n]+/).filter(function(v){return v!==" && !isNaN(v);}).map(Number);if(numbers.length<2){alert('Please enter at least two valid numbers.');return;}var n=numbers.length;var sum=0;for(var i=0;i<n;i++){sum+=numbers[i];}var mean=sum/n;var sqDiffSum=0;for(var j=0;j<n;j++){sqDiffSum+=Math.pow(numbers[j]-mean,2);}var variance;if(type==='sample'){variance=sqDiffSum/(n-1);}else{variance=sqDiffSum/n;}var sd=Math.sqrt(variance);document.getElementById('calculatorAnswer').style.display='block';var typeLabel=(type==='sample')?'Sample Standard Deviation (s)':'Population Standard Deviation (σ)';document.getElementById('mainResult').innerHTML=typeLabel + '
' + sd.toFixed(4);var detailHtml='Count (n): '+n+'
'+'Sum: '+sum.toFixed(4)+'
';if(showSteps){detailHtml+='Mean (x̄): '+mean.toFixed(4)+'
'+'Variance: '+variance.toFixed(4)+'
';}document.getElementById('subResults').innerHTML=detailHtml;}

Standard Deviation Calculator Use

This standard deviation calculator helps you find the spread of a dataset quickly and accurately. Whether you are analyzing scientific data, financial returns, or test scores, understanding the standard deviation is crucial for statistical significance.

To use the calculator, simply select whether your data represents a whole population or just a sample. Then, input your numbers separated by commas, spaces, or new lines. The tool will instantly provide the standard deviation, and you can optionally view the mean and variance.

Dataset Type
Choose "Sample" if your data is a subset of a larger group. Choose "Population" if you have data for every member of the group being studied.
Data Values
The list of numerical values you wish to analyze. Non-numeric characters are automatically filtered out.
Standard Deviation (s or σ)
The primary output, representing the average amount by which each value in the dataset differs from the mean.

How It Works

The standard deviation is the square root of the variance. It tells us how concentrated the data is around the mean (average). A low standard deviation indicates that the data points tend to be very close to the mean, while a high standard deviation indicates that the data points are spread out over a large range of values.

The formulas used by our standard deviation calculator are:

Sample Standard Deviation (s) = √[ Σ(xᵢ – x̄)² / (n – 1) ]
Population Standard Deviation (σ) = √[ Σ(xᵢ – μ)² / N ]

  • xᵢ: Each individual value in the dataset.
  • x̄ or μ: The arithmetic mean (average) of the values.
  • n or N: The total number of values in the dataset.
  • Σ: The sum of the calculated differences.

Calculation Example

Example: Suppose a teacher wants to find the sample standard deviation for test scores: 85, 90, 70, and 75.

Step-by-step solution:

  1. Calculate the Mean (x̄): (85 + 90 + 70 + 75) / 4 = 320 / 4 = 80.
  2. Subtract the Mean from each score and square the result:
    (85-80)² = 25
    (90-80)² = 100
    (70-80)² = 100
    (75-80)² = 25
  3. Sum the squares: 25 + 100 + 100 + 25 = 250.
  4. Divide by (n-1): 250 / (4-1) = 250 / 3 = 83.33 (This is the Variance).
  5. Square root: √83.33 = 9.1287.
  6. Result: The sample standard deviation is 9.1287.

Common Questions

What is the difference between sample and population standard deviation?

Use the population standard deviation when you have data for every individual in the group you are studying (e.g., every student in one specific class). Use the sample standard deviation when you are using a small group to represent a much larger population (e.g., 100 voters representing a whole country). The sample formula uses n-1 to correct for bias in estimating population variance.

Can standard deviation be negative?

No. Because the differences from the mean are squared before being summed, and the final step involves taking a square root, the standard deviation is always zero or positive. A standard deviation of zero means all values in the dataset are identical.

Why is standard deviation better than variance?

Variance is measured in squared units (like dollars squared), which is hard to visualize. The standard deviation calculator converts that value back into the original units of the data (like dollars), making it much easier to interpret and apply to real-world scenarios.

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