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Standard Deviation Calculator

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How to Use This Calculator

Step 1: Enter numbers separated by commas or spaces.

Step 2: Select Population (all data) or Sample (subset).

Step 3: Click Calculate for SD, mean, variance, count, sum.

Step 4: Review step-by-step breakdown.

What Is Standard Deviation?

Standard deviation measures spread from the average. Low SD = data clusters around mean. High SD = spread out.

Scores [90,91,89,90,92] have low SD (~1.1). Scores [60,70,80,90,100] have high SD (~15.8).

Population (σ) divides by N for all data. Sample (s) divides by N-1 (Bessel's correction) for subsets.

Standard Deviation Formula

Population: σ = √[Σ(xi-μ)²/N]
Sample: s = √[Σ(xi-x̄)²/(N-1)]

Example: Data: 4,8,6,5,3
Mean=5.2, Squared diffs: 1.44,7.84,0.64,0.04,4.84
Variance=14.80/5=2.96, SD=√2.96=1.7205

SD Interpretation

ContextLow SDHigh SD
Test scoresConsistentWide range
ManufacturingQuality consistentQuality issues
StocksStableVolatile
TemperatureConsistentExtreme variation
SurveysConsensusDivided

Examples

Example 1: Scores 85,90,78,92,88,76,95,83 → SD=5.97

Example 2: Sales (sample) 12000,15000,11000,14000,13500,16000 → SD=1,817

Example 3: Weights 50.2,49.8,50.1,50.0,49.9,50.3,49.7 → SD=0.20

Tips

Population for ALL data, Sample for subsets. When in doubt, use sample.

Watch for outliers — one extreme value dramatically increases SD.

68-95-99.7 rule for normal distributions.

Paste from spreadsheets — handles commas, spaces, tabs, newlines.

Frequently Asked Questions

When should I use population vs sample standard deviation?
Use population standard deviation (σ) when your data includes every member of the group — for example, test scores for all 30 students in a class, or every product in your inventory. Use sample standard deviation (s) when your data is a subset of a larger population — for example, surveying 200 voters to estimate opinion across a whole state. The sample formula divides by N-1 instead of N to correct for the bias inherent in estimating from a subset.
What is variance and how does it relate to standard deviation?
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of variance. The reason we take the square root is that variance is in squared units — if your data is in meters, variance is in meters-squared, which is hard to interpret. Standard deviation brings it back to the original unit (meters), making it much more practical for understanding how spread out your data is.
Can standard deviation be negative?
No. Standard deviation is always zero or positive. It equals zero only when every data point is identical — meaning there is no spread at all. Since the formula squares the differences from the mean and then takes a square root, the result can never be negative. A very small standard deviation means the data points are clustered tightly around the mean.
What is considered a good or bad standard deviation?
There is no universal good or bad standard deviation — it depends entirely on context. A standard deviation of 5 on a test scored out of 100 indicates consistent performance, but the same SD of 5 on dice rolls (range 1-6) would mean extreme variation. To compare meaningfully, use the coefficient of variation: divide SD by the mean and multiply by 100%. A CV under 15% is generally considered low variability.
How many numbers do I need to enter?
You need at least two numbers to calculate a meaningful standard deviation. With only one number, there is no spread to measure. For sample standard deviation specifically, you need at least two data points because the formula divides by N-1, and dividing by zero (when N=1) is undefined. For reliable statistical analysis, larger datasets (30+ values) give more meaningful results.
Can I paste data from a spreadsheet?
Yes. Copy a column of numbers from Microsoft Excel, Google Sheets, or any spreadsheet program and paste it directly into the input field. The calculator handles numbers separated by commas, spaces, tabs, or newlines, so pasted spreadsheet data works automatically. You do not need to reformat the data — just copy the column and paste.

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