Standard Deviation Calculator

Calculate standard deviation, variance, and mean with step-by-step solutions

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∑Math Input
4, 8, 6, 5, 3
10, 20, 30, 40, 50
2.5, 3.1, 4.7, 1.8

What is Standard Deviation?

Standard deviation measures how spread out data values are from the mean. A low standard deviation means data points cluster near the mean; a high standard deviation means data is more spread out.

Population Standard Deviation

Used when you have data for the entire population:

σ=∑i=1N(xi−μ)2N\sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}

Sample Standard Deviation

Used when you have a sample from a larger population (uses n−1n-1 for Bessel's correction):

s=∑i=1n(xi−xˉ)2n−1s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}

where μ\mu (or xˉ\bar{x}) is the mean and NN (or nn) is the number of data points.

How to Calculate Standard Deviation

Step-by-Step Process

  1. Find the mean xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}
  2. Subtract the mean from each data point: (xi−xˉ)(x_i - \bar{x})
  3. Square each difference: (xi−xˉ)2(x_i - \bar{x})^2
  4. Sum all squared differences: ∑(xi−xˉ)2\sum(x_i - \bar{x})^2
  5. Divide by nn (population) or n−1n-1 (sample) to get the variance
  6. Take the square root to get the standard deviation

Related Measures

MeasureFormulaMeaning
Meanxˉ=∑xin\bar{x} = \frac{\sum x_i}{n}Average value
Variances2=∑(xi−xˉ)2n−1s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}Squared spread
Standard Deviations=s2s = \sqrt{s^2}Spread in original units

Examples

Step 1: Mean: xˉ=4+8+6+5+35=265=5.2\bar{x} = \frac{4+8+6+5+3}{5} = \frac{26}{5} = 5.2
Step 2: Squared differences: (4−5.2)2=1.44(4-5.2)^2=1.44, (8−5.2)2=7.84(8-5.2)^2=7.84, (6−5.2)2=0.64(6-5.2)^2=0.64, (5−5.2)2=0.04(5-5.2)^2=0.04, (3−5.2)2=4.84(3-5.2)^2=4.84
Step 3: Sum: 1.44+7.84+0.64+0.04+4.84=14.81.44+7.84+0.64+0.04+4.84 = 14.8
Step 4: Variance: s2=14.85−1=3.7s^2 = \frac{14.8}{5-1} = 3.7
Step 5: Standard deviation: s=3.7≈1.924s = \sqrt{3.7} \approx 1.924
Answer: s≈1.924s \approx 1.924

Step 1: Mean: μ=10+20+303=20\mu = \frac{10+20+30}{3} = 20
Step 2: Squared differences: (10−20)2=100(10-20)^2=100, (20−20)2=0(20-20)^2=0, (30−20)2=100(30-20)^2=100
Step 3: Variance: σ2=100+0+1003=2003≈66.67\sigma^2 = \frac{100+0+100}{3} = \frac{200}{3} \approx 66.67
Step 4: Standard deviation: σ=66.67≈8.165\sigma = \sqrt{66.67} \approx 8.165
Answer: σ≈8.165\sigma \approx 8.165

Frequently Asked Questions

Population standard deviation divides by N (total data points), while sample standard deviation divides by n-1 (Bessel's correction) to give an unbiased estimate of the true population spread.

A high standard deviation indicates that data points are spread out over a wider range of values, meaning there is more variability in the data set.

Variance is the square of the standard deviation. It measures the average squared distance from the mean. Standard deviation is preferred for interpretation because it uses the same units as the data.

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