Standard Deviation Calculator

Calculate sample and population standard deviation, variance, mean, standard error, and absolute deviations for your data.

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Banker's Rounded Precision
Sample & Population Dual Modes
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Separate values with commas, spaces, semicolons, tabs, or newlines. Filters automatically.

Sample SD (s)
5.23723
(divisor: n - 1)
Population SD (Ļƒ)
4.89898
(divisor: N)
Mean (Ī¼): 18-1Ļƒ+1Ļƒ-2Ļƒ+2ĻƒHover dots below line to inspect deviationsMin: 10Max: 23
Sample Variance (sĀ²)
27.42857
Pop. Variance (ĻƒĀ²)
24
Mean (Ī¼)
18
Sum / Count (n)
144 / 8
Mean Abs. Dev (MAD)
4.5
Std. Error (SEM)
1.85164
Sample CV
0.29096
Population CV
0.27217
1. Sorted Dataset:
[10, 12, 16, 16, 21, 23, 23, 23] (n = 8)
2. Mean (Ī¼ or xĢ„):
Mean = Sum / n = (10 + 12 + 16 + 16 + 21 + 23 + 23 + 23) / 8 = 144 / 8 = 18
3. Calculate Deviations & Squares:
iValue (x_i)Deviation (x_i - mean)Squared (x_i - mean)Ā²Abs |x_i - mean|
110-8648
212-6366
316-242
416-242
521393
6235255
7235255
8235255
Sum144019236
4. Variance Calculations:
Sample Variance (sĀ²): Use divisor (n - 1)
sĀ² = Sum of Squares / (n - 1) = 192 / (8 - 1) = 27.42857
Population Variance (ĻƒĀ²): Use divisor (N)
ĻƒĀ² = Sum of Squares / N = 192 / 8 = 24
5. Standard Deviation Calculations:
Sample Standard Deviation (s):
s = āˆšsĀ² = āˆš(27.42857) = 5.23723
Population Standard Deviation (Ļƒ):
Ļƒ = āˆšĻƒĀ² = āˆš(24) = 4.89898
6. Secondary Metrics:
  • Mean Absolute Deviation (MAD): Average of absolute deviations
    MAD = Sum(|x_i - mean|) / n = 36 / 8 = 4.5
  • Standard Error of Mean (SEM): Spread of sample means
    SEM = s / āˆšn = 5.23723 / āˆš8 = 1.85164
  • Coefficient of Variation (CV): Normalized dispersion (SD / Mean)
    Sample CV = s / Mean = 5.23723 / 18 = 0.29096
    Population CV = Ļƒ / Mean = 4.89898 / 18 = 0.27217
Scientific References & Assumptions
Assumptions:
  • Data is drawn from a normally-distributed interval variable population.
  • Divisors of $n-1$ (Bessel's correction) are used for sample estimation to correct bias, and $N$ for populations.
  • Zero variance occurs when all values in the dataset are identical.
  • Standard error and coefficient of variation are undefined when sample size is 1 or mean is 0, respectively.
Sources & Citations:

Frequently Asked Questions

What is the difference between sample and population standard deviation?

Population standard deviation (Ļƒ) is used when you have the entire population dataset; it uses a divisor of N. Sample standard deviation (s) is used when your dataset is a sample representing a larger population; it uses Bessel's correction with a divisor of n - 1 to adjust for bias in estimating variance.

Why do we divide by n - 1 instead of n for the sample standard deviation?

Dividing by n - 1 (Bessel's correction) corrects the bias in the estimation of the population variance. Using n underestimates variance because the sample mean is closer to the sample data points than the population mean is.

Can standard deviation be negative?

No, standard deviation is always greater than or equal to zero. This is because it is the square root of the variance, and the variance is the average of squared differences, which are always non-negative.

What does it mean if the standard deviation is zero?

A standard deviation of zero means that all values in the dataset are identical. There is no variance or dispersion; every data point equals the mean.

How does the coefficient of variation (CV) help in comparing datasets?

The coefficient of variation (CV = SD / Mean) represents the relative standard deviation. Because it is unitless, it allows you to compare the relative variability of datasets with different units or widely different means.

How does the mean absolute deviation (MAD) compare to standard deviation?

While standard deviation squares the deviations from the mean (making it more sensitive to outliers), MAD takes the simple average of absolute deviations. MAD is less affected by extreme outliers than standard deviation.

What is the standard error of the mean (SEM)?

SEM measures the precision of the sample mean as an estimate of the true population mean. It represents the standard deviation of the theoretical distribution of sample means (s / āˆšn).

About the Standard Deviation Calculator

Quickly calculate population and sample standard deviation and variance. Enter your numbers with commas, spaces, or semicolons to see sorted lists, step-by-step arithmetic solutions, and visual distribution charts.

Mathematical Formula & Logic

Standard deviation is computed based on population or sample formulas: - Sample Standard Deviation: s = sqrt(Sum((x_i - mean)^2) / (n - 1)) (Uses Bessel's correction to estimate population variance without bias) - Population Standard Deviation: sigma = sqrt(Sum((x_i - mean)^2) / n) (Represents the absolute variance of the entire population) - Mean Absolute Deviation (MAD): MAD = Sum(|x_i - mean|) / n - Standard Error of the Mean (SEM): SEM = s / sqrt(n)

Step-by-Step Example

Worked Examples: 1. Basic Set: For the dataset: 10, 12, 23, 23, 16, 23, 21, 16. The count is 8, and the sum is 144. The mean is 144 / 8 = 18. The sum of squared deviations is 192. The sample variance is 192 / 7 = 27.42857. The sample standard deviation is sqrt(27.42857) = 5.23723. The population variance is 192 / 8 = 24. The population standard deviation is sqrt(24) = 4.89898. 2. Negatives: Consider the dataset: -5, -2, 0, 2, 5. The count is 5, sum is 0, mean is 0. The sum of squared deviations is 58. The sample variance is 58 / 4 = 14.5, yielding a sample standard deviation of 3.80789. The population variance is 58 / 5 = 11.6, yielding a population standard deviation of 3.40588.

Reference Data & Values

metricformuladescription
Sample Standard Deviation (s)s = āˆš[Ī£(x_i - xĢ„)Ā² / (n - 1)]Measures variation in a sample from a larger population (Bessel's correction)
Population Standard Deviation (Ļƒ)Ļƒ = āˆš[Ī£(x_i - Ī¼)Ā² / N]Measures variation across the entire population
Sample Variance (sĀ²)sĀ² = Ī£(x_i - xĢ„)Ā² / (n - 1)Estimated population variance from a sample
Population Variance (ĻƒĀ²)ĻƒĀ² = Ī£(x_i - Ī¼)Ā² / NAbsolute variance of the population
Mean Absolute Deviation (MAD)MAD = Ī£|x_i - xĢ„| / nAverage absolute distance from the mean, less outlier-sensitive
Standard Error of the Mean (SEM)SEM = s / āˆšnEstimates precision/variability of the sample mean

Frequently Asked Questions

Population standard deviation (Ļƒ) is used when the dataset represents the entire group under study, dividing by the count (N). Sample standard deviation (s) is used when the dataset is a random subset of a larger population, dividing by N - 1 (Bessel's correction) to prevent statistical bias.
Dividing by n - 1, known as Bessel's correction, corrects the bias in the estimation of the population variance. Since the sample mean is calculated from the sample itself, it tends to be closer to the sample data points than the true population mean, leading to an underestimate of variance unless n - 1 is used.
No, standard deviation cannot be negative. It represents a distance from the mean, and because the formula squares the individual deviations before summing and taking the square root, the result is always a non-negative number (zero or positive).
A standard deviation of zero indicates that there is no variation in the dataset. All data points are exactly identical to one another, meaning they do not deviate from the mean at all.
Bessel's correction is the use of n - 1 instead of n in the formula for the sample variance and sample standard deviation. It compensates for the fact that sample data is slightly less spread out than the entire population.
The standard error of the mean (SEM) measures the dispersion of sample means around the population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size (n).
The coefficient of variation (CV) is a relative measure of dispersion, calculated by dividing the standard deviation by the mean. It is often expressed as a percentage, enabling comparison of variability between datasets with different units or scales.
Mean Absolute Deviation (MAD) is the average of the absolute differences between each data point and the mean. Unlike standard deviation, it does not square the differences, making it less sensitive to extreme outliers.