Learning goals

What you should understand

  • Explain how variance and standard deviation use the same deviations from the mean.
  • Choose the sample or population denominator from the role of the data.
  • Interpret squared units for variance and original units for standard deviation.
  • Predict how shifting or rescaling every observation changes both measures.
Before you begin

You should know how to find an arithmetic mean, subtract signed values, square numbers, and distinguish a sample from a complete population.

One spread calculation in two unit systems

Squaring prevents positive and negative deviations from cancelling and gives larger deviations more influence. The final square root does not change the ordering of spreads; it changes the units back to those of the original observations.

Sample variances2=∑i=1n(xi−xˉ)2n−1s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}
Sample standard deviations=s2s=\sqrt{s^2}
Population varianceσ2=∑i=1N(xi−μ)2N\sigma^2=\frac{\sum_{i=1}^{N}(x_i-\mu)^2}{N}
Population standard deviationσ=σ2\sigma=\sqrt{\sigma^2}
  • Use n minus 1 only when the listed data are a sample used to estimate population spread.
  • Use N when the listed values are the complete population being described.
  • Both measures are nonnegative and equal zero only when every observation is identical.

Reusable process

Move from raw deviations to an interpretable spread

  1. 1
    Identify the data role

    Decide whether the values are a sample or the complete population before choosing the denominator.

  2. 2
    Find deviations from the matching mean

    Subtract the sample mean or population mean from each observation.

    di=xi−xˉordi=xi−μd_i=x_i-\bar{x}\quad\text{or}\quad d_i=x_i-\mu
  3. 3
    Square, add, and divide

    The average or adjusted average of squared deviations is variance.

    variance=sum of squared deviationsselected denominator\text{variance}=\frac{\text{sum of squared deviations}}{\text{selected denominator}}
  4. 4
    Take the nonnegative square root

    Standard deviation returns spread to the original measurement units.

    standard deviation=variance\text{standard deviation}=\sqrt{\text{variance}}
  5. 5
    Check scaling and units

    Multiplying all observations by c multiplies standard deviation by absolute c and variance by c squared.

    Var⁡(cX)=c2Var⁡(X),SD(cX)=∣c∣SD(X)\operatorname{Var}(cX)=c^2\operatorname{Var}(X),\qquad SD(cX)=|c|SD(X)

Spread comparison

Watch units change while the data stay the same

Use the scale control to multiply 1, 2, and 3 by a common factor. Compare how the population variance and standard deviation respond.

1, 2, 3⟶c, 2c, 3c1,\ 2,\ 3\quad\longrightarrow\quad c,\ 2c,\ 3c
1
Scaled data1, 2, 3
Population variance0.667
Standard deviation0.816

At scale 1, variance is about 0.667 and standard deviation is about 0.816.

Check the reasoning

At scale c, the population variance is 2c squared over 3, while the population standard deviation is absolute c times the square root of 2/3. The text fallback is: multiplying every value by 3 changes variance from 2/3 to 6 and standard deviation from about 0.816 to about 2.449.

Worked examples

Build from recognition to verification

Population spread for three values

Treat 2, 4, and 6 as the entire population.

2, 4, 62,\ 4,\ 6
  1. Find the mean: The center is 4.
    μ=4\mu=4
  2. Add squared deviations: The deviations are negative 2, 0, and 2.
    SS=4+0+4=8SS=4+0+4=8
  3. Divide and take the root: Use N = 3, then take the square root.
    σ2=8/3,σ=8/3≈1.633\sigma^2=8/3,\quad \sigma=\sqrt{8/3}\approx1.633

Answer

σ2=83,σ≈1.633\sigma^2=\frac83,\qquad \sigma\approx1.633

Check: Squaring the exact standard deviation returns 8/3, and multiplying variance by 3 returns the squared-deviation total 8.

Sample spread for evenly spaced values

Treat 1, 3, and 5 as a sample from a larger population.

1, 3, 51,\ 3,\ 5
  1. Find the sample mean: The center is 3.
    xˉ=3\bar{x}=3
  2. Add squared deviations: The squared deviations are 4, 0, and 4.
    SS=8SS=8
  3. Use n minus 1: Divide by 2 and take the square root.
    s2=4,s=2s^2=4,\quad s=2

Answer

s2=4,s=2s^2=4,\qquad s=2

Check: Two times the sample variance 4 returns SS = 8, and 2 squared returns the same variance.

Zero spread

Identical observations have no distance from their common mean.

7, 7, 7, 77,\ 7,\ 7,\ 7
  1. Find the mean: The common value is 7.
    μ=7\mu=7
  2. Check every deviation: Every deviation is zero.
    xi−μ=0x_i-\mu=0
  3. Calculate both measures: The squared total and its root are zero.
    σ2=0,σ=0\sigma^2=0,\quad \sigma=0

Answer

variance=standard deviation=0\text{variance}=\text{standard deviation}=0

Check: Every observation equals the mean, which is the exact condition for zero variance and zero standard deviation.

Rescaling changes the measures differently

Compare a population with the same values measured in units ten times as large.

1, 2, 3⟶10, 20, 301,\ 2,\ 3\quad\longrightarrow\quad10,\ 20,\ 30
  1. Find the original spread: The original population variance is 2/3.
    σX2=2/3\sigma_X^2=2/3
  2. Apply the scale factor: Variance receives the square of 10.
    σ10X2=102(2/3)=200/3\sigma_{10X}^2=10^2(2/3)=200/3
  3. Compare standard deviations: Standard deviation receives the absolute scale factor 10.
    σ10X=102/3\sigma_{10X}=10\sqrt{2/3}

Answer

variance multiplies by 100,standard deviation by 10\text{variance multiplies by }100,\quad\text{standard deviation by }10

Check: Direct deviations from 20 are negative 10, 0, and 10; their squared average is 200/3, matching the scaling identity.

Try it yourself

Practice the decision before checking the answer

  1. For sample data 4, 6, 8, find the variance and standard deviation.

    Show answer and explanation

    Answer: \bar{x}=6,\ SS=8,\ s^2=4,\ s=2

    The sample denominator is 2, and the square root of 4 is 2.

  2. If every value is increased by 5, what happens to variance and standard deviation?

    Show answer and explanation

    Answer: Neither measure changes.

    Adding a constant shifts the mean and every observation equally, leaving deviations unchanged.

  3. If every value is multiplied by negative 3, how do the measures change?

    Show answer and explanation

    Answer: Variance is multiplied by 9 and standard deviation by 3.

    Variance uses the squared scale factor; standard deviation uses its absolute value.

  4. Which measure is in centimeters when the data are lengths in centimeters?

    Show answer and explanation

    Answer: Standard deviation.

    Variance is in square centimeters; standard deviation is in centimeters.

Common mistakes

Notice the decision before the algebra

Treating the two measures as unrelated

Calculating variance and standard deviation from separate procedures.

Fix: Calculate variance once, then take its nonnegative square root for standard deviation.

Using original units for variance

Reporting a variance of 4 centimeters.

Fix: Use square centimeters for variance and centimeters for standard deviation.

Choosing n or n minus 1 by habit

Using the same denominator regardless of whether the data are a sample or a population.

Fix: Identify the role of the data before choosing the formula.

Assuming small spread means accurate data

Calling tightly clustered measurements unbiased or correct.

Fix: Treat spread, accuracy, bias, and study quality as different questions.