Deep Learning · MT2 · formulas

Mean, variance and standard deviation

The mean is a measure of location; variance and standard deviation quantify dispersion. Distinguish population and sample conventions before calculating these statistics.

\mu=\frac{1}{n}\sum_i x_iLaTeX: \mu=\frac{1}{n}\sum_i x_i
v=\frac{1}{n}\sum_i(x_i-\mu)^2LaTeX: v=\frac{1}{n}\sum_i(x_i-\mu)^2
s=\sqrt{v}LaTeX: s=\sqrt{v}

Mean, variance and standard deviation from ordinary arithmetic

Definition. The arithmetic mean measures the centre of a numerical dataset. Variance measures the average squared deviation from that mean, and standard deviation expresses dispersion in the original units.

\mu=\frac{1}{N}\sum_{i=1}^{N}x_i,\qquad \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2,\qquad \sigma=\sqrt{\sigma^2}LaTeX: \mu=\frac{1}{N}\sum_{i=1}^{N}x_i,\qquad \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2,\qquad \sigma=\sqrt{\sigma^2}

These are population formulas, or empirical moments treating the supplied dataset as the complete reference set. An unbiased sample-variance estimate instead divides the squared-deviation sum by N−1. Use the convention specified by the question; the two denominators are not interchangeable.

Question. Find the population mean, variance and standard deviation of [3, 3, 7, 7].

Given

Observations = [3,3,7,7]; N = 4.

Use population variance (divide by N).

To find

Find the population mean μ, variance σ² and standard deviation σ.

Formula — which one and why

Mean and spread. The mean is the sum divided by the number of values. Population variance averages squared distances from that mean; standard deviation is its square root.

N = number of values; xᵢ = value i; μ = mean; σ² = variance; σ = standard deviation. Σ means add the indicated terms.

\mu=\frac{1}{N}\sum_{i=1}^{N}x_iLaTeX: \mu=\frac{1}{N}\sum_{i=1}^{N}x_i
\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2LaTeX: \sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2
\sigma=\sqrt{\sigma^2}LaTeX: \sigma=\sqrt{\sigma^2}

Yaad rakhne ka cue: Mean = sabka total ÷ count; variance = squared distances ka mean.

Solution — step by step

Step 1: add all four observations.

3+3+7+7=20LaTeX: 3+3+7+7=20

Step 2: divide by the count to obtain the mean.

\mu=\frac{20}{4}=5LaTeX: \mu=\frac{20}{4}=5

Step 3: subtract the mean from each observation.

[3-5,3-5,7-5,7-5]=[-2,-2,2,2]LaTeX: [3-5,3-5,7-5,7-5]=[-2,-2,2,2]

Step 4: square each deviation.

[(-2)^2,(-2)^2,2^2,2^2]=[4,4,4,4]LaTeX: [(-2)^2,(-2)^2,2^2,2^2]=[4,4,4,4]

Step 5: add the squared deviations.

4+4+4+4=16LaTeX: 4+4+4+4=16

Step 6: divide by four for population variance.

\sigma^2=\frac{16}{4}=4LaTeX: \sigma^2=\frac{16}{4}=4

Step 7: take the nonnegative square root.

\sigma=\sqrt4=2LaTeX: \sigma=\sqrt4=2

Answer: mean = 5, population variance = 4 and population standard deviation = 2. Variance has squared units; standard deviation has the same units as the observations. With the sample convention, the variance would instead be 16/3.

Population mean and spread: [1, 2, 3]

Question. For the population [1, 2, 3], find mean, population variance and standard deviation.

Given

Values = [1, 2, 3]; N = 3. Use population division by N.

To find

Mean μ, variance σ² and standard deviation σ.

Formula — which one and why

The question specifies a population, so use N rather than the N−1 sample-variance divisor.

μ is the centre; σ² averages squared distances from it; σ converts spread back to the original units.

\mu=\frac1N\sum_i x_iLaTeX: \mu=\frac1N\sum_i x_i
\sigma^2=\frac1N\sum_i(x_i-\mu)^2LaTeX: \sigma^2=\frac1N\sum_i(x_i-\mu)^2
\sigma=\sqrt{\sigma^2}LaTeX: \sigma=\sqrt{\sigma^2}

Yaad rakhne ka cue: Mean, distance, square, mean again, then root.

Solution — step by step

Step 1: add all values.

1+2+3=6LaTeX: 1+2+3=6

Step 2: divide by the number of values.

\mu=\frac{6}{3}=2LaTeX: \mu=\frac{6}{3}=2

Step 3: find and square deviation 1.

(1-2)=-1LaTeX: (1-2)=-1
(-1)^2=1LaTeX: (-1)^2=1

Step 4: find and square deviation 2.

(2-2)=0LaTeX: (2-2)=0
(0)^2=0LaTeX: (0)^2=0

Step 5: find and square deviation 3.

(3-2)=1LaTeX: (3-2)=1
(1)^2=1LaTeX: (1)^2=1

Step 6: average the squared deviations.

\sigma^2=\frac{1+0+1}{3}\approx0.666666667LaTeX: \sigma^2=\frac{1+0+1}{3}\approx0.666666667

Step 7: take the nonnegative square root.

\sigma=\sqrt{0.666666667}\approx0.816496581LaTeX: \sigma=\sqrt{0.666666667}\approx0.816496581

Final answer: μ = 2; σ² ≈ 0.666666667; σ ≈ 0.816496581.

Population mean and spread: [2, 2, 2]

Question. For the population [2, 2, 2], find mean, population variance and standard deviation.

Given

Values = [2, 2, 2]; N = 3. Use population division by N.

To find

Mean μ, variance σ² and standard deviation σ.

Formula — which one and why

The question specifies a population, so use N rather than the N−1 sample-variance divisor.

μ is the centre; σ² averages squared distances from it; σ converts spread back to the original units.

\mu=\frac1N\sum_i x_iLaTeX: \mu=\frac1N\sum_i x_i
\sigma^2=\frac1N\sum_i(x_i-\mu)^2LaTeX: \sigma^2=\frac1N\sum_i(x_i-\mu)^2
\sigma=\sqrt{\sigma^2}LaTeX: \sigma=\sqrt{\sigma^2}

Yaad rakhne ka cue: Mean, distance, square, mean again, then root.

Solution — step by step

Step 1: add all values.

2+2+2=6LaTeX: 2+2+2=6

Step 2: divide by the number of values.

\mu=\frac{6}{3}=2LaTeX: \mu=\frac{6}{3}=2

Step 3: find and square deviation 1.

(2-2)=0LaTeX: (2-2)=0
(0)^2=0LaTeX: (0)^2=0

Step 4: find and square deviation 2.

(2-2)=0LaTeX: (2-2)=0
(0)^2=0LaTeX: (0)^2=0

Step 5: find and square deviation 3.

(2-2)=0LaTeX: (2-2)=0
(0)^2=0LaTeX: (0)^2=0

Step 6: average the squared deviations.

\sigma^2=\frac{0+0+0}{3}\approx0.000000000LaTeX: \sigma^2=\frac{0+0+0}{3}\approx0.000000000

Step 7: take the nonnegative square root.

\sigma=\sqrt{0.000000000}\approx0.000000000LaTeX: \sigma=\sqrt{0.000000000}\approx0.000000000

Final answer: μ = 2; σ² ≈ 0.000000000; σ ≈ 0.000000000. Standardization is undefined with this zero standard deviation.