Free lesson · Math & notation

Read sums, indices, averages and squared deviations

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Start with the idea

Expand a compact sum into a short list before generalising. The notation saves space, but every term still describes an ordinary arithmetic operation.

Symbols, units & horizon

xᵢ: observation i in the chosen units
n: count of observations, at least 2 for sample variance
x̄: arithmetic sample average
Σ with limits: sum over the specified indices
s²: sample variance in squared units
s: sample standard deviation in the units of x

When and why to use this

Use this when hand-checking statistics computed by a spreadsheet or a Python function.

The summation symbol Σ means “add these terms.” The index i tells you which observation to use; the lower and upper limits tell you where to start and stop. A bar above x means its arithmetic average. The notation xᵢ is the i-th value; it does not mean x raised to i.

\overline{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}, s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \overline{x})^{2}

Algebra and arithmetic

Expand the mean and variance sums

For n=3, $\sum_{i = 1}^{3} x_{i} = x_{1} + x_{2} + x_{3}$ . Divide by 3 to obtain the mean.
For x=(1,2,3), mean=2. The variance numerator expands to $(1 - 2)^{2} + (2 - 2)^{2} + (3 - 2)^{2} = 2$ . Divide by n−1=2.

Work it by hand

Sample variance=1; sample SD=√1=1 in the units of x.

Subtracting the mean centres observations. Squaring makes negative and positive deviations contribute positively. Variance has squared units; taking its square root gives standard deviation in the original units. The sample variance uses n−1 when estimating a population variance under the usual assumptions.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from statistics import mean, variance, stdev

def sample_summary(values):
    if len(values) < 2:
        raise ValueError("At least two observations required")
    return mean(values), variance(values), stdev(values)

print(sample_summary([1, 2, 3]))

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