Descriptive Statistics: Mean, Median, and Spread

• Compute the sample mean x̄, the median, and the sample variance s² / standard deviation s from a real, finite dataset.
• Explain precisely why x̄ is a statistic — a number computed from observed data — and not the same mathematical object as the expectation μ of a random variable's theoretical distribution.
• Justify why the sample variance divides by n − 1 rather than n, and compute both versions on a small dataset to see the difference concretely.
• Choose between the mean and the median as a measure of center for a given dataset, based on skewness and the presence of outliers.
• Interpret the standard deviation as a measure of spread in the same units as the original data, and contrast it with the variance.

Everything covered so far in this discipline's probability half — sample spaces, random variables, the PMF and PDF, expectation and variance — describes a theoretical object: a fully specified probability distribution, whose expectation μ = E[X] and variance σ² = Var(X) are exact numbers determined entirely by the distribution's mathematical definition. If you know a random variable X is Binomial(n, p), you can compute E[X] = np and Var(X) = np(1 − p) exactly, without ever running an experiment, because the distribution itself is fully known.

Real data does not work this way. A biologist measuring the wingspan of 40 sparrows, a platform engineer logging the response time of 10,000 API requests, or a pollster surveying 1,200 voters is never handed the underlying distribution — only a finite batch of numbers actually observed. The population's true mean μ and true variance σ² are, in essentially every real situation, unknown and unknowable directly; all that exists is the sample: a list of numbers, x₁, x₂, …, xₙ, drawn (we hope, under some sampling scheme) from that population. Descriptive statistics is the discipline of taking that finite list and computing summary numbers from it — the sample mean x̄, the sample median, the sample variance s² — that describe the data actually in hand.

This might look, at first glance, like the same arithmetic already used to define expectation: sum values and divide by a count. It is closely related, but it is not the same object, and confusing the two is one of the most consequential category errors a newcomer to statistics can make. μ is a fixed (if unknown) property of a distribution; x̄ is a computed number that depends on which particular sample happened to be drawn, and would come out slightly different if the sampling were repeated. This entire cluster of concepts — descriptive statistics, sampling distributions, point estimation, confidence intervals, hypothesis testing — exists precisely to formalize the relationship between the two: x̄ is used as an estimate of μ, and the rest of this discipline is about quantifying exactly how much to trust that estimate. Stanford's CS109 devotes its statistics unit to exactly this transition — from probability, where the distribution is given and outcomes are derived, to statistics, where outcomes are given and the distribution must be inferred.

The sample mean x̄

Given a sample of n observed values x₁, x₂, …, xₙ, the sample mean is

x̄ = (1/n) · Σᵢ xᵢ = (x₁ + x₂ + ⋯ + xₙ) / n

This is arithmetically the same formula used to compute the average of any list of numbers. What makes it a statistic rather than a parameter is what it is being used for: x̄ is treated as an estimate of the population mean μ — the true, generally unknown average over the entire population the sample was drawn from. If a different sample of n sparrows had been caught and measured, a different x̄ would result, even though the underlying population (and its true μ) never changed. This sample-to-sample variability is the subject of the very next concept in this cluster (sampling distributions); for now, the essential point is definitional: μ is a fixed number describing a distribution; x̄ is a random variable's realized value, computed from data, used to estimate μ.

The median

The median is the middle value of the data once sorted: for n odd, it is the single middle observation; for n even, it is the average of the two middle observations. Unlike the mean, the median does not weight every observation by its numeric value — it only cares about rank order. Consequence: the median is far more robust to outliers than the mean. Consider five household incomes (in thousands): 40, 45, 50, 55, 500. The mean is (40+45+50+55+500)/5 = 138 — a number that does not represent any household in the data, dragged upward entirely by one extreme value. The median is 50 — squarely in the middle of where four of the five households actually sit. Neither statistic is "wrong"; they answer different questions ("what's the arithmetic center of mass?" versus "what's the typical middle value?"), and the gap between them is itself informative — a large mean-median gap is a signature of a skewed distribution.

Sample variance and standard deviation

Spread is measured by how far, typically, observations sit from the center. The sample variance is

s² = (1/(n − 1)) · Σᵢ (xᵢ − x̄)²

and the sample standard deviation is s = √(s²), reported in the same units as the original data (unlike variance, which is in squared units).

The formula looks almost identical to the theoretical variance σ² = E[(X − μ)²], with one deliberate difference: the division is by n − 1, not n. This is Bessel's correction, and it exists for a concrete reason. x̄ is computed from the very same sample used to compute the deviations (xᵢ − x̄); by construction, x̄ is the value that makes Σᵢ(xᵢ − x̄) exactly zero — it is the point of minimum total squared deviation for this particular sample. Using the sample's own mean as the reference point systematically makes the observed squared deviations a little smaller, on average, than they would be around the true (unknown) population mean μ. Dividing by n instead of n − 1 would therefore produce an estimate of σ² that is systematically too small — a biased estimator (the next concept in this cluster treats bias formally). Dividing by n − 1 exactly corrects this systematic underestimate, making s² an unbiased estimator of σ²: E[s²] = σ².

Concretely, of the n deviations (x₁ − x̄), …, (xₙ − x̄), only n − 1 are free to vary — the last one is always determined by the constraint that they must sum to zero (since x̄ is their average). This is the sample's degrees of freedom: n observations, minus 1 for having already used the data once to estimate x̄, leaves n − 1 independent pieces of information to estimate spread.

Statistic vs. parameter: the central distinction

This vocabulary is worth stating with full precision, since the rest of this discipline depends on it:

A parameter describes the population or distribution and is (outside of contrived toy problems) not directly observable. A statistic is any quantity computed purely from the sample data. x̄ is a statistic used to estimate the parameter μ; it is never literally equal to μ except by coincidence. This is not a pedantic distinction — an entire vocabulary of later concepts (bias, consistency, standard error, confidence intervals) exists only because x̄ ≠ μ in general, and quantifying that gap is the whole game.

Example 1 — computing x̄, median, s² and s by hand

Problem: A quality-control engineer samples 8 light bulbs from a production run and records their lifetimes in hours: 980, 1005, 995, 1010, 970, 1000, 1015, 990. Compute the sample mean, median, sample variance, and sample standard deviation.

Sample mean. Sum = 980+1005+995+1010+970+1000+1015+990 = 7965. n = 8. x̄ = 7965/8 = 995.625 hours.

Median. Sorted: 970, 980, 990, 995, 1000, 1005, 1010, 1015. n = 8 (even), so median = average of the 4th and 5th values = (995 + 1000)/2 = 997.5 hours. (Close to x̄ here — no strong skew.)

Sample variance. Deviations from x̄ = 995.625: −15.625, 9.375, −0.625, 14.375, −25.625, 4.375, 19.375, −5.625. Squared: 244.14, 87.89, 0.39, 206.64, 656.64, 19.14, 375.39, 31.64. Sum of squares ≈ 1621.87. Divide by n − 1 = 7: s² ≈ 231.7 hours².

Sample standard deviation. s = √231.7 ≈ 15.22 hours.

Interpretation. Typical bulbs in this sample run about 995–996 hours, with a typical deviation from that center of roughly 15 hours. Note that dividing by n = 8 instead would have given s² ≈ 1621.87/8 ≈ 202.7 — noticeably smaller, illustrating the systematic downward bias that Bessel's correction repairs.

Example 2 — mean vs. median under skew

Problem: A startup reports the following annual salaries (in thousands of dollars) for its 10 employees: 65, 68, 70, 72, 75, 78, 80, 82, 85, 410 (the last is the founder-CEO). Compute both the mean and median, and discuss which better represents a "typical" employee's salary.

Mean. Sum = 65+68+70+72+75+78+80+82+85+410 = 1085. x̄ = 108.5 (thousand).

Median. Sorted list is already sorted above; n = 10 (even), median = average of 5th and 6th values = (75 + 78)/2 = 76.5 (thousand).

Discussion. The mean, 108.5K, exceeds every single employee's salary except the CEO's — it is not representative of anyone's actual pay. The median, 76.5K, sits right in the middle of the nine ordinary employees' salaries and is a far better single-number summary of what a "typical" employee earns. This is the standard justification for why government statistics on income (median household income, not mean) use the median: income distributions are right-skewed by a long tail of very high earners, exactly like this toy example.

Example 3 — a quick Python simulation of the n vs. n − 1 divisor

Problem: Demonstrate numerically, by repeated sampling from a known population, that dividing by n − 1 gives an unbiased estimate of σ² while dividing by n does not.

plaintext
import random

random.seed(0)
# True population: uniform on integers 1..100, so true variance sigma^2 is known exactly.
population = list(range(1, 101))
true_var = sum((x - sum(population)/len(population))**2 for x in population) / len(population)

n = 5
trials = 200_000
sum_s2_n_minus_1 = 0.0
sum_s2_n = 0.0

for _ in range(trials):
    sample = random.choices(population, k=n)
    xbar = sum(sample) / n
    sq_dev = sum((x - xbar) ** 2 for x in sample)
    sum_s2_n_minus_1 += sq_dev / (n - 1)
    sum_s2_n += sq_dev / n

print("true sigma^2:      ", round(true_var, 2))
print("avg s^2 (÷ n-1):    ", round(sum_s2_n_minus_1 / trials, 2))
print("avg s^2 (÷ n):      ", round(sum_s2_n / trials, 2))

Running this (small samples of n = 5, drawn with replacement, repeated 200,000 times) shows the average of the ÷(n − 1) statistic landing very close to the true σ², while the average of the ÷n statistic lands systematically below it — a direct numerical confirmation of the bias Bessel's correction is designed to remove, and a preview of exactly how "an estimator's expected value" will be evaluated formally in the next concept in this cluster.

• "x̄ and μ are just two names for the same thing." They are not. μ is a fixed property of the population or distribution; x̄ is computed from one particular sample and would change if the sample were retaken. Example 1's x̄ = 995.625 is specific to those 8 bulbs — a different 8 bulbs from the same production run would almost certainly give a different x̄, even though the production run's true μ never moved.
• "The mean is always the right measure of center." Example 2 shows a case where the mean (108.5K) is worse than useless as a "typical value" summary because a single extreme outlier drags it far from where nearly all the data actually sits — the median (76.5K) is the better summary there. The right choice depends on the data's shape, not a fixed rule.
• "Dividing by n − 1 instead of n is just an arbitrary convention." It is not arbitrary — it corrects a specific, provable, systematic downward bias that results from estimating deviations around x̄ (which is itself fit to minimize those very deviations) rather than the true, unknown μ. Example 3 demonstrates this bias numerically: the ÷n version consistently underestimates the true variance across repeated sampling, while ÷(n − 1) does not.
• "A small standard deviation means the data has no outliers." Standard deviation summarizes typical spread across the whole sample; a single, extreme outlier can inflate s substantially even when the rest of the data is tightly clustered, which is exactly why some fields use the median and a robust spread measure (like the interquartile range) instead of the mean and standard deviation when heavy skew or outliers are suspected.
• "Variance and standard deviation are interchangeable." They are related but not interchangeable in interpretation: s² is in squared units (hours² in Example 1) and is what enters directly into formulas like the one for sample variance itself; s is in the original units (hours) and is what should be quoted when communicating "typical spread" to someone reading a report.

Descriptive statistics take a finite, real, observed dataset and summarize it with the sample mean x̄, median, and sample variance s² / standard deviation s. The single most important conceptual point is that these are statistics — numbers computed from actual data, which vary from sample to sample — not the same mathematical objects as the parameters μ and σ² that describe a theoretical distribution exactly and are generally unknown in practice. x̄ estimates μ; s² (dividing by n − 1, not n, to correct a systematic bias from using x̄ itself as the reference point) estimates σ². The median offers a center-measure that is robust to outliers and skew in a way the mean is not, and the gap between mean and median is itself a useful diagnostic. This statistic/parameter distinction is the foundation on which sampling distributions, point estimation, and confidence intervals — the next three concepts in this cluster — are all built.