How much can one observation move the center?

Intuition

The mean is a balance point: every observation contributes to the sum. The median asks where the ordered middle lies. A mode asks which observed value occurs most often. They answer different questions, so disagreement can be useful information. Try moving the selected 7 to 20 before reading on.

Definition

For a nonempty dataset $x_1,\ldots,x_n$, the mean is $\bar x=n^{-1}\sum_i x_i$. Order the observations as $x_{(1)}\leq\cdots\leq x_{(n)}$. For odd $n$, the median is $x_{((n+1)/2)}$; for even $n$, average $x_{(n/2)}$ and $x_{(n/2+1)}$. Modes maximize observed frequency.

Mechanics

Select an observation by its numbered button, then drag its number-line handle, use arrow keys, or enter a value. Add or remove observations and watch the ordered list and histogram update. Values may repeat: the frequency view keeps counts while the number line can show coincident points. Reset restores [1,2,2,3,7].

Derivation

The mean uniquely minimizes $\sum_i(x_i-a)^2$: expanding around $\bar x$ gives $\sum_i(x_i-a)^2=\sum_i(x_i-\bar x)^2+n(a-\bar x)^2$. By contrast, a median minimizes the sum of absolute distances; for an even dataset a whole interval can minimize that objective, while our reported convention averages the middle pair.

Worked examples

For $[1,2,2,3,7]$, the sum is 15, so the mean is 3. The middle observation is 2 and its frequency is 2: median and single mode are both 2. Replacing 7 by 20 gives mean $28/5$ and the same median 2. For $[1,1,3,3]$, both 1 and 3 are modes.

Executable reference code

Run both files. The pedagogical program exposes the arithmetic; the NumPy companion uses array operations. Literal assertions check the worked example rather than testing a simulation against a desired statistical outcome.

"""Runnable summaries reference; exact mechanics / NumPy companion."""
from fractions import Fraction as F
from collections import Counter
x = [F(v) for v in [1,2,2,3,7]]
a = sorted(x)
mean = sum(x)/len(x)
median = a[len(a)//2]
counts = Counter(x)
modes = [v for v,c in counts.items() if c == max(counts.values())]
assert (mean,median,modes)==(F(3),F(2),[F(2)])
print(mean, median, modes)

"""Runnable summaries reference; exact mechanics / NumPy companion."""
import numpy as np
x=np.array([1,2,2,3,7],dtype=float)
values, counts=np.unique(x,return_counts=True)
modes=values[counts==counts.max()]
assert x.mean()==3 and np.median(x)==2 and modes.tolist()==[2]
print(x.mean(),np.median(x),modes)

Explore

Compare the outlier, singleton, multiple-mode and distinct-value presets. Predict the result before moving a point. Find two different datasets with the same mean but different medians; then explain why one number cannot reconstruct the observations.

Common mistakes and counterexamples

Do not choose a tied mode arbitrarily. When every distinct value occurs once, every value ties for maximal frequency; this explorer reports no repeated value and no uniquely determined mode. Some textbooks call every tied value a mode. This declared convention avoids implying a preferred value. A singleton has its own value as all three summaries.

Numerical considerations

Inputs contain at most two decimal places and lie between −20 and 20; datasets have 1–24 observations. Sorting, sums, medians and frequency equality use exact rational arithmetic. Rendering converts these exact values to binary64 coordinates. Rounded inspector companions use ≈, while observed fractions use =. The plot reports data values, not unknown population parameters.

Engineering and ML connections

Latency, sensor readings and feature distributions often have a long right tail. A mean can reveal total load, while a median describes a typical ordered observation. A mode is useful for discrete categories but depends on bin choices for continuous data; this chapter uses actual repeated values rather than inventing histogram modes.

Exercises

Predict, calculate and explain before checking the corrective feedback. Each solution states the relevant rule and its conditions.

Summary and checkpoints

Check that you can order observations, distinguish three definitions, calculate an even median, retain mode ties and explain an outlier. A summary compresses observations; always retain enough of the dataset to inspect what it hides.