Lesson 71 — Measures of central tendency: mean, median, mode

Data: 5, 6, 7, 7, 7, 8, 9. Calculate the mean, median, and mode.

Grades of 8 students: 4, 5, 6, 6, 7, 8, 9, 10. Calculate the mean, median, and mode.

Monthly salaries ($k): 2, 2, 3, 4, 5, 50. Compare mean and median. Which better represents the typical salary?

Ages of 7 participants: 18, 19, 20, 21, 22, 23, 24. Calculate the mean, median, and mode.

Loading times (s): 0.5; 0.7; 0.8; 0.9; 1.1; 1.5; 7.0. Calculate the mean and median. Is the median more informative than the mean in this case?

Car colors in a parking lot: 12 white, 8 black, 5 gray, 5 red. Which measure of central tendency is appropriate?

Data: 1, 1, 2, 3, 5, 5, 7. Determine the mode(s). How is this distribution classified?

Frequency table: $x$ = 4, 5, 6, 7, 8 with frequencies $f$ = 2, 3, 5, 3, 2. Calculate the arithmetic mean.

Grouped data: intervals $[0,10)$, $[10,20)$, $[20,30)$ with frequencies 5, 12, 3. Calculate the mean using midpoints.

A class has a mean age $\bar{x} = 17.5$ years. A new 20-year-old student joins and the new mean becomes $17.75$ years. How many students were there originally?

Calculate mean, median, and mode for: 3, 3, 4, 5, 6, 6, 6, 9.

Calculate mean, median, and mode(s) for: 10, 10, 11, 12, 13, 14, 14, 15, 19.

Why does IBGE prefer to use the median (and not the mean) to describe the per capita household income of Brazil?

Emergency room waiting time: most are seen in 1 to 2 hours, but some serious cases wait more than 10 hours. Which measure to use to describe typical waiting time? Justify.

A manufacturer wants to declare the typical useful life of its LED bulbs. Suggest which measure of central tendency to use and justify.

An election poll asks 1,000 voters which party they intend to vote for. Which measure of central tendency will identify the preferred party?

For a unimodal distribution with right skew (long positive tail), what is the typical order between mode, median, and mean? Explain intuitively.

Uniform distribution in $[0, 10]$. Determine the mean, median, and discuss the mode. What does this say about symmetric distributions?

ENEM grades have a distribution close to normal. Is the mean or median more adequate to describe typical performance? Justify.

An investor wants to know the most common number of rooms in apartments in a neighborhood. Which measure to use?

Page loading time: 95% of requests respond in less than 300 ms, but 1% take more than 5 s. Why do reliability engineers prefer median (P50) and percentiles (P95, P99) instead of the mean?

Why for a continuous unimodal symmetric distribution are the three measures of central tendency equal? Explain geometrically.

A/B testing: checkout time for site A has mean 12 s and median 9 s. Site B has mean 10 s and median 10 s. Which site has better experience for the typical user? Justify.

Company A reports only an average salary of R$ 10k. Company B reports mean of R$ 8k and median of R$ 7k. What might the absence of the median in A be hiding?

In K-means, the centroid of a cluster is the mean. What is the effect of an outlier on the centroid? How does K-medoids (which uses the median point) mitigate this problem?

Quality control: parts with mean diameter $\bar{d} = 10.05$ mm and approximately symmetric distribution. To what value would you expect the median to be close? Why?

In machine learning, MSE as a loss function implies the model learns to estimate the mean conditional. MAE implies the model estimates the median conditional. Explain why this follows from the variational characterization of central measures.

A meta-analysis with 50 studies reports the median effect size instead of the mean. Why is the median preferred in meta-analysis?

Why does the boxplot use the median as the central line (and IQR as box width) instead of using mean and standard deviation?

In federated learning, why does replacing the mean of gradients with the median increase resistance to malicious clients (Byzantine attacks)?

For the log-normal distribution ($\ln X \sim N(\mu, \sigma^2)$): mode $= e^{\mu-\sigma^2}$, median $= e^\mu$, mean $= e^{\mu+\sigma^2/2}$. Verify the ordering mode less than median less than mean for $\sigma > 0$.

Salaries ($k): 4, 4, 5, 5, 6, 7, 7, 8 (8 employees). A CEO with a salary of $60k is added (without removing anyone). Calculate mean and median before and after. Which measure changed more?

Grades of 30 students on a test, grouped: $[60,70)$: 3 students; $[70,80)$: 8 students; $[80,90)$: 12 students; $[90,100]$: 7 students. Calculate the mean estimated by midpoints.

Show that $\sum_{i=1}^{n}(x_i - \bar{x}) = 0$.

Show that $\sum_{i=1}^{n}(x_i - c)^2$ is minimized at $c = \bar{x}$ for any sequence $x_1, \ldots, x_n \in \mathbb{R}$.

Show that $\sum_{i=1}^{n}|x_i - c|$ is minimized at $c = \text{median}$. (Hint: analyze what happens when shifting $c$ to one side or the other of the median, counting how many $x_i$ remain above and below.)

Show that if $y_i = ax_i + b$ (linear transformation), then $\bar{y} = a\bar{x} + b$.

Does the mean satisfy $\overline{f(x_i)} = f(\bar{x})$ in general? And the median? Investigate with $f(t) = t^2$ and the data $x = \{1, 2, 3\}$.

Cauchy distribution: $f(x) = 1/[\pi(1+x^2)]$. Calculate the median. Show that the mean does not exist (the integral $\int_{-\infty}^{+\infty} x f(x)\,dx$ diverges).

Show that if we swap the largest value of a dataset for an even larger value, the median does not change, but the mean increases.

Two groups have means $\bar{x}_1$ and $\bar{x}_2$ with sizes $n_1$ and $n_2$. Derive the formula for the combined mean of the two groups.

Jensen's inequality states that for convex $\varphi$, $\varphi(E[X]) \leq E[\varphi(X)]$. Apply with $\varphi(t) = t^2$ to obtain an inequality between $\bar{x}^2$ and $\overline{x^2}$. What does this imply about the variance?