The question

Act I · One variable

You have 171 numbers. Now what?

Annual earnings for 171 full-time working women aged 30. A column of 171 dollar amounts tells you nothing at a glance.

Univariate summary is how we compress that column into a handful of numbers — and a few pictures — that reveal its center, its spread, and its shape.

Univariate = a single variable. Before any modeling, you summarize. This chapter is the toolkit: summary statistics for the numbers, charts for the shape.

Where we’re going

• Summarize a variable with the mean, median, spread, and shape statistics
• Visualize numerical data — box plots, histograms, density, line charts
• Compare categories with bar and pie charts
• Transform skewed data with logarithms; rescale with z-scores
• Smooth time series with moving averages and seasonal adjustment

Six ideas anchor this chapter

• Mean vs. median — the average, pulled by outliers; vs. the honest middle value
• Standard deviation — the typical distance of a point from the mean
• Quartiles & IQR — \(Q_1\), \(Q_2\), \(Q_3\) split the data in four; IQR \(= Q_3 - Q_1\)
• Skewness — asymmetry; \(> 1\) means a long right tail
• Kurtosis — tail heaviness; excess \(> 0\) means more outliers than a bell curve
• Z-score — \(z_i = (x_i - \bar{x})/s\), a value rescaled to “standard deviations from the mean”

For skewed economic data, the median is usually the more honest “typical” value.

From numbers to a shape

Act II · The investigation

Start with the numbers: center, spread, shape

import pandas as pd                          # tabular data
from scipy import stats                      # skewness & kurtosis

# dataset: 171 full-time working women aged 30, 2010 (earnings in $)
url = "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_EARNINGS.DTA"
data = pd.read_stata(url)                    # read the Stata .dta into a DataFrame

data[["earnings"]].describe().round(0)       # count, mean, std, min, quartiles, max
stats.skew(data["earnings"]), stats.kurtosis(data["earnings"])  # shape: 1.71, 4.32 (excess)

↗ Try it in Colab

pandas gives center and spread in one call; scipy.stats adds shape. scipy returns EXCESS kurtosis by default (raw − 3), so 0 is the bell-curve baseline.

The mean sits above the median — a telltale of skew

• Mean \(= \$41{,}413\) but median \(= \$36{,}000\) — a $5,413 gap (15%).
• Std. dev. \(= \$25{,}527\) — 62% of the mean: enormous spread.
• \(Q_1 = \$25{,}000\), \(Q_3 = \$49{,}000\) → IQR \(= \$24{,}000\) holds the middle half.
• Skewness \(= 1.71\) (\(> 1\) = highly skewed); excess kurtosis \(= 4.32\) (heavy tails).

A handful of high earners (max $172,000) drag the mean up; report the median as “typical.”

The box plot draws those statistics in one picture

Box = \(Q_1\) to \(Q_3\) (the middle 50%); red line = median; dots on the right = high-earner outliers.

The median sits left of center and the right whisker is long — the chart is the skew.

A histogram shows the whole distribution — bin width is a choice

import pandas as pd                          # tabular data
import matplotlib.pyplot as plt               # plotting

url = "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_EARNINGS.DTA"
earnings = pd.read_stata(url)["earnings"]    # the single variable

fig, ax = plt.subplots(1, 2, figsize=(12, 5))         # two panels side by side
ax[0].hist(earnings, bins=12, color="#22d3ee", edgecolor="black")  # wider bins → coarse shape
ax[1].hist(earnings, bins=25, color="#22d3ee", edgecolor="black")  # narrower bins → fine detail
plt.show()                                   # render the histograms

↗ Try it in Colab

Wide bins give the shape; narrow bins give the detail

The same 171 earnings at two bin widths. Both show the long right tail; the narrower bins reveal clustering around $25k–$45k.

Always try more than one bin width — each answers a different question.

For categories, a bar chart ranks the pieces

U.S. health spending, 2018: hospital care alone is ~1/3 of $3,653B — more than physicians and drugs combined.

Sort categories before plotting so the ranking reads instantly.

Categorical data: count, then show the shares

import pandas as pd                          # tabular data
import matplotlib.pyplot as plt               # plotting

url = "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_FISHING.DTA"
data = pd.read_stata(url)                    # 1,182 recreational fishers

counts = data["mode"].value_counts()         # frequency of each fishing site
counts / counts.sum() * 100                  # relative frequency (% of total)
plt.pie(counts, labels=counts.index, autopct="%1.1f%%")  # each slice = a site's share
plt.show()                                   # render the pie chart

↗ Try it in Colab

value_counts() builds the frequency table; normalize=True (or dividing by the sum) gives relative frequencies. Boat-based modes (charter 38% + private 35%) are ~74% of choices.

Boat fishing dominates the 1,182 choices

Charter (38.2%) + private (35.4%) = 74% of fishers choose a boat; pier and beach split the rest.

Pie charts show shares; for many categories a sorted bar chart compares them more cleanly.

Logs tame right-skewed economic data

import numpy as np                           # natural logarithm
import pandas as pd                          # tabular data
from scipy import stats                      # skewness

url = "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_EARNINGS.DTA"
data = pd.read_stata(url)                    # load the 171 earnings

data["ln_earnings"] = np.log(data["earnings"])   # log-transform the skewed variable
stats.skew(data["earnings"])                      # 1.71  → strong right skew
stats.skew(data["ln_earnings"])                   # -0.91 → nearly symmetric

↗ Try it in Colab

Logs convert multiplicative, right-skewed data (income, prices, wealth) toward symmetry. On the log scale the mean (10.46) and median (10.49) nearly coincide. Logs need positive data.

The log scale collapses the long tail toward a bell

Panel A: earnings in dollars, strongly right-skewed. Panel B: log(earnings), roughly symmetric — skew falls from 1.71 to −0.91.

Many tests assume a bell shape; the log transform is how skewed economic data gets there.

Time series need smoothing to reveal the trend

import pandas as pd                          # tabular data

url = "https://raw.githubusercontent.com/quarcs-lab/data-open/master/AED/AED_MONTHLYHOMESALES.DTA"
data = pd.read_stata(url).query("year >= 2005")   # monthly U.S. home sales, 2005+
data = data.sort_values("daten")             # ensure time order

data["ma11"] = data["exsales"].rolling(11, center=True).mean()  # 11-month moving average
data[["exsales", "ma11"]].describe().round(0)     # raw (noisy) vs smoothed series

↗ Try it in Colab

A moving average averages each point with its neighbors, cancelling month-to-month noise. Seasonal adjustment instead removes the recurring summer-peak / winter-trough pattern.

Smoothing reveals the 2007–2011 housing crash

Raw monthly sales zig-zag with the seasons; the 11-month moving average (red) and seasonal adjustment (green) expose the boom–bust–recovery cycle.

A moving average cancels seasonal noise; seasonal adjustment removes the predictable pattern itself.

Reading it right

Act III · The resolution

Choose the summary that matches the shape

Objection. “Just report the mean and a histogram — that’s what a summary is.”

Response. The right summary depends on the shape. For skewed earnings, the mean overstates the “typical” value, so lead with the median and IQR. A histogram’s story changes with its bin width. And a raw time series hides its trend until you smooth it.

Match the tool to the data: skew → median & logs; categories → bars; time → moving averages.

What to carry forward

• Summarize center (mean/median), spread (std. dev., IQR), and shape (skew, kurtosis).
• For skewed economic data the median is the honest center; the mean is pulled by outliers.
• Pick the chart for the data type: histogram/box/line (numerical), bar/pie (categorical).
• Logs symmetrize right-skewed data; z-scores put different variables on one scale.
• For time series, moving averages smooth noise and seasonal adjustment removes recurring patterns.

Summarize the numbers, then draw the shape — and let the data pick the tool.