Univar. Stats

Univariate Statistics

  • Measure distribution of a single variable
  • 4 types of measures:
    • Central Tendency
    • Dispersion
    • Distribution Shape
    • Graphical Representation
  • Define “univariate”: analyze one variable at a time.
  • Give 2–3 real-world examples (height, test score, income).
  • State learning objective: summarize and interpret a single variable.

Central Tendencies

  • Mean: Average of all values
  • Median: Middle values when sorted
  • Mode: Most frequent value
  • Explain when to prefer each measure:
    • Mean: use for symmetric data.
    • Median: robust to outliers or skew.
    • Mode: useful for categorical or multimodal data.
  • Note median for even n = average of two middle values.
  • Suggest quick demo comparing mean vs median with an outlier.

Dispersion

  • Range: Difference between max and min
  • Variance: Average of squared differences from mean
  • Standard Deviation: Square root of variance
  • Emphasize purpose: quantify spread beyond center.
  • Range: simple but sensitive to outliers.
  • Variance/SD: summarize spread around mean; SD returns to original units.

Variance

\[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \]

  • \(s^2\) is the sample variance
  • \(n\) is the number of observations in the sample
  • \(x_i\) is the \(i\)th observation in the sample
  • \(\bar{x}\) is the sample mean
  • Walk through formula: deviations, square, sum, divide by n-1.
  • Explain Bessel’s correction (n-1) for unbiased estimation.
  • Contrast with population variance (divide by N).

Variance Use

  • Variance is used to measure how spread out the values in a data set are
  • A high variance indicates that the data points are very spread out from the mean
  • Interpret: larger variance = more spread.
  • Limitation: units are squared, less intuitive than SD.
  • Advice: compare variances only for same-scale variables.

Standard Deviation

\[ s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} \]

  • \(s\) is the sample standard deviation
  • \(n\) is the number of observations in the sample
  • \(x_i\) is the \(i\)th observation in the sample
  • \(\bar{x}\) is the sample mean
  • SD = sqrt(variance); same units as data.
  • Intuition: typical distance from the mean (summary metric).
  • Mention sensitivity to outliers.

Standard Deviation Use

  • Turns variance into a more understandable metric
  • Standard deviation explains how much the data deviates from the mean
  • Each standard deviation away from the mean accounts for a certain percentage of the data
  • Introduce empirical rule (68/95/99.7) for approx. normal data.
  • Warn: empirical rule assumes normality; may not hold if skewed.
  • Suggest alternatives: IQR, MAD for non-normal data.

Distribution Shape

  • Skewness: Measure of symmetry in data
  • Kurtosis: Measure of the “tailedness” of the data
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

mean = 0
std = 1
sample_num = 1000
data = np.random.normal(mean, std, sample_num)
data = [round(x, 2) for x in data]
mode = pd.Series(data).mode()[0]
median = pd.Series(data).median()

plt.hist(data, bins=30, density=True, alpha=0.6, color="g")

xmin, xmax = plt.xlim()
x = np.linspace(xmin, xmax, 100)
p = np.exp(-0.5 * (x - mean) ** 2 / std**2) / (std * np.sqrt(2 * np.pi))

plt.plot(x, p, "k", linewidth=2)
title = "Gaussian Distribution\nMean = {:.2f}, Median = {:.2f}, Mode {:.2f}\nstd = {:.2f}".format(
    mean, median, mode, std
)

plt.axvline(mean, color="b", linestyle="dashed", linewidth=2, label="Mean")
plt.axvline(median, color="r", linestyle="dashed", linewidth=2, label="Median")
plt.axvline(mode, color="y", linestyle="dashed", linewidth=2, label="Mode")

plt.legend()

plt.title(title)
plt.xlabel("Value")
plt.ylabel("Frequency")

plt.show()
Figure 1
  • Run code to show histogram + theoretical normal curve.
  • Highlight vertical lines: mean (blue), median (red), mode (yellow).
  • Demo ideas: change mean/std or sample size to show variability.

Skewness: Normal, Gaussian, Bell Curve

  • 68% of data falls within 1 standard deviation
  • 95% of data falls within 2 standard deviations
  • 99.7% of data falls within 3 standard deviations
  • Mean = Median = Mode
  • Reinforce empirical rule and its normality assumption.
  • Use to connect SD with probability mass for symmetric data.
  • Note equality of mean/median/mode is a property of symmetry.

Skewness: Positive/Negative

  • Positive Skewness: Tail is farther on the larger side of the x-axis
  • Negative Skewness: Tail is farther on the smaller side of the x-axis

Skewness Representation
  • Explain effect on mean vs median:
    • Right-skew (positive): mean > median.
    • Left-skew (negative): mean < median.
  • Give examples of right-skewed data: income, wait times.

Kurtosis

  • Measure of the “tailedness” of the data
  • The more the Kurtosis, the more the data is in the tails
  • Normal Distribution has a Kurtosis of 3
  • Higher Kurtosis: More data in the tails
  • Lower Kurtosis: Less data in the tails
  • Measures how many outliers.
  • Define kurtosis and “excess kurtosis” (kurtosis − 3).
  • Interpret: high kurtosis → heavier tails → more extremes.
  • Warn: kurtosis sensitive to a few outliers.

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