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

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.

Let’s Code

• Go here: https://raw.githubusercontent.com/nkmwicz/data-for-students/refs/heads/main/soccer-players.csv