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Introduction to Statistics (STAT 101) · Module 1 of 7

Exploring One-Variable Data

Types of variables, describing distributions by shape, center and spread, outliers, and z-scores.

In this module
  1. Variables & Describing Distributions
  2. Spread, Boxplots & Outliers
  3. z-Scores & Transformations
  4. Check yourself

Variables & Describing Distributions

  • Categorical variables place individuals in groups (e.g. eye color); quantitative variables are numbers you can average (e.g. height).
  • Describe a quantitative distribution with SOCS: Shape (symmetric, skewed left/right, unimodal/bimodal), Outliers, Center, and Spread, always in context.
  • Skewed right: a long tail toward high values, and the mean is pulled above the median. Skewed left: the mean is below the median.
Worked example

Household incomes in a town are strongly skewed right. Which is the better measure of a "typical" income, and why?

Solution. The median. A few very high incomes pull the mean up, but the median is resistant to extreme values.

Spread, Boxplots & Outliers

  • Standard deviation measures typical distance from the mean (not resistant). IQR = Q3 − Q1 measures the spread of the middle 50% (resistant).
  • The 1.5 × IQR rule: a value is an outlier if it is below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.
  • A boxplot shows the five-number summary: min, Q1, median, Q3, max, with outliers plotted separately.
Boxplot with an outlier
valuefence 4548Q1medianQ3
Q1 = 20, median = 25, Q3 = 30, so IQR = 10 and the upper fence is 30 + 1.5 × 10 = 45. The value 48 lies beyond the fence and is plotted as an outlier; the whisker stops at the largest non-outlier (44).
Worked example

Q1 = 20 and Q3 = 30. Is a value of 48 an outlier?

Solution. IQR = 10, so the upper fence is 30 + 1.5 × 10 = 45. Since 48 > 45, it is an outlier.

z-Scores & Transformations

  • z = (x − mean) / SD tells how many standard deviations a value lies above (+) or below (−) the mean. z-scores let you compare values from different distributions.
  • Adding a constant to every value shifts the center but not the spread. Multiplying every value by a constant multiplies both the center and the spread.
Worked example

A student scores 85 on a test with mean 70 and SD 10, and 80 on another with mean 65 and SD 5. Which score is better relative to the class?

Solution. First: z = (85 − 70)/10 = 1.5. Second: z = (80 − 65)/5 = 3.0. The second score is relatively better.

Common mistake: Always describe a distribution in the CONTEXT of the problem. "The distribution is skewed right" earns less credit than "Commute times are skewed right, with most under 20 minutes."

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. In a strongly right-skewed distribution, typically:

2. Which measure of spread is resistant to outliers?

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