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Introductory Statistics Unit 1: Describing one-variable data

Before any formulas, statistics starts with describing data: what kind of variable it is, what its distribution looks like, and where a value sits relative to the rest.

Covers: Categorical vs quantitative, graphs, center & spread, z-scores, normal distribution

Study guide

Statistics starts by classifying variables. Categorical variables place individuals into groups (eye color, major); quantitative variables record numbers that make sense to average (height, income). Categorical data are shown with bar charts and pie charts; quantitative data with dotplots, stemplots, histograms and boxplots.

When describing a quantitative distribution, always address shape, center, spread and outliers, in context. Shape can be symmetric, skewed right (long tail to the right) or skewed left, and unimodal or bimodal. Center is measured by the mean or median; spread by the standard deviation, range or IQR. In a right-skewed distribution the mean is pulled above the median. Use the median and IQR for skewed data or data with outliers, since they resist extreme values. The 1.5·IQR rule flags outliers below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

A z-score, z = (x − mean)/SD, measures how many standard deviations a value lies from the mean, which lets you compare values from different distributions. Adding a constant to every value shifts the center but not the spread; multiplying scales both.

Normal distributions are symmetric and bell-shaped. The empirical rule says about 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations of the mean. For other cutoffs, convert to z-scores and use a table or technology to find areas.

To study two categorical variables, organize the counts in a two-way table and compare conditional distributions, meaning percentages computed within each group. If the conditional distributions differ, the variables are associated. A segmented bar graph shows each group's conditional distribution as a bar divided into parts that total 100%. A mosaic plot does the same but also makes each bar's width proportional to the group's size, so the area of each piece represents a count.

A larger proportion doesn't mean a larger count when group sizes differ, because count = proportion × group total. Convert to counts before comparing how many individuals are in each group. Each section of a boxplot (minimum to Q1, Q1 to median, median to Q3, Q3 to maximum) holds about 25% of the data. That fact lets you reason about where a median or percentile falls, including for combined groups.

When you compare distributions, address shape, center, spread and outliers for each group, and use comparative words such as "greater than" or "about the same as," with values and context.

Key ideas

• Categorical variables place individuals in groups; quantitative variables are numbers you can average.
• Describe a distribution with shape, center, spread and outliers, always in context.
• Skewed right pulls the mean above the median; skewed left pulls it below.
• Use mean and standard deviation for roughly symmetric data, median and IQR for skewed data or outliers.
• A z-score says how many standard deviations a value is from the mean.

Formulas and rules

Common mistakes

• Describing a distribution without context (no variable name or units).
• Using the mean to summarize strongly skewed data.

Words to know

Categorical vs quantitative variable

Categorical puts individuals into groups; quantitative records numbers that make sense to average.

Describing a distribution

Shape, center, spread and outliers, in context.

Right-skewed distribution

Long tail to the right; mean > median.

z-score formula

z = (x − mean) / SD

1.5 × IQR outlier rule

An outlier is below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Empirical rule

In a normal distribution about 68%, 95% and 99.7% of values fall within 1, 2 and 3 SDs of the mean.

Categorical variable

Places individuals into groups.

Quantitative variable

Records numbers that make sense to average.

Histogram

Bars showing counts of quantitative data in intervals.

Skewed left

Long tail to the left; mean < median.

Standard deviation

Typical distance of values from the mean.

Interquartile range (IQR)

Q3 − Q1: the spread of the middle 50%.

Resistant measure

Not strongly affected by outliers, like the median and IQR.

Normal distribution

Symmetric, bell-shaped, described by mean μ and SD σ.

Percentile

The percent of values at or below a given value.

Dotplot, stemplot, boxplot

Graphs for quantitative data; boxplots show the five-number summary.

Sample practice questions

51 more questions like these, from level 1 to 5, are in the practice set.