AP Statistics Unit 1: exploring one-variable data

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Unit 1 of AP Statistics teaches you to describe data clearly. These skills appear across the whole exam.

Types of variables

A categorical variable puts individuals into groups. A quantitative variable takes numerical values you can average.

Displaying data

• Bar charts and pie charts for categorical data.
• Dotplots, stemplots, histograms and boxplots for quantitative data.

Describing a distribution: SOCS

• Shape: symmetric, skewed left or right, unimodal or bimodal.
• Outliers: values far from the rest, often checked with the 1.5 × IQR rule.
• Centre: mean or median.
• Spread: standard deviation, range or interquartile range.

Always describe distributions in the context of the data.

Choosing summary statistics

For skewed data or data with outliers, use the median and IQR, which are resistant. For roughly symmetric data, the mean and standard deviation work well.

z-scores and the normal distribution

• A z-score says how many standard deviations a value is from the mean: z = (x − μ)/σ.
• The empirical rule: about 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean in a normal distribution.

Common mistakes

• Describing a graph without context.
• Calling a histogram a bar chart.
• Using the mean for strongly skewed data.

Worked examples

Example 1: the five-number summary

Find the mean, median and IQR of 2, 4, 4, 5, 7, 9, 10, and check for outliers.

• Mean = 41/7 ≈ 5.86. Median = 5 (the 4th value).
• Lower half 2, 4, 4 gives Q1 = 4; upper half 7, 9, 10 gives Q3 = 9; IQR = 5.
• Fences: 4 − 7.5 = −3.5 and 9 + 7.5 = 16.5. No values lie outside, so no outliers.

Example 2: a z-score

A student scores 85 on a test with mean 70 and standard deviation 10. Interpret the z-score.

• z = (85 − 70)/10 = 1.5: the score is 1.5 standard deviations above the mean.

Example 3: the empirical rule

Heights are approximately normal with mean 170 cm and standard deviation 8 cm. What percentage are between 154 and 186 cm?

• 154 and 186 are 2 standard deviations below and above the mean, so about 95%.

Example 4: comparing distributions

Class A scores have median 70 and IQR 10; class B has median 78 and IQR 20. Compare them.

• Class B has a higher centre (median 78 vs 70) and more variability (IQR 20 vs 10). Always compare centre and spread, in context.

Shapes you should recognise

• Symmetric: mean and median are close.
• Skewed right: a long tail to the right; the mean is pulled above the median. Incomes are a classic example.
• Skewed left: a long tail to the left; the mean is below the median. Scores on an easy test are often like this.
• Bimodal: two peaks, often a sign that two groups are mixed together.

Practice questions

Try each one before opening the answer.

Frequently asked questions

When should I use a histogram versus a boxplot?

Histograms show the shape in detail. Boxplots make it easy to compare several groups and to spot outliers.

Do I always need context?

Yes. Every description should mention what the data measures, with units where possible.

Key takeaways

• Describe distributions using shape, outliers, centre and spread, in context.
• Use median and IQR for skewed data, mean and standard deviation for symmetric data.
• The 1.5 × IQR rule identifies outliers.
• z-scores measure distance from the mean in standard deviations.
• The empirical rule applies to approximately normal data.

Where to go next

Continue to the next skills in the course.

• All nine AP Statistics units
• AP Statistics formula sheet, explained
• Is AP Statistics hard?

Choosing the right graph

Unit 1 asks you to choose and interpret displays. For categorical data, use bar charts, and pie charts only when the parts make up a whole. For quantitative data, dotplots suit small data sets, stemplots keep the actual values, histograms show shape for larger data sets, and boxplots make comparisons between groups easy. Exam questions often ask what one display shows that another does not, for example that a histogram shows whether a distribution is bimodal while a boxplot cannot.

Describing distributions well

A good description mentions shape, centre, spread and unusual features, always in context. For example: "The distribution of commute times is skewed right, with a median of about 25 minutes and an IQR of about 15 minutes. There is one high outlier at about 90 minutes." Each feature is specific and tied to the data.

Transforming data

Adding a constant to every value shifts the centre but does not change the spread. Multiplying every value by a constant multiplies both the centre and the spread. This matters when converting units, such as from minutes to hours.

More practice

Try each one before opening the answer.

One more worked example: percentiles

A student's height is at the 80th percentile. What does that mean?

• About 80% of the group is the same height or shorter than this student.
• Percentiles describe position within a distribution, which is useful when distributions are not normal.

More questions students ask

What is the difference between standard deviation and IQR?

Standard deviation measures typical distance from the mean and is affected by outliers. IQR is the range of the middle 50% of the data and is resistant to outliers.

Quick check

Study tip

Describe every graph you see in your textbook out loud using shape, centre, spread and outliers. It quickly becomes a habit.

A final word

Unit 1 sets habits for the whole course. If you learn now to describe data clearly, in context and with the right summary statistics, every later unit will feel more familiar, because the same language returns again and again.