What you'll learn
Before doing any advanced statistics, you have to be able to describe a single set of data clearly — its shape, centre, spread and any unusual values. This is the focus of Exploring One-Variable Data (Unit 1 of AP Statistics). For AP Statistics you need to understand types of variables, how to display and describe distributions, measures of centre and spread, identifying outliers, and comparing distributions. This guide covers variable types, graphical displays, summary statistics, the normal distribution, and how to describe and compare data. By the end you should be able to describe a distribution fully and choose appropriate summary statistics.
Key terms and definitions
Categorical variable — A variable that places individuals into groups or categories.
Quantitative variable — A numerical variable that can be measured or counted.
Distribution — The pattern of values a variable takes, including their frequencies.
Median — The middle value of ordered data; a measure of centre.
Mean — The average; a measure of centre affected by outliers.
Standard deviation — A measure of the typical distance of values from the mean.
Interquartile range (IQR) — The range of the middle 50% of the data (Q3 − Q1).
Outlier — A value that is unusually far from the rest of the data.
Core concepts
Types of variables
Data comes in two main types. Categorical variables place individuals into groups (for example, eye colour or brand). Quantitative variables are numerical (for example, height or test score) and can be discrete (counted) or continuous (measured). The type of variable determines which graphs and summaries are appropriate — you cannot take a mean of categorical data, for example.
Displaying distributions
Different displays suit different data:
- Categorical data: bar charts and frequency tables show the counts or proportions in each category.
- Quantitative data: dotplots, stem-and-leaf plots, histograms and boxplots show the distribution of values.
A key AP skill is choosing the right display and reading it correctly.
Describing a distribution: shape, centre, spread and outliers
Whenever you describe a quantitative distribution, describe four things (a common exam requirement, remembered as SOCS — Shape, Outliers, Centre, Spread):
- Shape — is it symmetric, skewed left, skewed right, unimodal or bimodal?
- Centre — a typical value, given by the mean or median.
- Spread — how variable the data is, given by the range, IQR or standard deviation.
- Outliers — any unusual values far from the rest.
Always describe the distribution in context, referring to the actual variable and units.
Measures of centre and spread
The two common measures of centre are the mean and the median:
- The mean is the average, but it is sensitive to outliers and skew.
- The median is the middle value and is resistant to outliers.
For spread, the standard deviation measures typical distance from the mean (used with the mean, for symmetric data), while the interquartile range (IQR) measures the spread of the middle 50% (used with the median, for skewed data or when there are outliers). Choose the pair that matches the shape: mean and standard deviation for roughly symmetric data, median and IQR for skewed data or data with outliers.
Identifying outliers
A common rule for identifying outliers uses the IQR: a value is an outlier if it is below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. This is used to draw boxplots and to flag unusual values. Being able to apply this rule and mark outliers on a boxplot is expected.
The normal distribution and z-scores
Some distributions are approximately normal — symmetric and bell-shaped. For a normal distribution, the empirical rule (68–95–99.7) states that about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. A z-score measures how many standard deviations a value is from the mean: z = (value − mean) ÷ standard deviation. Z-scores let you compare values from different distributions and find proportions using the normal model.
Comparing distributions
When comparing two distributions, you must compare shape, centre, spread and outliers, and do so explicitly and in context — for example, "the median score for Class A is higher than for Class B" — rather than describing each separately. Making direct comparisons, not just listing features, is what earns full marks on the exam.
Describing shape in more detail
Describing the shape of a distribution well is an important skill. A distribution is symmetric if the two halves are roughly mirror images, skewed right if it has a longer tail towards high values, and skewed left if the longer tail is towards low values. It is unimodal with one clear peak, bimodal with two, or uniform if all values are about equally common. The direction of skew also tells you about the mean and median: in a right-skewed distribution the mean is pulled above the median by the high tail, while in a left-skewed distribution the mean is below the median. Recognising skew therefore helps you choose the right measure of centre, and describing shape precisely, with the correct terms, is expected on the exam.
Frequency tables, relative frequency and percentiles
You should also be comfortable working with frequency and relative frequency (the proportion or percentage in each class), and with percentiles. A percentile tells you the percentage of the data at or below a value — for example, a score in the 90th percentile is higher than about 90% of the scores. The median is the 50th percentile, and the quartiles Q1 and Q3 are the 25th and 75th percentiles. Cumulative relative frequency graphs (ogives) can be used to read off percentiles. Being able to interpret these measures, and to convert between counts, proportions and percentiles, is part of describing one-variable data fully and is frequently tested.
Worked examples
Example 1: Choosing centre and spread
A data set is strongly skewed right with a few high outliers. Which measures of centre and spread should you use? Use the median and interquartile range (IQR), because they are resistant to outliers and skew, whereas the mean and standard deviation would be pulled by the extreme high values.
Example 2: Calculating a z-score
A test has a mean of 70 and a standard deviation of 8. What is the z-score of a student who scored 86? z = (value − mean) ÷ standard deviation = (86 − 70) ÷ 8 = 16 ÷ 8 = 2. The student scored 2 standard deviations above the mean.
Example 3: Applying the empirical rule
Scores are approximately normal with mean 100 and standard deviation 15. About what percentage of scores are between 85 and 115? 85 and 115 are 1 standard deviation below and above the mean, so by the empirical rule about 68% of scores lie in this range.
Example 4: Identifying an outlier
For a data set, Q1 = 20 and Q3 = 40, so IQR = 20. Is a value of 75 an outlier? The upper boundary is Q3 + 1.5 × IQR = 40 + 1.5 × 20 = 40 + 30 = 70. Since 75 is greater than 70, it is an outlier.
Common mistakes and how to avoid them
A very common error is using the mean and standard deviation for skewed data. When data is skewed or has outliers, use the median and IQR instead, because they are resistant.
Students often describe a distribution without mentioning all four features. Always cover shape, outliers, centre and spread (SOCS), and always in context.
Another mistake is comparing distributions by describing each one separately. You must make explicit comparisons ("A's median is higher than B's"), not two separate descriptions.
When finding outliers, remember the 1.5 × IQR rule and apply it to both Q1 and Q3. Forgetting to multiply by 1.5, or checking only one end, gives wrong results.
Finally, when using z-scores and the empirical rule, make sure the distribution is approximately normal first — these tools assume a bell-shaped distribution.
Exam technique for "Exploring One-Variable Data"
Whenever you describe a distribution, use SOCS (shape, outliers, centre, spread) and always describe it in context with the variable and units. This structure is expected on free-response questions.
Match your summary statistics to the shape: mean and standard deviation for symmetric data, median and IQR for skewed data or data with outliers. Justify your choice.
Be ready to calculate z-scores, apply the empirical rule, and use the 1.5 × IQR rule for outliers. When comparing distributions, make direct comparisons rather than separate descriptions. Show your reasoning and use correct statistical vocabulary throughout.
Quick revision summary
- Variables are categorical (groups) or quantitative (numerical); choose displays and summaries to match.
- Describe distributions using SOCS: Shape, Outliers, Centre, Spread — always in context.
- Mean/standard deviation for symmetric data; median/IQR for skewed data or data with outliers (these are resistant).
- Outlier rule: below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
- For approximately normal data, use the empirical rule (68–95–99.7) and z-scores = (value − mean) ÷ standard deviation.
- When comparing distributions, make explicit comparisons of shape, centre, spread and outliers.