This ap statistics one variable lesson explains exploring one-variable data in one printable page, with definitions, a method and a worked example.
One-variable data analysis describes a single variable. Categorical variables use bar charts and relative frequencies; quantitative variables are described by shape, center, spread and unusual features using histograms, boxplots, the mean and standard deviation, or the median and IQR. Z-scores and the empirical rule place a value within a normal distribution.
AP Statistics One Variable Lesson: key facts
- Describe every quantitative distribution with shape, center, spread and unusual features, in context.
- Use the median and IQR for skewed data or outliers, and the mean and standard deviation for symmetric data.
- A value is an outlier if it is below Q1 – 1.5 IQR or above Q3 + 1.5 IQR, or if it is more than 2 standard deviations from the mean.
- A z-score is (x – mean) / s; the empirical rule gives 68%, 95% and 99.7% within 1, 2 and 3 standard deviations.
What is the difference between categorical and quantitative variables?
A categorical variable sorts each individual into a group, such as favorite lunch or eye color. A quantitative variable is a number where arithmetic makes sense, such as minutes of travel. A zip code is categorical because adding zip codes means nothing. Show categorical data in a bar chart and compare groups with relative frequencies, count divided by total.
How do you describe a distribution?
Always address shape (symmetric, skewed right, skewed left, unimodal, bimodal), center, spread and unusual features, all in context. Skewed right means a long tail toward large values.
| Distribution | Center | Spread |
|---|---|---|
| Roughly symmetric, no outliers | mean \(\,\overline{x}\) | standard deviation \(s\) |
| Skewed or has outliers | median | IQR |
The median and IQR are resistant to outliers. The mean and s are not.
How do you build a five-number summary and a boxplot?
- Order the data and find the median.
- Q1 is the median of the lower half and Q3 the median of the upper half (leave out the median when n is odd).
- Compute \(\text{IQR}=Q_3-Q_1\). Values below \(Q_1-1.5\,\text{IQR}\) or above \(Q_3+1.5\,\text{IQR}\) are outliers.
- Draw the box from Q1 to Q3, a line at the median, and whiskers to the most extreme values that are not outliers.
Example. The 24 commute times have minimum 5, Q1 12, median 17, Q3 26 and maximum 58. So \(\text{IQR}=26-12=14\), and the fences are \(12-1.5(14)=-9\) and \(26+1.5(14)=47\). The value 58 is above 47, so it is an outlier and the upper whisker stops at 41. The second rule flags a value more than 2 standard deviations from the mean: \(\,\overline{x}\approx 20.3\) and \(s\approx 12.3\) give an upper limit of 44.9, and again only 58 is flagged.
What are z-scores, percentiles and the empirical rule?
A 40-minute commute has \(z=\frac{40-20.3}{12.3}\approx 1.60\). In an approximately normal distribution about 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean. With mean 70 and standard deviation 8, a score of 86 has \(z=\frac{86-70}{8}=2\), so about 97.5% of scores fall below it.
How do you compare two distributions?
Use side-by-side boxplots, then write full sentences that name both groups and cover center, spread and shape. Section B has the higher median, and its IQR of 21 is almost double Section A’s IQR of 11.
Exploring One-Variable Data: frequently asked questions
When should I use the median instead of the mean?
Use the median when the distribution is skewed or has outliers, because it is resistant. A single value of 58 among 24 commute times pulls the mean up to 20.3 but leaves the median at 17.
Why does AP Statistics divide by n – 1 in the standard deviation?
Data are almost always a sample, and dividing by n – 1 gives a better estimate of the spread of the population. Calculators report this as Sx. The value measures the typical distance from the mean.
How do quartiles work when n is odd?
Leave the median out, then take the median of the lower half for Q1 and of the upper half for Q3. For 11 ordered values, Q1 is the median of the first five.
What does a z-score of -1.5 mean?
The value is 1.5 standard deviations below the mean. In an approximately normal distribution that is near the 7th percentile.
What must a comparison of two distributions include?
Name both groups and compare center, spread and shape with comparison words such as higher, more variable or more skewed, and give numbers from the summaries. Mention any outliers in context.
How to use this ap statistics one variable lesson
Use this ap statistics one variable lesson before the worksheet, then check what you learned with the quiz.
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