# Summary Statistics for One Quantitative Variable

> AP Statistics · AP 2024-2026 Statistics CED
> Source: https://www.owlsprep.com/study/ap-statistics-u10-summary-statistics-for-one-quantitative/

We cover core measures of center, spread, formal outlier detection, and guidance for selecting appropriate summary statistics for symmetric vs skewed distributions, fully aligned to AP Statistics Unit 1 CED requirements.

**Prerequisites:** [Basics of quantitative vs categorical variables](https://www.owlsprep.com/study/ap-statistics-u10-intro-to-data-types/); [Shape of distributions (symmetric, skewed left/right)](https://www.owlsprep.com/study/ap-statistics-u10-distribution-shapes/)

## Learning objectives

- Calculate measures of center (mean, median) for ungrouped quantitative data
- Compute measures of spread (range, IQR, standard deviation) and interpret their real-world context
- Identify outliers using the formal 1.5×IQR rule
- Select the most appropriate summary statistics for skewed vs symmetric distributions

## Measures of Center: Mean and Median

The two most common measures of center describe the typical value of a quantitative dataset. The mean is the arithmetic average, calculated by summing all values and dividing by the total number of observations. The median is the middle value of a sorted dataset, and is unaffected by extreme outliers.

**Resistant Statistic** — A statistic that does not change significantly when extreme outlier values are added to the dataset. The median is resistant, while the mean is not.

**Worked example:** Calculate the mean and median for the following 7 ordered AP Statistics test scores: 62, 75, 78, 81, 83, 87, 94

1. Step 1: Sum all 7 scores to get the total: 62 + 75 + 78 + 81 + 83 + 87 + 94 = 560
2. Step 2: Divide the total sum by the number of observations n=7: 560 / 7 = 80, which is the sample mean
3. Step 3: For n=7 (odd), the 4th ordered value is the middle point, which is 81, the median

**Check your understanding**

Confirm your understanding of center calculations

1. For the 5 values 2, 4, 6, 8, 10, what is the mean?

   - 5
   - 6
   - 7
   - 8

   *Why:* Sum of values is 30, divided by n=5 gives 6, which matches the median for this perfectly symmetric dataset.

## Measures of Spread: Range, IQR, Standard Deviation

Measures of spread quantify how far individual data points are distributed away from the center of the dataset. Range is the simplest measure, calculated as maximum minus minimum value. IQR describes the spread of the middle 50% of data, while standard deviation quantifies the typical squared deviation from the mean.

**Interquartile Range** — The difference between the 75th percentile (Q3) and 25th percentile (Q1) of an ordered dataset, fully resistant to outliers.

*Notation:* IQR = Q3 - Q1

**Worked example:** For the ordered test scores 62, 75, 78, 81, 83, 87, 94, calculate the range, IQR, and sample standard deviation

1. Step 1: Range = max - min = 94 - 62 = 32
2. Step 2: Q1 = median of lower half [62,75,78] = 75, Q3 = median of upper half [83,87,94] = 87, so IQR = 87 -75 =12
3. Step 3: Sum of squared deviations from mean 80 = 324 +25 +4 +1 +9 +49 +196 = 608
4. Step 4: Divide sum by n-1 = 6, take square root: s = sqrt(608/6) ≈ 10.07

> **Exam tip:** The AP Exam explicitly requires you to use n-1 (not n) for sample standard deviation calculations to earn full credit.

## Outlier Identification Using the 1.5×IQR Rule

The 1.5×IQR rule is the formal, College Board-approved method to flag outliers in a quantitative dataset. Any value that falls outside the calculated lower and upper fences is classified as an outlier.

**Worked example:** Check the test score dataset for outliers, then add a new score of 50 and recheck

1. Step 1: Calculate 1.5 × IQR = 1.5 ×12 =18
2. Step 2: Lower fence = Q1 - 18 =75 -18 =57, Upper fence = Q3 +18 =87 +18 =105
3. Step 3: All original scores fall between 57 and 105, so no outliers exist in the original dataset
4. Step 4: The new score of 50 falls below the lower fence of 57, so it is classified as an outlier

> **mnemonic**
>
> 1.5 is the official outlier multiplier: any point more than one and a half IQRs away from the nearest quartile is flagged as unusual.

## Choosing Summary Statistics for Different Distribution Shapes

Your choice of summary statistics must match the shape of the distribution you are describing. For symmetric distributions with no extreme outliers, mean and standard deviation are the preferred metrics. For skewed distributions or datasets with known outliers, median and IQR are far more representative of the typical value.

**Exam command terms**

AP Exam FRQs often ask you to justify your choice of summary statistics, look for these common cues:

- **Justify your choice** — You must explicitly state that the metric you selected is resistant (or not) and explain why it fits the distribution shape

- **Compare the two distributions** — You must reference both a measure of center and a measure of spread for both groups to earn full comparison points

## Common pitfalls

- **Wrong:** Using n instead of n-1 to calculate sample standard deviation
  - Why it fails: This returns population standard deviation, which is not the value the AP Exam expects for sample data
  - Correct: Always divide sum of squared deviations by n-1 for all sample standard deviation calculations
- **Wrong:** Calculating median or quartiles on unsorted data
  - Why it fails: Unordered values will return incorrect middle points that do not represent the true center
  - Correct: Always sort all data points from smallest to largest before calculating median, Q1, or Q3
- **Wrong:** Using mean and standard deviation to describe a heavily skewed dataset
  - Why it fails: Extreme values pull the mean and inflate standard deviation, making them unrepresentative of typical values
  - Correct: For skewed distributions, report median and IQR as your preferred summary statistics
- **Wrong:** Adding 1.5×IQR to the mean instead of Q1/Q3 to calculate fences
  - Why it fails: The outlier rule uses quartiles, not the overall center, to define the acceptable range of non-outlier values
  - Correct: Calculate lower fence as Q1 minus 1.5×IQR, upper fence as Q3 plus 1.5×IQR
- **Wrong:** Defining standard deviation as the average absolute deviation from the mean
  - Why it fails: Standard deviation uses squared deviations, not absolute deviations, so this definition is mathematically incorrect
  - Correct: Define standard deviation as the square root of the average squared deviation from the sample mean

## Cheatsheet

| Statistic | Category | Resistant? | Best Use Case |
| --- | --- | --- | --- |
| Mean | Center | No | Symmetric data with no outliers |
| Median | Center | Yes | Skewed data or data with outliers |
| IQR | Spread | Yes | Skewed data or data with outliers |
| Sample Standard Deviation | Spread | No | Symmetric data with no outliers |

## What's next

Mastering these summary statistics is the non-negotiable foundation for all later inferential statistics work in AP Stats, including confidence intervals and hypothesis testing for population means. You will next apply these calculations to build visual representations of quantitative data, including boxplots and dotplots, and learn how to compare summary statistics across two or more groups to draw formal, evidence-based conclusions. This skill is tested in nearly every AP Stats FRQ set, so ensure you can quickly calculate and justify your choice of metrics for any given dataset before moving forward.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ap-statistics-u10-summary-statistics-for-one-quantitative/