Summary Statistics for One Quantitative Variable
AP Statistics· 12 min read
1. Measures of Center: Mean and Median★★☆☆☆⏱ 10 min
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.
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
Confirm your understanding of center calculations
For the 5 values 2, 4, 6, 8, 10, what is the mean?
5
6
7
8
Reveal answer
6 —Sum of values is 30, divided by n=5 gives 6, which matches the median for this perfectly symmetric dataset.
2. Measures of Spread: Range, IQR, Standard Deviation★★★☆☆⏱ 12 min
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.
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.
3. Outlier Identification Using the 1.5×IQR Rule★★★☆☆⏱ 8 min
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.
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
4. Choosing Summary Statistics for Different Distribution Shapes★★☆☆☆⏱ 7 min
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.
5. Common Pitfalls
Wrong move:
Using n instead of n-1 to calculate sample standard deviation
Why:
This returns population standard deviation, which is not the value the AP Exam expects for sample data
Correct move:
Always divide sum of squared deviations by n-1 for all sample standard deviation calculations
Wrong move:
Calculating median or quartiles on unsorted data
Why:
Unordered values will return incorrect middle points that do not represent the true center
Correct move:
Always sort all data points from smallest to largest before calculating median, Q1, or Q3
Wrong move:
Using mean and standard deviation to describe a heavily skewed dataset
Why:
Extreme values pull the mean and inflate standard deviation, making them unrepresentative of typical values
Correct move:
For skewed distributions, report median and IQR as your preferred summary statistics
Wrong move:
Adding 1.5×IQR to the mean instead of Q1/Q3 to calculate fences
Why:
The outlier rule uses quartiles, not the overall center, to define the acceptable range of non-outlier values
Correct move:
Calculate lower fence as Q1 minus 1.5×IQR, upper fence as Q3 plus 1.5×IQR
Wrong move:
Defining standard deviation as the average absolute deviation from the mean
Why:
Standard deviation uses squared deviations, not absolute deviations, so this definition is mathematically incorrect
Correct move:
Define standard deviation as the square root of the average squared deviation from the sample mean
6. Quick Reference 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 |
When this came up on past exams
AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.
- 2023 · Paper 1
FRQ on household income summary stats
- 2022 · Paper 2
MCQ comparing mean vs median
- 2021 · Paper 1
Outlier identification task
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.