AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-7)
Unit 1 · Topic 1.7
1.7 Summary Statistics for One Quantitative Variable
Summary statistics put numbers on center, variability and position. You'll calculate the mean, median, quartiles, range, IQR and standard deviation, identify outliers with two common rules, see how changing units affects each measure and choose the right summary for skewed data.
Key terms
- mean (x̄) and median
- standard deviation (s)
- interquartile range (IQR)
- percentile
- outlier rules (1.5 × IQR, 2 standard deviations)
- resistant measure
Measures of center
The mean is the sum of the values divided by how many there are. For a sample it's written x̄ = (Σxᵢ)/n. The median is the middle value after you sort the data. With an odd number of values it's the middle one; with an even number it's the average of the two middle ones.
The mean is the balance point of the data. The median splits the data in half by count.
Measures of position
Sort the data. The minimum and maximum are the smallest and largest values. The first quartile, Q1, is the median of the lower half of the data, and the third quartile, Q3, is the median of the upper half. When n is odd, leave the overall median out of both halves (this is the method most calculators use; some software uses slightly different rules).
About 25% of the values are at or below Q1 and about 75% are at or below Q3, so Q1 and Q3 fence in the middle 50%. The median is also called Q2.
The pth percentile is the value with p% of the data at or below it. Q1 is the 25th percentile and Q3 is the 75th.
Measures of variability
The range is maximum − minimum. The interquartile range is IQR = Q3 − Q1, the spread of the middle half.
The standard deviation measures a typical distance between the values and the mean. For a sample, s = √[Σ(xᵢ − x̄)² / (n − 1)]. To compute it by hand: find each deviation xᵢ − x̄, square them, add them up, divide by n − 1 and take the square root. The number before the square root, s², is the sample variance. s is never negative, and it's 0 only when every value is the same.
Outliers and resistance
Two common rules flag outliers. The 1.5 × IQR rule calls a value an outlier if it's below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. The 2-standard-deviation rule calls a value an outlier if it's more than 2s away from the mean. The two rules can disagree, so say which one you're using.
A statistic is resistant if a few extreme values barely change it. The median and IQR are resistant. The mean, standard deviation and range are not, because one huge value pulls them a lot.
That's why you choose summaries by shape. For a strongly skewed distribution or one with outliers, the median and IQR describe the typical value and spread better. For a roughly symmetric distribution without outliers, the mean and standard deviation work well.
Changing units
If you add the same number to every value, the measures of center and position shift by that number, but measures of spread (range, IQR, s) stay the same. If you multiply every value by a positive number, the measures of center, position and spread are all multiplied by it.
Converting °C to °F uses both: F = 1.8C + 32. A mean of 22 °C becomes 1.8(22) + 32 = 71.6 °F, but a standard deviation of 2.74 °C becomes only 1.8(2.74) ≈ 4.93 °F. The + 32 doesn't affect spread.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Full set of summary statistics with an outlier
Commute times (minutes) for 11 students: 5, 8, 10, 12, 12, 15, 18, 20, 22, 25, 48. Find the mean, median, Q1, Q3, IQR and standard deviation. Use the 1.5 × IQR rule to check for outliers.
Show the solutionHide the solution
- Step 1: Mean: the sum is 195, so x̄ = 195/11 ≈ 17.7 minutes.
- Step 2: Median: with 11 sorted values, the 6th is the middle one, 15 minutes.
- Step 3: Q1 is the median of the lower five values (5, 8, 10, 12, 12), which is 10. Q3 is the median of the upper five (18, 20, 22, 25, 48), which is 22.
- Step 4: IQR = 22 − 10 = 12 minutes.
- Step 5: Standard deviation (calculator, 1-Var Stats): s ≈ 11.8 minutes.
- Step 6: Fences: 1.5 × 12 = 18. Lower fence 10 − 18 = −8. Upper fence 22 + 18 = 40. 48 > 40, so 48 is an outlier. No value is below −8.
Answer: x̄ ≈ 17.7 min, median = 15 min, Q1 = 10, Q3 = 22, IQR = 12 min, s ≈ 11.8 min. The 48-minute commute is an outlier by the 1.5 × IQR rule.
- Example 2Calculator allowed
Which statistics are resistant?
Remove the 48-minute commute from the data above. The new values are x̄ = 14.7, median = 13.5 and s ≈ 6.4. Which statistics changed the most, and what does that show?
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- Step 1: The mean dropped from 17.7 to 14.7, by 3.0 minutes.
- Step 2: The median dropped from 15 to 13.5, by only 1.5 minutes.
- Step 3: The standard deviation dropped from about 11.8 to about 6.4, almost by half.
- Step 4: One extreme value moved the mean and standard deviation much more than the median.
Answer: The mean and especially the standard deviation changed a lot; the median barely moved. The median is resistant; the mean and s are not, so the median and IQR are better summaries for these skewed data.
- Example 3Calculator allowed
Trap: changing units and spread
A class's quiz scores have mean 14 and standard deviation 3. The teacher adds 5 bonus points to everyone, then doubles each score. What are the new mean and standard deviation?
Show the solutionHide the solution
- Step 1: Adding 5 shifts the mean to 19 but leaves the standard deviation at 3.
- Step 2: Doubling multiplies both: mean 2 × 19 = 38, standard deviation 2 × 3 = 6.
- Step 3: The trap is adding 5 to the standard deviation. Adding a constant never changes spread.
Answer: New mean = 38 points; new standard deviation = 6 points.
Common mistakes
- Forgetting to sort the data before finding the median or quartiles.
- Dividing by n instead of n − 1 for the sample standard deviation.
- Adding a constant to the standard deviation or IQR when every value shifts. Shifts change center, not spread.
- Using the mean and standard deviation to summarize strongly skewed data or data with outliers.
On the exam
- Use technology (1-Var Stats) for s on the exam, but know the formula's meaning: a typical distance from the mean.
- When asked to show a value is an outlier, show the work: compute the fence (for example, Q3 + 1.5 × IQR = 40) and compare.
- If asked which measure of center to use, justify with shape or outliers: "the median, because the distribution is skewed right and the mean is pulled toward the long tail."
Connected topics
Videos
Check yourself
4 questions on 1.7 Summary Statistics for One Quantitative Variable. Pick an answer to see if you got it, and why.
The numbers of hours six students volunteered last month are 4, 6, 6, 8, 10 and 14. What is the sample standard deviation of these values?
The battery lives of a sample of phones have a mean of 50 hours and a standard deviation of 6 hours. Using the rule that a value is an outlier if it is more than 2 standard deviations from the mean, which battery life would be an outlier?
The daily high temperatures in a city for one month have a mean of 20°C and a standard deviation of 5°C. The temperatures are converted to degrees Fahrenheit using F = 1.8C + 32. What are the mean and standard deviation in °F?
A 2-year-old's height is at the 80th percentile for children of that age. Which is the correct interpretation?
0 of 4 answered