Distributions, center, spread and outliers
Read frequency-based displays and compare what center and spread reveal about a distribution.
What you will tackle
- Read frequency tables, histograms, dot plots and box plots
- Calculate and interpret mean, median and range
- Compare standard deviations without unnecessary calculation
- Explain effects of outliers and changes applied to all values
Displays summarize different information
A frequency table records how many times each value or interval occurs; do not average just the row labels. A dot plot preserves individual values, a histogram groups them into intervals, and a box plot marks quartiles and median. A box plot generally does not reveal the exact mean or every observation.
Location and spread answer different questions
The mean uses every value, while the median depends on middle position. An extreme high observation can pull the mean upward more than the median. The range uses the extremes. Standard deviation measures dispersion around the mean: tightly clustered data have smaller standard deviation than a more dispersed set with the same center. Adding c to all values shifts center by c without changing spread; multiplying by k scales standard deviation by |k|.
| Interval | Frequency |
|---|---|
| 0 ≤ x < 10 | 3 |
| 10 ≤ x < 20 | 6 |
| 20 ≤ x < 30 | 2 |
| 30 ≤ x < 40 | 1 |
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
For 2, 3, 3, 4, 8 find mean, median and range. Replace 8 with 28 and compare the changes.
Show a starting hint
Sort the data before locating the median.
Compare with the explained answer
- The sum is 20, so mean = 4; the middle value is 3 and range is 6.
- The replacement sum is 40, so mean = 8; median remains 3 and range becomes 26.
- The extreme value affects the mean and spread much more than the median.
Answer / response criteria: Initially mean 4, median 3, range 6; afterwards mean 8, median 3, range 26.
Guided 2
Read the histogram: counts in [0,10),[10,20),[20,30),[30,40) are shown by its bars. Find the total, the number below 20 and the interval containing the median. Can the exact mean be recovered?
Compare with the explained answer
- The bar counts sum to 3 + 6 + 2 + 1 = 12.
- Below 20 there are 3 + 6 = 9. Positions 6 and 7 both lie in [10,20).
- Grouping hides exact observations, so an exact mean is not determined.
Answer / response criteria: 12 total; 9 below 20; median interval [10,20); exact mean unavailable.
Independent practice
Independent 1
Values 1, 2 and 4 have frequencies 2, 3 and 1. Find the mean and median. Compare the standard deviations of {4,5,6} and {0,5,10}.
Compare with the explained answer
- There are 6 observations totaling 1·2 + 2·3 + 4·1 = 12, so mean = 2.
- The ordered list is 1,1,2,2,2,4, giving median 2.
- Both comparison sets have mean 5; {0,5,10} is farther from 5 and has larger standard deviation.
Answer / response criteria: Mean 2; median 2; {0,5,10} has larger standard deviation.
Independent 2
Read the dot plot rather than treating each occupied position as one observation. Determine the frequency at 1,2,4, then the mean and median.
Compare with the explained answer
- The dots show frequencies 2,3,1.
- The sum is 2·1 + 3·2 + 1·4 = 12 for 6 observations, so mean = 2.
- The third and fourth sorted values are both 2, so median = 2.
Answer / response criteria: Frequencies 2,3,1; mean 2; median 2.
Application practice
Application 1
A box plot reports minimum 10, Q1 14, median 18, Q3 22 and maximum 40. Find its range and interquartile range, estimate the fraction between Q1 and Q3, and say whether its mean is determined.
Compare with the explained answer
- Range = 40 − 10 = 30; interquartile range = 22 − 14 = 8.
- The middle two quartiles contain approximately half the observations, with ties affecting exact endpoint counts.
- Those five summary values do not determine the mean.
Answer / response criteria: Range 30; IQR 8; about half in the middle interval; mean not determined.
Application 2
A museum uses the displayed histogram for visitor waiting times in minutes. What fraction waited at least 20 minutes? Explain why reporting an exact maximum of 40 minutes would be unsupported.
Compare with the explained answer
- The last two bins contain 2 + 1 = 3 of 12 visitors.
- The fraction is 1/4 or 25%.
- The final interval is [30,40), so its values are below 40 and their exact maximum is not shown.
Answer / response criteria: 25%; 40 is a bin boundary, not a measured maximum.
What can you explain now?
A museum uses the displayed histogram for visitor waiting times in minutes. What fraction waited at least 20 minutes? Explain why reporting an exact maximum of 40 minutes would be unsupported.
Compare with the explanation
25%; 40 is a bin boundary, not a measured maximum.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide