4.1 Representing and Interpreting Data
Key Takeaways
- Dot plots preserve every data value; histograms bucket values into equal-width bins; box plots show the five-number summary (min, Q1, median, Q3, max) and outliers beyond 1.5 × IQR.
- Use the median as the measure of center when the data are skewed or contain outliers; use the mean for symmetric, outlier-free data because outliers pull the mean toward the tail.
- IQR (Q3 − Q1) measures the spread of the middle 50% and is resistant to outliers; range (max − min) is sensitive to the extremes.
- Two-way tables hold joint frequencies (cell counts), marginal frequencies (row/column totals), and conditional relative frequencies (cell divided by a row or column total).
- An outlier rule flags any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR; on a box plot these plot as individual points beyond the whiskers.
4.1 Representing and Interpreting Data
Quick Answer: The Florida Algebra 1 EOC expects you to build and read dot plots, histograms, and box plots, compute mean/median and IQR/range, spot outliers, and interpret two-way frequency tables. These skills sit under benchmarks MA.912.DP.1.1, MA.912.DP.1.2, and MA.912.DP.1.4 within the 31–38% "Expressions, Functions, and Data Analysis" reporting category.
Graphical Displays
Three display types dominate this benchmark. A dot plot stacks one dot per value above a number line; it preserves every data point and shows clusters and gaps. A histogram buckets numerical data into consecutive, equal-width bins and uses bar height for frequency; bars touch because the variable is continuous. A box plot summarizes five numbers — minimum, first quartile (Q1), median, third quartile (Q3), and maximum; the box spans Q1–Q3, the whiskers extend to the extremes within the 1.5 × IQR fence, and any point beyond the fence plots as an outlier.
| Display | Best for | What it hides |
|---|---|---|
| Dot plot | Small data sets, every value visible | Hard to read with large n |
| Histogram | Shape and distribution of large sets | Individual values are lost |
| Box plot | Center, spread, outliers | Shape within the box |
Shape cues: a distribution is symmetric when left and right sides mirror, skewed right when a long tail pulls toward larger values (mean > median), and skewed left when the tail runs toward smaller values (mean < median). The EOC asks which measure of center is "more appropriate" — choose the median when skew or outliers are present because the mean is pulled toward extremes; choose the mean for symmetric, outlier-free data.
Measures of Center and Spread
The mean is the arithmetic average: sum of values divided by count. The median is the middle value of an ordered list (or the average of the two middle values when n is even).
For spread, the range is max − min and is sensitive to outliers. The interquartile range (IQR) is Q3 − Q1 and is resistant to outliers because it ignores the tails. The EOC outlier rule: a value is an outlier if it lies below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
Worked example. Data: 4, 6, 7, 9, 10, 12, 45.
- Ordered: 4, 6, 7, 9, 10, 12, 45 (n = 7)
- Median = 9 (the middle value)
- Q1 = median of lower half (4, 6, 7) = 6
- Q3 = median of upper half (10, 12, 45) = 12
- IQR = 12 − 6 = 6
- 1.5 × IQR = 9, so the upper cutoff is Q3 + 9 = 21. Because 45 > 21, 45 is an outlier.
- Mean = 93/7 ≈ 13.3 — pulled up by the outlier, while the median (9) resists it. Report the median.
Two-Way Frequency Tables
A two-way frequency table cross-tabulates two categorical variables. Each cell holds a joint frequency (a count satisfying both categories); row and column totals are marginal frequencies. Dividing a cell by the grand total gives a joint relative frequency; dividing a cell by its row or column total gives a conditional relative frequency. The denominator is the trap.
Example: a survey of 200 Algebra 1 students on whether they study with music.
| Plays music | No music | Total | |
|---|---|---|---|
| Passed | 70 | 50 | 120 |
| Did not pass | 30 | 50 | 80 |
| Total | 100 | 100 | 200 |
- Marginal: P(passed) = 120/200 = 0.60.
- Joint: P(passed AND music) = 70/200 = 0.35.
- Conditional: P(passed | music) = 70/100 = 0.70, compared with P(passed | no music) = 50/100 = 0.50. The conditional comparison is the kind of inference the EOC asks you to make; it does not require formal significance testing.
Interpreting Categorical and Numerical Data
When the EOC says "interpret," it wants a sentence that ties a number back to context. Do not stop at "the median is 9" — write "the median study time was 9 hours per week, meaning half the students studied fewer than 9 hours." For histograms, describe shape, center, and spread (SOCS): "roughly symmetric, centered near 8, spread from 2 to 15." For box plots, compare two distributions by overlapping versus separated boxes — separated boxes suggest a real difference; heavily overlapping boxes suggest similarity.
Common Exam Traps
- Mean vs median on skewed data. A problem lists salaries with one CEO outlier; the wording "average salary" tempts you to the mean, but the median is the better representative.
- Confusing range with IQR. Range uses the extremes; IQR uses only the middle 50%. A question asking which measure is "resistant to outliers" is pointing at IQR, not range.
- Conditional vs joint frequency. P(passed AND music) divides by 200; P(passed | music) divides by 100 (the music column total). Mis-reading the denominator is the most common frequency-table error.
- Reading Q1 and Q3 from a box plot. Q1 is the left edge of the box, Q3 the right edge — not the whisker ends. Whisker ends are the minimum and maximum within the 1.5 × IQR fence.
- Histogram bin boundaries. A value of 10 falls in a bin labeled 10–14 only if the convention is left-inclusive; read the bin boundaries carefully before counting.
A dot plot of quiz scores is roughly symmetric with no outliers. Which measure of center is most appropriate, and why?
A data set has Q1 = 20 and Q3 = 44. Using the 1.5 × IQR rule, which value would be flagged as an outlier?
In a two-way table of 200 students, 70 passed and play music, and 100 students total play music. Which calculation gives the conditional relative frequency of passing given that a student plays music?