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?
The mean, because symmetric data with no outliers let the mean reflect the typical value without being pulled by extremes.
The median, because dot plots always require the median regardless of shape.
The range, because spread is the only relevant measure for symmetric data.
The mode, because categorical data always use the mode.
A data set has Q1 = 20 and Q3 = 44. Using the 1.5 × IQR rule, which value would be flagged as an outlier?
A value of 70, because 70 is greater than the median.
A value of 80, because 80 exceeds the upper cutoff of 44 + 1.5 × (44 − 20) = 80.
A value of 30, because 30 lies inside the box.
A value of 20, because 20 equals Q1.
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?
70 divided by 200, the joint relative frequency.
100 divided by 200, the marginal relative frequency of playing music.
70 divided by 100, the cell count divided by the music column total.
70 divided by 120, the cell count divided by the passing row total.
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