The Digital SAT frequently presents data in visual forms — histograms, box plots, and dot plots — and asks you to extract information or compare distributions. You need to read these displays accurately and understand what they reveal about shape, centre, and spread.
Core Concepts
Histograms
A histogram uses bars to show the frequency (count) of data within intervals (bins).
- Height of bar = frequency of that interval.
- No gaps between bars (unlike bar charts).
- You can estimate the total, mean, and shape.
Reading a histogram: Total data points = sum of all bar heights.
Box Plots (Box-and-Whisker)
A box plot displays five key values:
- Minimum (left whisker end)
- Q1 (left edge of box) — 25th percentile
- Median (line inside box) — 50th percentile
- Q3 (right edge of box) — 75th percentile
- Maximum (right whisker end)
IQR (Interquartile Range) — measures the spread of the middle 50%.
Dot Plots
Each data point is shown as a dot above a number line. Easy to read exact values.
Shape of Distributions
- Symmetric: data evenly distributed around the centre (mean ≈ median).
- Skewed right (positively): tail extends to the right (mean > median).
- Skewed left (negatively): tail extends to the left (mean < median).
Comparing Distributions
Compare centre (median), spread (IQR or range), and shape.
Strategy Tips
Tip 1: For Histograms, Count Carefully
Add bar heights for the total. For "how many above 80?" add the appropriate bars.
Tip 2: For Box Plots, Read the Five Numbers
Every box plot question boils down to reading min, Q1, median, Q3, max.
Tip 3: Median from Histogram
Find the total count, then count from the left bars until you reach the middle value.
Tip 4: IQR for Spread Comparison
A smaller IQR means the middle 50% of data is more concentrated.
Worked Example: Example 1
A box plot shows min=20, Q1=35, median=50, Q3=65, max=90. What is the IQR?
Worked Example: Example 2
A histogram has bars: 0-10: 3, 10-20: 5, 20-30: 8, 30-40: 4. How many data points total? What interval contains the median?
Total = 20. Median position: 10th and 11th values. Counting: first 3 in 0-10, next 5 in 10-20 (cumulative 8), next 8 in 20-30 (cumulative 16). The 10th and 11th values are in 20-30.
Worked Example: SAT-Style
Two box plots: Class A has median 75 and IQR 20. Class B has median 70 and IQR 35. Which class has higher scores and which has more consistent scores?
Class A: higher median (75 > 70) and smaller IQR (20 < 35) → higher and more consistent.
Worked Example: Example 4
A dot plot shows a cluster at 5-7 with a few dots at 15-20. Is the distribution skewed?
Yes — skewed right (tail to the right). Mean > median.
Practice Problems
Problem 1
A box plot has Q1=40, Q3=70. What is the IQR?
Problem 2
A histogram with total 50 data points: find the interval containing the 25th value.
Problem 3
Compare two box plots: one with range 60 and IQR 15 vs. one with range 40 and IQR 30. Which is more spread out overall? Which has more concentrated middle 50%?
Problem 4
A distribution has mean = 80 and median = 85. Is it skewed left or right?
Want to check your answers and get step-by-step solutions?
Common Mistakes
- Confusing frequency with data value in histograms. The height is the count, not the value.
- Not reading box plot lines correctly. The line inside the box is the median, not the mean.
- Thinking IQR includes all data. IQR only covers the middle 50% (Q1 to Q3).
- Misidentifying skew direction. The tail (not the peak) indicates the direction of skew.
Key Takeaways
Histograms: bar height = frequency. Sum heights for total.
Box plots: show min, Q1, median, Q3, max. IQR = Q3 − Q1.
Shape: symmetric, skewed left, or skewed right.
Skew direction: tail points in the direction of skew. Mean is pulled toward the tail.
Compare distributions using centre (median), spread (IQR), and shape.