Graphical Displays SAT Notes

Statistical displays compress data into patterns. Read titles, axes, units, scales, and category or interval definitions before calculating. The graph type reveals what comparisons are valid.

Choose the display that matches the data

Purpose and structure of common graphical displays
Display	Best use	Visual clue
Line graph	Change across ordered time or another continuous sequence	Successive points are connected
Bar graph	Compare distinct categories	Separated bars represent categories
Histogram	Show a numerical distribution across intervals	Adjacent bars represent continuous bins
Frequency table	Show exact counts for values or intervals	Rows pair values or classes with counts
Relative-frequency table	Compare proportions when totals differ	Entries are fractions, decimals, or percents

Line, bar, and histogram displays

These original, data-driven examples show the distinct structures of three common SAT displays.

Enrollment over six yearsA line graph showing enrollment rising overall, with a small decrease in year 4.

View chart data

Series	Year	Students
Enrollment	1	420
Enrollment	2	480
Enrollment	3	560
Enrollment	4	530
Enrollment	5	610
Enrollment	6	650

Preferred study formatA bar graph comparing four separate study-format categories.

View chart data

Series	Format	Students
Students	Videos	34
Students	Notes	27
Students	Practice	42
Students	Cards	21

Distribution of test scoresA histogram with adjacent bars for score intervals of width 10.

View chart data

Series	Score interval midpoint	Frequency
Scores	50	2
Scores	60	5
Scores	70	11
Scores	80	13
Scores	90	7

Read axes, scales, and intervals accurately

A reliable graph-reading routine

Identify
Read the title and determine what population, measurement, or time span is represented.
Decode axes
Read each label, unit, tick interval, and any broken or nonzero scale.
Locate
Match the requested category, time, or bin to its plotted value.
Calculate
Only after reading values should you find differences, rates, proportions, or predictions.
Check
Confirm that the result uses the correct units and comparison base.

Relative frequency

\[\text{relative frequency}=\frac{\text{class frequency}}{\text{total frequency}}\]

Relative frequencies across all nonoverlapping classes sum to $$1$$ , or $100\%$ .

Absolute and percent change

\[\text{absolute change}=\text{new}-\text{old},\qquad \text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}(100\%)\]

Absolute change keeps the original unit; percent change compares that difference with the old value.

Worked example

Read and compare a line graph

The enrollment graph shows $$480$$ students in year $$2$$ and $$610$$ in year $$5$$ . Find the absolute and percent increase.

Absolute change
$$610-480=130$$ students.
Percent base
Use the year- $$2$$ value as the old value: $130/480\approx0.2708$ .
Convert
The increase is approximately $27.1\%$ .

The absolute increase is

\(130\)

students, or about

27.1\%

Shape, center, and outliers

Distribution shapes

Compare the direction of each tail and its typical effect on the mean relative to the median.

Right-skewed distributionMost observations are lower, with a tail extending toward larger values.

The long right tail tends to pull the mean above the median.

View chart data

Series	Value	Frequency
Frequency	1	12
Frequency	2	10
Frequency	3	7
Frequency	4	5
Frequency	5	3
Frequency	6	2
Frequency	7	1

Symmetric distributionFrequencies mirror across the center.

For a symmetric distribution, the mean and median are typically equal or very close.

View chart data

Series	Value	Frequency
Frequency	1	2
Frequency	2	5
Frequency	3	9
Frequency	4	12
Frequency	5	9
Frequency	6	5
Frequency	7	2

Left-skewed distributionMost observations are higher, with a tail extending toward smaller values.

The long left tail tends to pull the mean below the median.

View chart data

Series	Value	Frequency
Frequency	1	1
Frequency	2	2
Frequency	3	3
Frequency	4	5
Frequency	5	7
Frequency	6	10
Frequency	7	12

Skew direction and typical mean–median relationship
Shape	Tail direction	Typical relationship
Right-skewed	Toward larger values	Mean $$>$$ median
Approximately symmetric	Balanced tails	Mean $\approx$ median
Left-skewed	Toward smaller values	Mean $$<$$ median

Common graph and distribution mistakes

Ignoring a nonzero axis baseline or unusual tick interval.
Treating a histogram bin height as the value rather than its frequency.
Using the bin midpoint when a question asks for the number of observations in the whole interval.
Confusing absolute change with percent change.
Comparing raw frequencies when group totals differ and relative frequencies are needed.
Naming skew from the location of the peak instead of the direction of the tail.
Assuming every point in a histogram is known; a bin reports only an interval.

Mini check

Check your understanding

A histogram has $$12$$ observations in the interval $70\le x<80$ . Which statement must be true?

Exactly $$12$$ values lie in that interval.
Every value equals $$75$$ .
The mean is $$75$$ .
No value equals $$80$$ .

Show answer and explanation

Answer: Exactly $$12$$ values lie in that interval.

The bar height gives the interval frequency. It does not identify the individual values or the overall mean.

Key takeaways

What to remember

Use line graphs for ordered change, bar graphs for categories, and histograms for numerical intervals.
Read labels, units, scales, and bin boundaries before computing.
Relative frequency is class frequency divided by total frequency.
Absolute and percent change answer different questions.
The long tail names the skew and usually pulls the mean toward it.
Graph-based conclusions must stay within what the display actually encodes.

Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.

Go to Practice Review Flashcards

Shape	Tail direction	Typical relationship
Right-skewed	Toward larger values	Mean $\(>\)$ median
Approximately symmetric	Balanced tails	Mean $\approx$ median
Left-skewed	Toward smaller values	Mean $\(<\)$ median