Build the concept, interpret evidence, and apply it in a separate assessment pool.
Learn: Evidence and decisions
Check the scale before reading the picture
Read at least two labeled ticks on each axis. Points that are equally spaced on the page may represent unequal numerical intervals.
Average rate uses the full interval
For cumulative distance against time, the average speed over an interval is the change in distance divided by the elapsed time.
A segment rate is local
Use only the two endpoints of the requested interval. A cumulative graph’s steepness can change, so the rate on one segment may differ from the overall average.
Frequency, density and grouped data
A histogram groups numerical observations into adjacent intervals. With unequal interval widths, frequency density is the height and frequency is the rectangular area. A wide, short bar can contain more observations than a narrow, tall bar. The grouping limits what can be determined about exact values and medians.
Process diagrams encode sequence
Follow each arrow from the original population to the next stage. Counts at a later stage belong to a subset of the original group. For a conditional rate, divide by the group that reaches that stage. Check that mutually exclusive outgoing groups sum to the incoming group.
Two vertical axes need two separate readings
Match each series to its named axis in the legend. Read its values and units from that axis. Two lines can meet on the page while representing different numerical quantities. Combine the underlying values only through a meaningful relationship, such as revenue multiplied by margin.
Read logarithmic ticks as ratios
On a logarithmic axis, equal distances represent equal multiplicative factors. Read the printed values before calculating. A straight segment models proportional change only when the accompanying text permits interpolation. The midpoint between two positive values on a log axis is their geometric mean, not their arithmetic mean.
Learning Objectives
Read numerical axis values and detect unequal intervals.
Calculate average rate over an entire interval.
Calculate rate over one specified segment.
Distinguish distance traveled from speed.
Recover histogram frequencies from bin width and height.
Recognize when grouped data are insufficient for an exact statistic.
Trace branching processes and choose the correct conditional denominator.
Read and combine quantities from two independently scaled vertical axes.
Distinguish absolute differences, growth factors and percentage changes on a logarithmic scale.
Find a mean from specified plotted observations and compare changes over unequal intervals.
Prerequisite Knowledge
Coordinate pairs and units.
Rate of change as vertical change divided by horizontal change.
Worked Examples
Read a distribution by area, not height
The straight segments form a symmetric density model for delivery time. The total area under the model is 1. Determine the proportion of deliveries taking between 2 and 4 days.
Symmetric distribution of delivery time■ Density
Read the axis labels, scale and units before comparing values.
Read the horizontal variable: delivery time. The vertical axis is density, not a count of deliveries.
Use symmetry about 2 days: the interval from 2 to 4 days contains half the total area.P(2≤T≤4)=21=50%
Check using the triangular region, whose base is 2 days and height is 0.5 per day.Area=21×2×0.5=21
Conclusion: The required proportion is 50%. A curve height alone is not the probability of an interval. This example uses symmetry and elementary area; no calculus is needed.
Find overall average speed
The graph shows cumulative distance on the vertical axis and elapsed time on the horizontal axis.
Cumulative distance over time■ Distance
Read the axis labels, scale and units before comparing values.
Use total distance change divided by elapsed time.4−0100−0=25 km/h
Conclusion: This is the average speed over the complete interval.
Find speed on one interval
The requested segment is identified by its two endpoint times and distances.
Cumulative distance over time■ Distance
Read the axis labels, scale and units before comparing values.
Use the two values at the requested endpoints.4−2100−60=20 km/h
Conclusion: This is the speed over the requested segment.
Follow a branching process and keep the denominator
Applications pass through two screening stages. Find the overall approval rate and the stage 2 approval rate.
Two-stage application screening
Follow the arrows in order. Counts at later stages are subsets of the groups that reach those stages.
Check conservation at each split: the outgoing groups total the incoming group.100+60=160,75+25=100
Overall approval uses the original group; conditional approval uses the group reaching stage 2.16075=46.875%,10075=75%
Combining stage probabilities gives the overall approval probability.160100×10075=16075
Conclusion: A process diagram encodes sequence and group membership. Do not add percentages from successive stages or use the original total for a conditional question.
Compare absolute change, growth rate and share
Both services gained subscriptions between Q1 and Q4. Compare their absolute and percentage increases, then compare their shares of the combined total.
Quarterly subscriptions by service■ Service A■ Service B
Read the axis labels, scale and units before comparing values.
Read both endpoints on the same scale.ΔA=120−80=40,ΔB=100−40=60
Divide each increase by its own original value.gA=8040=50%,gB=4060=150%
Use the combined total when calculating a share, rather than a growth rate.sB,Q1=12040=31,sB,Q4=220100=115
Conclusion: Growth compares a series with its own starting value. Share compares it with the combined total in the same period. The 150% growth rate is three times 50%, but the absolute increase is only 1.5 times as large.
Two axes: equal height does not mean equal quantity
Revenue and operating margin use different units and scales. Determine how operating profit changes from Q1 to Q4.
Read the axis labels, scale and units before comparing values.
Identify the axis for each series before reading values: revenue from the left, margin from the right.
Calculate profit separately in each period.PQ1=40×0.20=8,PQ4=80×0.10=8
Explain the offsetting changes.4080=2,2010=21,2×21=1
Conclusion: Operating profit stays at $8 million. A visual crossing of two differently scaled series does not establish equality of their values. Compare quantities only after reading their units and scales.
Equal steps on a log scale represent equal ratios
The plotted cell count is measured hourly. Compare the growth factors, percentage increases and absolute increases between successive observations.
Read the actual tick values. Each equal vertical step multiplies count by ten.10100=1001000=100010000=10
A factor of ten is a 900% increase.(10−1)×100%=900%
The absolute increases are different even though the plotted segments have equal slopes.100−10=90,1000−100=900,10000−1000=9000
Conclusion: A straight line on a logarithmic vertical scale indicates constant proportional change when time intervals are equal. It does not indicate constant absolute change. Zero and negative counts cannot be plotted on this log scale.
Average from selected observations is not an endpoint average
A line graph reports a batch’s measured mass (grams) on integer days 1 through 5: 28, 31, 35, 38 and 43. Find the mean on days 2 through 4, then compare the increase from day 1 to day 3 with the increase from day 3 to day 5.
Batch mass: selected observations■ Measured mass
Read the axis labels, scale and units before comparing values.
Use the three plotted values named for the mean; do not substitute the endpoints or average the full interval.331+35+38=3104=3432 g
Compute the two changes from their specified endpoints.35−28=7,43−35=8
Compare the changes as a ratio, not just by visual segment length.87
Conclusion: The selected-observation mean is 34 2/3 g, and the first increase is 7/8 of the second. The values at unselected days do not belong in the mean.
Unequal bin widths: compare area, not height
The first two waiting-time intervals are 5 minutes wide; the last is 10 minutes wide. Read the number of customers from frequency density.
Customer waiting-time distribution
Bar area represents frequency. Height is frequency divided by interval width. Intervals include the lower bound and exclude the upper bound.
Multiply each bar’s density by its interval width.4(5)=20,6(5)=30,4(10)=40
The last bar contains the most customers although the middle bar is tallest. Find the proportion waiting at least 10 minutes.20+30+4040=94
Conclusion: The tallest bar has the highest density. The largest area has the most observations. Grouped data do not reveal individual waiting times.
Think Like a GMAT Test-Taker
Write each point as (time, distance) before calculating.
For an average, use the full start-to-end interval; for a segment, use only that segment’s endpoints.
Keep the units in the final rate.
Before comparing histogram bars, inspect their widths and the vertical-axis definition.
Check population conservation at every process split before combining outcomes.
Read both requested statements before calculating; note whether each asks for an average, change, ratio, or approximate range.
Use exact plotted values only when the prompt identifies those observations; do not interpolate between points without a stated model.
Common Traps
Dividing the final distance by a time interval that does not start at zero.
Reading a point from the wrong tick interval.
Averaging segment speeds without weighting by time.
Reading the tallest histogram bar as the largest frequency despite unequal widths.
Adding pass rates from successive screening stages.
Inferring individual observations from grouped intervals.
Reading both series from the left axis.
Inferring equality of quantities from a visual crossing on different scales.
Averaging quarterly margins without weighting by revenue.
Interpreting equal log-axis steps as equal additions.
Confusing a growth factor with a percentage increase.
Assuming interpolation is justified when the question supplies only observations.