9.3 Frequency Distributions & Distribution Properties

Reading a List as a Distribution

A frequency distribution counts how often each value (or each interval) occurs. GMAT Quantitative Reasoning will give you a small table, a bar graph, or a short list and ask for a mean, a median, a mode, a comparison of mean and median, or a conceptual description of shape. Official QR language includes interpreting graphs. It does not include fitting a density curve, computing a z-score from a table, or running a normality test. The no-calculator, 21-question, 45-minute section still wants arithmetic you can finish by hand.

Build every table question from the same columns: value, frequency, value × frequency, cumulative frequency. The product column is the contribution to the sum. The cumulative column finds the median.

Worked frequency table

• n = 20 (sum of the frequencies), not 4 (the number of rows).
• Sum of scores = 510, so mean = 510 / 20 = 25.5.
• Mode = 20, the score with frequency 6.
• Median: n is even, so average the 10th and 11th ordered scores. Cumulative counts: positions 1–4 are 10s, positions 5–10 are 20s, positions 11–15 are 30s. The 10th score is a 20; the 11th is a 30. Median = (20 + 30) / 2 = 25.

The mean 25.5 sits slightly above the median 25 because of the extra weight in the 40s. That is a mild right-hand pull, not a drama. GMAT items usually make skew obvious with one extreme class.

Rules you can reuse:

• Mean from a table: Σ(value × frequency) / Σ(frequency). Never average the listed values while ignoring frequencies, and never average the frequencies themselves.
• Median from a table: walk the cumulative count until you pass position (n + 1) / 2 (odd n) or the two central positions (even n).
• Mode from a table: the value with the largest frequency. If two columns tie, the table is bimodal.

If the table uses intervals (0–9, 10–19, ...) rather than exact values, the exam typically asks a modal class, a median class, or a comparison — not fake precision. If midpoints are supplied, treat those midpoints as the value column and proceed.

Second table, odd n. Values 2, 4, 6, 8 with frequencies 3, 5, 5, 2.

• n = 15, so the median is the 8th ordered value: (15 + 1) / 2 = 8.
• Cumulative frequencies: 3, then 8, then 13, then 15. The 8th value is the last of the 4s. Median = 4.
• Sum = 2×3 + 4×5 + 6×5 + 8×2 = 6 + 20 + 30 + 16 = 72. Mean = 72 / 15 = 4.8.
• Modes 4 and 6 (both frequency 5) — bimodal.

Mean 4.8 > median 4, so the 6s and 8s pull the mean right of the middle position.

Evenly Spaced Sets

An evenly spaced set (arithmetic sequence) has a constant gap: consecutive integers, consecutive even integers, multiples of 7, every third year.

For any evenly spaced list:

1. Mean = median
2. Mean = (first + last) / 2
3. Sum = n × (first + last) / 2
4. n = (last − first) / step + 1

Example. Sum of multiples of 7 from 21 through 140 inclusive.

• n = (140 − 21) / 7 + 1 = 17 + 1 = 18
• Mean = (21 + 140) / 2 = 80.5
• Sum = 18 × 80.5 = 1,449

Example. Mean of all multiples of 5 from 15 through 85 inclusive.

• First 15, last 85, step 5
• Mean = (15 + 85) / 2 = 50
• You do not need n if the question only asks for the mean. Even spacing makes the mean the midpoint of the ends.

Example. Mean of all odd integers between 40 and 160, that is 41 through 159.

• Mean = (41 + 159) / 2 = 100

If a list is evenly spaced, it is symmetric. Mean, median, and the midpoint of the range all coincide. If a GMAT list is "the first 20 positive even integers," use this machinery; do not add 2 + 4 + ... + 40 by hand. The first 20 positive even integers run from 2 through 40, mean = (2 + 40) / 2 = 21, sum = 20 × 21 = 420.

A geometric list such as {2, 4, 8, 16, 32} is not evenly spaced. Median is 8; mean is 62 / 5 = 12.4. The large right-hand values pull the mean above the median. Do not apply the (first + last) / 2 shortcut unless the gap is constant.

Skew: The Mean Walks Toward the Tail

Skew is a shape word: one tail is longer than the other. You do not need a skewness coefficient. You need the direction of the tail and what it does to the mean.

• Symmetric: left and right match. Mean = median. Evenly spaced lists and balanced bell-shaped lists are the usual GMAT examples.
• Right-skewed (positive skew): a long tail of large values. Those large values pull the mean to the right. Mean > median. Classic story: household income, house prices, a team with one star salary.
• Left-skewed (negative skew): a long tail of small values. Mean < median. Classic story: an easy exam with a pile of high scores and a few very low ones.

Memory hook: the mean is not resistant; it is dragged toward the long tail. The median stays with the middle position. The mode sits at the peak of the frequency graph, which is often on the opposite side from a long thin tail.

GMAT check. Salaries with median $75,000 and mean $92,000. The mean is higher, so a right tail of high earners is the supported story. You cannot conclude a standard deviation from those two numbers. You cannot say that exactly half the people earn more than the mean (half earn more than the median, not more than the mean). You cannot call the distribution symmetric.

If a graph is given (bar chart, histogram, or a simple box-style picture of a list), read it as ordered data with frequencies, not as a calculus curve. A tall bar is a high frequency. A bar sitting far to the right with small height is a right-tail class that inflates the mean.

Bell-Shaped on the GMAT — Conceptual Only

GMAC may describe a set of scores as approximately bell-shaped, clustered symmetrically about the mean, or roughly normal. That language means:

• The graph is high in the middle and tapers on both sides.
• Mean and median (and, typically, mode) sit together at the center.
• Most values are near the mean; far values are rare.

It does not turn Quant into a statistics course. You will not be scored on memorizing 68%, 95%, and 99.7% as an official GMAC identity. Some prep books quote that empirical visualization for a textbook normal curve; treat it as optional background, not as a formula GMAC requires you to compute with. You will not use z-tables. You should not invent a percent of scores inside "one standard deviation" unless the problem states a count or a fraction.

What you should do instead:

• If a list is described as symmetric about 50 with SD 4, still compare spread with the Section 9.2 rules. Do not assume 68% of the list is between 46 and 54 unless the item gives that information.
• If two bell-shaped lists share a mean and one has a larger SD, that list is the wider bell — more mass farther from the mean.
• If an item gives a graph, read heights as frequencies and use the table method.
• Regression, hypothesis tests, and z-tables are not on GMAT Quantitative Reasoning.

Quartiles and Percentiles at GMAT Level

Ordered data can be cut into pieces.

• The p-th percentile is a value such that about p% of the ordered list lies at or below it. The median is the 50th percentile.
• Quartiles cut the list into four parts:
  • Q1 (first quartile) ≈ 25th percentile — median of the lower half
  • Q2 = median = 50th percentile
  • Q3 (third quartile) ≈ 75th percentile — median of the upper half
• Interquartile range (IQR) = Q3 − Q1, the width of the middle 50%. Like the median, IQR ignores the extreme tails, so a single outlier that wrecks the range often leaves IQR alone.

GMAT-level method on a small list: sort, find the median, then find the median of each half. Official Quant items either use a list where the halves are obvious (even n, so the overall median is an average of two values and each half is a clean block) or they ask a conceptual comparison rather than a one-point IQR fight. Do not spend the 128-second budget arguing about whether to include the overall median in both halves when the item only needs a shape conclusion.

Example. Ordered list of 8 values: 4, 6, 7, 9, 12, 14, 15, 20.

• n even, median = (9 + 12) / 2 = 10.5 = Q2
• Lower half: 4, 6, 7, 9 → Q1 = (6 + 7) / 2 = 6.5
• Upper half: 12, 14, 15, 20 → Q3 = (14 + 15) / 2 = 14.5
• IQR = 14.5 − 6.5 = 8
• Range = 20 − 4 = 16, twice the IQR, because 20 sits out in the right tail

The mean of this list is 87 / 8 = 10.875, slightly above the median 10.5 — a mild right pull from 20. IQR stayed with the middle 50%; range felt the tail.

Box plots as graphs of a list

A box plot (box-and-whisker) is a drawing of five numbers from an ordered list: min, Q1, median, Q3, max. Official QR's "interpret graphs" umbrella covers this if the figure is given. You are not learning a new statistic; you are reading a picture of a list.

If the box is narrow and the whiskers are long, the middle 50% is tight but the tails are not — range is large, IQR is small. That is the same "range ignores the interior / IQR ignores the tails" contrast from Section 9.2, drawn as a graph.

You will not compute regression through a box plot, and you will not need a 1.5 × IQR outlier fence as a scored formula. If a point is drawn as a far dot, treat it as an extreme list value that moves the mean and the range more than it moves the median and the IQR.

A Full Pass on One List

A 12-person team has salaries, in thousands of dollars: 40, 40, 42, 43, 45, 47, 50, 52, 55, 60, 90, 240.

• n = 12, even. Median = average of 6th and 7th = (47 + 50) / 2 = 48.5
• Sum = 804, mean = 67
• Mode = 40 (frequency 2; every other value appears once)
• Mean 67 > median 48.5 → right skew from 90 and especially 240
• Lower half 40, 40, 42, 43, 45, 47 → Q1 = (42 + 43) / 2 = 42.5
• Upper half 50, 52, 55, 60, 90, 240 → Q3 = (55 + 60) / 2 = 57.5
• IQR = 15; range = 240 − 40 = 200
• The 240 is almost invisible to the median and IQR and dominates the mean and the range

That is the entire GMAT statistics story: small lists, a table or a graph, center versus spread, and a mean that cannot ignore a tail. Walk the item in that order — identify the display, get the mean from a sum, get the median from a position, then name the skew from mean versus median — and you will not reach for tools GMAC does not test.