🔢 Quant

Statistics & Averages

Mean via sums, medians by position, weighted averages, frequency tables, and what standard deviation responds to.

~5h

to master

The five measures, and what each one ignores

Measure	Definition	Blind spot
Mean	sum ÷ count	distorted by outliers
Median	middle of the sorted list (average of the middle two if count is even)	ignores how far values sit
Mode	most frequent value (can be several)	ignores everything else
Range	max − min	uses only two values
Standard deviation	typical distance from the mean	hardest to compute — GMAT asks for comparisons, not decimals

Repeated values count every time they appear: the mean of $6, 4, 7, 10, 4$ is $\frac{31}{5} = 6.2$ , with both 4s in the sum.

Work with the sum, not the mean

📐Core Rule

$\text{sum} = \text{mean} \times \text{count}$ — the most useful identity in the topic. Four people average 80 miles → total 320; three drove $72 + 78 + 83 = 233$ → the fourth drove $87$ . Every "find the missing value" question is this one move.

The same identity powers average-shift problems: if

n

days average 50 units and today's 90 units lifts the average to 55, then old sum

+ 90 =

new sum:

50n + 90 = 55(n + 1)

→

n = 7

. Practice it on .

Weighted average: with group sizes $n_1, n_2$ and means $m_1, m_2$ , the overall mean is $\frac{n_1 m_1 + n_2 m_2}{n_1 + n_2}$ — it lands between the group means, closer to the larger group. It never equals the simple midpoint unless the groups match in size.

Median: position, not value

Sort first — always. For the five daily ticket counts $7, 4, 9, 4, 6$ , the sorted list $4,4,6,7,9$ has median 6. With an even count, average the two middle entries. The median can equal, exceed, or trail the mean; skewed data drags the mean toward the tail while the median barely moves. "About half the values are below the median" is usually true but fails with heavy ties — a list can have most values equal to its median.

Mean vs median questions ("for which value are they equal?") are case checks: for each candidate, recompute both — the mean moves smoothly with the new value, the median only jumps when the order changes. See .

Evenly spaced lists

For consecutive integers — or any arithmetic sequence — mean $=$ median $=$ the midpoint of first and last. Sliding a whole list up by $k$ slides its mean by exactly $k$ ; the spread (range, SD) doesn't move. Ten consecutive odd integers starting 7 above another list's start have a mean exactly... well, that's a one-line deduction once you trust the midpoint rule.

Standard deviation, conceptually

Find the mean. 2. Square each deviation from it. 3. Average the squares. 4. Square-root.

What the GMAT actually tests:

Tighter clustering = smaller SD — compare $\{6, 6, 6.5, 7.5, 9\}$ vs $\{0, 7, 8, 10, 10\}$ (same mean 7; the first is far tighter).
Adding a constant to every value: SD unchanged. Multiplying every value by $k$ : SD scales by $|k|$ .
Adding values at the mean shrinks SD; adding values far from the mean grows it.
SD $= 0$ exactly when all values are equal.

⚠️GMAT Trap

Range and SD are different animals. Two sets can share a range while one has nearly all its weight at the endpoints (big SD) and the other clusters at the centre (small SD). Never infer SD from range — or from the mean.

💡Exam Tip

Frequency tables are compressed lists. A value with frequency 7 appears 7 times in mean and median bookkeeping. For the median, find which value spans the middle position ( $\frac{n+1}{2}$ for odd $n$ ) by accumulating frequencies.

Checklist

Mean questions → convert to sums immediately
Sort before taking any median
Weighted mean leans toward the bigger group
Evenly spaced → mean = median = midpoint
SD: shift-invariant, scale-by- $|k|$ , clustering beats range

Sample Questions

22 practice questions

Medium

Over her first $n$ games of a season, a bowler averaged 140 points per game. After scoring 188 points in her next game, her average for the season rose to 146 points per game. What is the value of $n$ ?

Hard

The five numbers 4, 9, 10, 13, and $x$ have an average (arithmetic mean) equal to their median. Which of the following could be the value of $x$ ?

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