9.2 Range, Variance & Standard Deviation

Spread Is Not Location

Mean, median, and mode answer "where is the list?" Range, variance, and standard deviation answer "how spread out is it?" Two lists can share a mean of 50 and look nothing alike: {49, 50, 51} versus {0, 50, 100}. Official Guide Math Review 3.4 puts these tools on Quantitative Reasoning. The live section is still 21 Problem Solving questions in 45 minutes with no calculator, so GMAC tests small lists and comparisons you can settle without a messy square root. You will not run a spreadsheet SD on 80 rows. You may compute SD on a 4- or 5-number list, or — more often — say which list is more spread out from its mean.

Advanced statistics are out of scope: no regression, no hypothesis tests, no z-tables, and no population-versus-sample debate. The Quant formula for the list in front of you is the average of squared deviations:

Variance = (sum of (xi − mean)²) / n

Standard deviation = √(variance)

Use n in the denominator, not n − 1. The exam is describing the list you were given, not estimating a hidden population.

Range = Max − Min

Range = largest value − smallest value

Properties that actually get tested:

• Range is at least 0. Range equals 0 if and only if every number is identical.
• Range uses only two numbers. Every interior value is ignored.
• One outlier sets the range by itself.

Example. List P = {3, 10, 10, 10, 11, 12}. Range = 12 − 3 = 9, even though four of the six values sit in a two-point band. List Q = {3, 4, 7, 8, 11, 12}. Range is also 9, but the interior is much more spread. Equal ranges do not imply equal spread. That is why range is a weak summary and why comparison items exist.

All four ranges equal 20. Their standard deviations are not equal.

Adding a constant c to every value slides max and min by the same amount, so range is unchanged. Multiplying every value by k stretches the gap: new range = |k| × old range. The absolute value matters because range is a positive difference. Multiplying by −3 reverses order, but the gap is three times as wide.

Variance: Mean Squared Deviation

Spread around the mean is not the average of signed deviations — those already add to 0. Squaring makes every deviation positive and penalizes far values more than near values. A point 10 away from the mean contributes 100 to the sum of squares; a point 2 away contributes 4.

Procedure for a small GMAT list:

1. Find the mean (sum / n).
2. Subtract the mean from each value.
3. Square each deviation.
4. Average those squares. That average is the variance.
5. Take the positive square root. That is the standard deviation (SD).

Worked example with a clean integer SD. List R = {0, 0, 6, 6}.

• Mean = 12 / 4 = 3
• Deviations: −3, −3, 3, 3
• Squared: 9, 9, 9, 9. Sum = 36
• Variance = 36 / 4 = 9
• SD = √9 = 3

Compare list S = {0, 3, 3, 6}, which has the same mean 3 and the same range 6.

• Deviations: −3, 0, 0, 3
• Squared: 9, 0, 0, 9. Sum = 18
• Variance = 18 / 4 = 4.5
• SD = √4.5, which is less than 3

You do not need the decimal for √4.5. The sum of squares is 18 versus 36, so S is less spread out from the mean than R. Pairs equally spaced from the mean, parked at the endpoints, produce a larger SD than clustered pairs with the same range.

Worked example you can leave under a radical. List A = {2, 4, 6, 8}.

• Mean = 20 / 4 = 5
• Deviations: −3, −1, 1, 3
• Squared: 9, 1, 1, 9. Sum = 20
• Variance = 20 / 4 = 5
• SD = √5

If a second list has variance 8, it has the larger SD because square root preserves order for nonnegative numbers. Stop before extracting the root unless the answer choices force a decimal.

SD = 0 if and only if every number equals the mean, which is the same as "every number is identical." {7, 7, 7, 7} has SD 0. Any other list has SD > 0.

Compare Before You Compute

Because squared deviations grow fast, values far from the mean dominate SD. GMAC writes comparison questions around that fact. Fix the mean and the range, then watch where the other points sit.

• A pair sitting on the endpoints (as far from the mean as the range allows) contributes the maximum possible squared deviation.
• A pair sitting next to the mean contributes almost nothing.
• A list with the same min and max but with extra copies of the mean is more concentrated than a list that parks most of its mass at the ends.

All four lists below have n = 4, mean = 5, min 0, max 10, and therefore range 10.

The last list has the largest SD: both pairs sit equally far from the mean, at the extremes. The first list has the smallest SD among lists that actually use the full range, because the interior pair sits on the mean.

Five-number version the exam loves. Every list below has mean 20.

{0, 0, 20, 40, 40} and {0, 20, 20, 20, 40} share the same mean and the same range. The first has sum of squared deviations 1,600; the second has 800. Twice the variance means a larger SD, and no square-root arithmetic is required.

The clustered list {18, 19, 20, 21, 22} has a much smaller range as well as a much smaller SD. If two lists have different ranges, the one with the larger range is often more spread out — but not always, because range ignores the interior. A list {0, 50, 50, 50, 100} has range 100; a list {0, 0, 0, 100, 100} also has range 100, and the second list has the larger SD because more values sit at the ends.

Adding a Constant Versus Multiplying

Shift: y = x + c

Adding (or subtracting) a constant slides every point. Distances to the new mean are identical to the old distances.

• Range unchanged
• Each (yi − new mean) = (xi − old mean)
• Variance unchanged
• SD unchanged

{10, 20, 30} and {110, 120, 130} have the same range and the same SD. The lists are the same dots, moved 100 units right. A GMAT item that converts units by adding a constant, or that adds the same bonus points to every score, is a shift. Spread does not change.

Scale: y = kx

Multiplying every value by k stretches (or shrinks) every gap by |k|.

• New range = |k| × old range
• New deviations = k × old deviations
• New variance = k² × old variance (squares turn k into k²)
• New SD = |k| × old SD

The absolute value matters: multiplying by −2 flips the list but doubles the spread. SD is never negative. Dividing by 2 is multiplying by 1/2, so SD halves.

Combined rule: y = kx + c

The +c does nothing to SD or range. The ×k multiplies both by |k|. If every element of a list with SD 4.2 is replaced by 3x − 7, the new SD is 3 × 4.2 = 12.6. The −7 is a decoy. Compute |k| times the old SD and stop.

Inserting a value equal to the mean

The new deviation is 0, so the numerator of variance (sum of squared deviations) stays the same while n increases. Variance falls, and SD falls. The mean is unchanged. This is a classic true/false: "a number equal to the average is included."

Inserting a value far from the mean usually increases SD, and it increases range if the new value is a new min or max. Inserting a value inside the current min–max leaves range alone; the SD effect then depends on how far the new point is from the mean. Replacing a value at the mean with a value at an endpoint increases SD. Replacing an endpoint with the mean decreases SD.

Example. Ten numbers have mean 50 and SD 8. Add an eleventh number equal to 50. New mean = (10 × 50 + 50) / 11 = 50. The extra squared deviation is 0, the denominator rises from 10 to 11, and SD decreases. Range is unchanged because 50 is not a new extreme.

No-Calculator Habits

1. If the question only needs a comparison, do not extract a square root. Compare sums of squared deviations, or even compare how many points sit far from the mean.
2. If the lists have different means, subtract each list's own mean (or subtract a convenient constant from every value — a shift does not change SD) so you are looking at deviations.
3. If one list is a scaled copy of another, use the |k| rule instead of recomputing.
4. If two lists share min and max, hunt for mass at the ends versus mass in the middle.
5. Every |deviation| is at most the range, so SD cannot exceed the range. A reported SD larger than the range is impossible.
6. SD = 0 only for a constant list. If even one value differs, SD is positive.

A last comparison that catches testers who only look at range: {0, 0, 10, 10} versus {0, 4, 6, 10}. Same min, same max, same mean of 5. The first list has sum of squares 100; the second has 52. The first list is more spread out from the mean even though a casual glance at 4 and 6 might feel "more interior variety." Variety near the mean is not what SD rewards. SD rewards distance from the mean.