Measures of Central Tendency
Standard Deviation & Range
GMAT does not ask you to compute standard deviation from scratch. It tests your conceptual understanding of what SD measures and how it changes.
Measures spread around the mean. SD increases when values are more spread out. Adding the same constant to all values does NOT change SD. Multiplying all values by $k$ multiplies SD by $|k|$.
Simplest measure of spread. Can be misleading if outliers exist.
Weighted Average
GMAT loves mixing groups: "Class A has 20 students with avg 80, Class B has 30 students with avg 90 — what is the combined average?" Answer: weighted, closer to 90 since Class B is larger.
Data Set Changes
| Operation | Effect on Mean | Effect on SD |
|---|---|---|
| Add constant $k$ to all values | Mean + k | No change |
| Multiply all values by $k$ | Mean × k | SD × |k| |
| Add a value equal to current mean | No change | SD decreases |
| Add an outlier | Shifts toward outlier | SD increases |
10 Statistics Traps
1. Median requires sorting first
Always sort values before finding the median. {5,1,8,2} → sorted {1,2,5,8} → median = 3.5.
2. Even vs odd count for median
Odd count: take middle. Even count: average the two middle values.
3. Mean ≠ weighted average when group sizes differ
Averaging two group means ignores group size. Always use $\frac{w_1x_1+w_2x_2}{w_1+w_2}$.
4. Adding constant doesn't change SD
Shift all values by 5 → mean shifts, but SD stays the same.
5. Range is not standard deviation
Range = max − min. SD measures average squared deviation from the mean. Very different.
6. Larger range ≠ larger SD
{0, 50} has range 50 but SD = 25. {10, 20, 30} has range 20 but could have similar SD.
7. Mean is affected by outliers; median is not
A single very large value pulls the mean up but may not change the median at all.
8. Sum = Mean × n
If average of 8 numbers is 15, their sum = 120. This 'Sum trick' is used in dozens of GMAT problems.
9. Multiplying by $k$: SD multiplies by $|k|$
Double all values → SD doubles. But SD is always non-negative.
10. Mean of consecutive integers = median
For any evenly spaced set (AP), mean = median = average of first and last terms.
10 GMAT Practice Questions
The average (mean) of five numbers is 40. If a sixth number, 100, is added, what is the new mean?
(B) 50. Sum of 5 numbers = 5×40 = 200. New sum = 200+100 = 300. New mean = 300/6 = 50.
What is the median of the set {3, 7, 1, 9, 4, 6, 2}?
(B) 4. Sorted: {1, 2, 3, 4, 6, 7, 9}. Middle value (4th of 7) = 4.
Class A has 20 students with an average score of 70. Class B has 30 students with an average score of 80. What is the combined average score?
(D) 76. Total sum = 20×70 + 30×80 = 1400+2400 = 3800. Total students = 50. Weighted avg = 3800/50 = 76.
A set of numbers has mean 50 and standard deviation 10. If 5 is added to every number, what are the new mean and standard deviation?
(B) Mean=55, SD=10. Adding a constant shifts the mean (+5) but does NOT change the spread (SD stays 10).
What is the range of the set {−3, 0, 4, 7, −1, 11, 2}?
(D) 14. Max = 11, Min = −3. Range = 11 − (−3) = 14.
The average of x, y, and z is 15. The average of x and y is 12. What is the value of z?
(D) 21. Sum of x+y+z = 45. Sum of x+y = 24. Therefore z = 45−24 = 21.
Is the standard deviation of set A greater than the standard deviation of set B?
(1) Set A: {1, 2, 3, 4, 5} and Set B: {10, 11, 12, 13, 14}
(2) The range of A is 4 and the range of B is 4.
(D) Each alone sufficient. (1): Both sets are equally spaced with same spacing. SD of A = SD of B (adding 9 to all values doesn't change SD). So A is NOT greater than B — definitively NO. Sufficient. (2): Equal range and same type of data → same SD. Answer definitively NO. Sufficient. Both alone → (D).
A data set has 8 values with mean 20. If two of the values (24 and 16) are removed, what is the new mean?
(C) 20. Original sum = 8×20 = 160. Remove 24+16=40. New sum = 120. New mean = 120/6 = 20. The removed values averaged to 20 — same as the mean — so mean is unchanged.
If each number in a set is multiplied by 3, which of the following changes?
I. The mean
II. The median
III. The standard deviation
(D) I, II, and III. Multiplying by 3: mean×3, median×3, SD×3. All three measures scale by the multiplier.
The average of a list of 5 numbers is 30. A number is added and the new average is 32. What number was added?
(C) 42. Original sum = 150. New sum = 6×32 = 192. Number added = 192−150 = 42.
Lesson Summary — Key Takeaways
Sum = Mean × n
Recover the total sum from the average. Then adjust for added/removed values.
Median needs sorted data
Sort first, then find the middle. For even count, average the two middle values.
Adding a constant: no SD change
SD measures spread, not position. Shifting all values leaves spread identical.
Weighted average: proportional to group size
The combined mean is pulled toward the larger group's mean.