Statistics & Data Interpretation — 700-800

Statistics & Data Interpretation: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

At this level the questions are about center and spread: mean, median, standard deviation. You will almost never be asked "what is the mean?" You will be asked what happens to the mean, or what the mean forces some other value to be. Three archetypes cover nearly every hard item.

Archetype 1: The Mean Is a Total in Disguise

The single most useful sentence in this topic:

$\text{sum} = n \times \text{mean}$

Every hard item that mentions a mean is really handing you a total. Convert immediately, work with the total, convert back at the end.

• Missing value. Eleven seedlings, mean 38 g. Total is $11 \times 38 = 418$. Add the ten known masses, subtract, and the eleventh appears.
• Removed value. Twelve sheets, mean 5.5. Total 66. Remove an 8.8 sheet: new total 57.2 across eleven sheets, so the new mean is $57.2/11 = 5.2$. The classic error is dividing by 12 again.
• Corrected entry. 30 scores with mean 81.4, but a 67 was typed as 76. The total is $9$ too high, so subtract 9 from the total (not from the mean) and re-divide.
• Added values. 15 values with mean 52, then 55 and 66 join. New total $780 + 121 = 901$, new count 17, new mean 53.

The trap in every one of these: the question usually asks for a change or a second statistic, and the intermediate total or the intermediate mean is sitting right there in the answer choices.

Archetype 2: How Mean and Median React to a Change

Hard items love "which of the following describes the effect on the mean and on the median?" Memorize how each statistic responds:

Change	Mean	Median
Add a value equal to the current mean	unchanged	usually moves
Change only the largest (or smallest) value	moves by $\dfrac{\text{change}}{n}$	unchanged
Remove an extreme value	moves a lot	moves half a step
Remove two values that average to the mean	unchanged	may be unchanged

Two facts do the heavy lifting. First, the mean feels every value, so an extreme value drags it; the median only feels position. Second, when you add or remove a value the list changes parity, so the median position shifts by half a step even if nothing near the middle changed.

Never answer this archetype from intuition. Compute both statistics before and after. The distractors are built precisely from "the median must move too" and "the mean must move too."

Archetype 3: Weighted Means and the Lever

Two groups, two means, one combined mean. The average of two averages is almost never the combined average — it is only correct when the groups are the same size. This single idea generates a whole family of hard items.

The reliable method is totals:

$\text{combined mean} = \frac{n_1 m_1 + n_2 m_2}{n_1 + n_2}$

Worked example. 14 students average 78 and 21 students average 88. Total $= 14(78) + 21(88) = 2940$, so the combined mean is $2940/35 = 84$ — not $83$, which is what averaging 78 and 88 gives.

The lever shortcut. The combined mean always lands closer to the larger group. If the combined mean sits $a$ above group 1's mean and $b$ below group 2's mean, then

$\frac{n_1}{n_2} = \frac{b}{a}$

The sizes are proportional to the opposite distances. Group means 50 and 62 with a combined mean of 54? The distances are 4 and 8, so the sizes are in ratio $8:4 = 2:1$ — group 1 is bigger, which is why the combined mean drifted toward 50. Writing the ratio as $1:2$ is the single most common error on this archetype, and it is always an answer choice.

The same lever solves "how many more must be added": if you need the mean to reach a target, remember that new members enlarge the denominator too. Set up $\dfrac{S + vt}{n + t} = \text{target}$ and solve — never $vt = \text{shortfall}$.

The Fourth Statistic: Standard Deviation

You will never compute one. You will only compare two, and the rule is always the same: standard deviation measures typical distance from the mean.

• Same mean and same median tells you nothing about spread.
• More distinct values does not mean more spread.
• A distribution piled up at the center has a small SD; one hollowed out at the center with weight at both ends has a large SD, even when both are symmetric with identical mean and median.

Statistics & Data Interpretation: Traps & Speed

Part 2 of 3 — The Distractor Species

Hard-tier statistics items are not hard arithmetic. They are easy arithmetic with one extra step, and the answer choices are stocked with the values you produce along the way. Learn the species and you stop losing points you already earned.

Species 1: The Intermediate Value

This is the most common wrong answer in the entire topic. You solve for the missing value, the new mean, the second group's mean — and then the question asks for the change, the median, or the difference. Your correct intermediate result is sitting among the choices, waiting.

Before you bubble, re-read the last clause of the question. "By how much does it differ," "what is the median," "how many more" — these are not the quantity you just computed.

Species 2: The Average of Averages

Any time two groups with different sizes are combined, the unweighted average of the two means is an answer choice. It is correct only when the groups are the same size, which hard items are careful never to allow.

Species 3: The Even-Count Median

With an even number of values the median is the average of the two middle values, not one of them. Adding or removing a single value flips a list between odd and even, so the median almost always creeps by half a step. Distractors are built from picking a single middle-looking value instead of averaging.

Species 4: "Both Must Move" (and "Neither Moves")

When a data set is edited, students assume the mean and median move together. They usually do not:

• Change only the largest value: the mean moves by $\frac{\text{change}}{n}$, the median does not move at all.
• Add a value equal to the current mean: the mean holds still, the median shifts.
• Remove a symmetric pair (two values averaging to the mean): the mean holds still, and if both lie outside the middle, so does the median.

Answer choices systematically swap the two changes — attaching the mean's movement to the median. Compute both. Every time.

Species 5: Spread Confused With Center

"They have the same mean and the same median, so the standard deviations are equal" is always wrong. Center and spread are independent. Judge spread by asking one question: on average, how far from the mean does a value sit?

Speed Technique 1: Work in Totals, Not Means

Never average a list twice. Convert to a total once, edit the total, divide once at the end.

After 6 rounds her mean is 74.5; she wants a 7-round mean of at least 76. Total so far $= 6(74.5) = 447$; required total $= 7(76) = 532$; required round $= 532 - 447 = 85$. Two multiplications, one subtraction.

Speed Technique 2: Deviations From an Anchor

To average 87, 91, 84, 90, 88, anchor at 88: the deviations are $-1, +3, -4, +2, 0$, summing to $0$. The mean is exactly 88. This is faster and far less error-prone than adding five two-digit numbers, and it is how you should check any mean you compute under time pressure.

Speed Technique 3: Transformations Move Center, Scaling Moves Spread

If every value becomes $ax + b$:

• Mean becomes $a(\text{mean}) + b$
• Median becomes $a(\text{median}) + b$
• Spread (the range, or the standard deviation) is stretched by a factor of $|a|$, because multiplying every value by $a$ multiplies every gap between values by $|a|$. The $+b$ slides every value the same distance, so it changes nothing about spread.

Order matters. "Increase by 5, then triple" is $3(x+5)$; "triple, then increase by 5" is $3x + 5$. Both results appear among the choices.

Speed Technique 4: Read Frequency Tables by Cumulative Count

For a frequency table with $N$ entries, find the median by position, not by scanning the value column. Build the running total, then locate position $\frac{N+1}{2}$ (odd) or average positions $\frac{N}{2}$ and $\frac{N}{2}+1$ (even). Adding values at the top of the distribution shifts the middle position, but the median only changes if that shift crosses a category boundary — sometimes it does, sometimes it does not, and the item is written to punish assuming either way.

Speed Technique 5: Constraint Items — Push One Variable to Its Limit

"Five positive integers, mean 16, median 15, smallest 9, largest 27, what is the greatest possible second-largest value?" Write the ordered list $9, b, 15, d, 27$. The mean fixes the total, so $b + d$ is a constant. To maximize $d$, drive $b$ to its minimum legal value. Maximizing one member of a fixed-sum pair always means minimizing the other; choosing the split evenly, or pushing the wrong variable, produces the distractors.

Statistics & Data Interpretation: Timed Drill

Part 3 of 3 — 5 Questions, About 90 Seconds Each

Work these under a clock. Before each one, run the checklist:

1. Convert every mean to a total. $\text{sum} = n \times \text{mean}$.
2. Underline the actual question. Change, or level? Median, or mean? Count, or amount?
3. If two groups combine, weight them — and remember the lever runs backward: sizes are proportional to the opposite distances.
4. If the data set is edited, compute both statistics twice. Never predict the median's behavior from the mean's.
5. If the answer must be a whole number, decide which direction the constraint forces you to round before you look at the choices.

If you find yourself averaging two averages, stop — you have almost certainly walked into the item's main trap.