One Variable Data: Distributions and Measures of Center and Spread Pattern - Statistical Inference

One-Variable Data: Statistical Inference

This pattern gives you a grouped frequency table — where data is organized into ranges (like 50–59, 60–69, etc.) — and asks you to find which range contains the median. You cannot compute the exact median from grouped data, but you can determine which interval it falls in.

The Key Technique: Cumulative Counting

1. Find the total number of data points by summing all the frequencies.
2. Find the position of the median: for $n$ data points, the median is at position $\dfrac{n+1}{2}$ (if $n$ is odd) or between positions $\dfrac{n}{2}$ and $\dfrac{n}{2}+1$ (if $n$ is even).
3. Build a running cumulative total by adding frequencies row by row from the first group onward.
4. The median falls in the first group where the cumulative total reaches or exceeds the median position.

Worked Example 1

The table summarizes the scores of 25 students on a physics exam. Which of the following could be the median score?

A) 13 $\quad$ B) 65 $\quad$ C) 76 $\quad$ D) 84

SOLUTION

Step 1 — Total: $3 + 4 + 8 + 6 + 4 = 25$ students
Step 2 — Median position: $\dfrac{25+1}{2} = 13$
Step 3 — Cumulative count:

Score Range	Frequency	Cumulative
50–59	3	3 (positions 1–3)
60–69	4	7 (positions 4–7)
70–79	8	15 (positions 8–15)
80–89	6	21 (positions 16–21)
90–99	4	25 (positions 22–25)

Position 13 falls in the 70–79 group (positions 8–15).
The only answer in the 70–79 range is 76.
Answer: C) 76

Why the wrong answers are tempting:
A) 13 is the position of the median, not a score — the SAT's favorite trap here.
B) 65 is in the 60–69 range (positions 4–7) — too early.
D) 84 is in the 80–89 range (positions 16–21) — too late.

Worked Example 2

A biologist recorded the lengths, in centimeters, of 31 fish. The results are summarized below. Which could be the median length?

A) 16 $\quad$ B) 18 $\quad$ C) 22 $\quad$ D) 27

SOLUTION

Total $= 5 + 7 + 6 + 8 + 5 = 31$
Median position $= \dfrac{31+1}{2} = 16$
Cumulative: 5, then 12, then 18

After 10–14: cumulative $= 5$ (positions 1–5)
After 15–19: cumulative $= 12$ (positions 6–12)
After 20–24: cumulative $= 18$ (positions 13–18) ← position 16 falls here

The median is in the 20–24 range. The only choice in that range is 22.
Answer: C) 22

A) 16 is the position of the median — the same trap as before.
B) 18 is in the 15–19 range (positions 6–12).
D) 27 is in the 25–29 range (positions 19–26).

Worked Example 3 — Even Number of Data Points

A store recorded the ages of 30 customers. The frequency table is below. Which could be the median age?

SOLUTION

Total $= 30$ (even). Median is average of positions 15 and 16.
Cumulative: after 18–24: 4; after 25–34: 13; after 35–44: 23

Both positions 15 and 16 fall in the 35–44 range (positions 14–23).
The median is between 35 and 44.

Common Gotchas

• Confusing the position with the value. The median is at position 13, but the answer is a score in the 70–79 range, not the number 13. The SAT almost always puts the position number as a wrong answer.
• Picking the wrong interval. If you lose track while counting cumulatively, you might land in the wrong group. Write the running total next to each row.
• Forgetting the formula. For $n$ data points: position $= \dfrac{n+1}{2}$ (odd) or average of $\dfrac{n}{2}$ and $\dfrac{n}{2}+1$ (even). For even $n$, both middle positions usually fall in the same interval on the SAT.

What to Do on Test Day

• Sum the frequencies to get the total $n$.
• Compute the median position: $\dfrac{n+1}{2}$.
• Build cumulative totals row by row until you pass the median position.
• The answer must be a value within the interval you land on — not the position number, not the frequency.
• If the position number appears as an answer choice, it is a trap. Eliminate it immediately.
• These take about 30–45 seconds if you stay organized. Write the cumulative totals in the margin.