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

One-Variable Data: Measures of Center and Spread

 

This pattern tests whether you can compute or identify the mean and median of a data set, read a box plot, and correctly build or read a frequency table. No advanced formulas are needed — the SAT keeps the arithmetic manageable.

 

Key Formulas

The mean (average) of a data set is:

$\text{Mean} = \dfrac{\text{sum of all values}}{\text{number of values}}$

The median is the middle value when all values are arranged in order. For $n$ values:

• If $n$ is odd, the median is the value at position $\dfrac{n+1}{2}$.
• If $n$ is even, the median is the average of the values at positions $\dfrac{n}{2}$ and $\dfrac{n}{2}+1$.

 

Reading a Box Plot

A box plot shows five key values at a glance:

• Minimum — the left end of the left whisker
• Q1 (first quartile) — the left edge of the box
• Median — the vertical line inside the box
• Q3 (third quartile) — the right edge of the box
• Maximum — the right end of the right whisker

The SAT often asks "What is the median?" — just find the vertical line inside the box and read its position on the number line.

 

Worked Example 1 — Median from a Box Plot

The box plot summarizes the scores of 25 students on a biology quiz. What is the median score?

A) 50
B) 75
C) 85
D) 95

SOLUTION

The median is the vertical line inside the box. In this plot, that line sits at 75.
Answer: B) 75

A) 50 is the minimum (left whisker end), not the median.
C) 85 is Q3 (right edge of the box).
D) 95 is the maximum (right whisker end).
The most common mistake is confusing the minimum or Q1 with the median. Always look for the line inside the box.

 

Worked Example 2 — Another Box Plot

The box plot shows the heights, in centimeters, of 50 seedlings in a nursery. What is the median height?

A) 15
B) 8
C) 14
D) 22

SOLUTION

Find the vertical line inside the box. It sits at 15 on the horizontal axis.
Answer: A) 15

B) 8 is the minimum (left whisker). C) 14 is the range ($22 - 8 = 14$), not a box-plot feature. D) 22 is the maximum (right whisker).

 

Frequency Tables

A frequency table lists each distinct value and how many times it appears. The SAT often gives you raw data and asks you to pick the correct frequency table. The traps in the wrong answer choices are:

• Swapped columns — the value column shows the count and vice versa.
• Products instead of counts — the "frequency" column shows $\text{value} \times \text{count}$ instead of just the count.
• Both swapped and multiplied — a combination of the above two errors.

To avoid these traps, count the occurrences yourself. For the data set 7, 8, 9, 9, 7, 10, 8, 9, 8, 10, 9, you would tally: 7 appears 2 times, 8 appears 3 times, 9 appears 4 times, 10 appears 2 times.

 

Worked Example 3 — Computing the Mean

A student's scores on five quizzes were 5, 8, 10, 15, and 17. What is the mean score?

A) 10
B) 12
C) 11
D) 17

SOLUTION

Sum $= 5 + 8 + 10 + 15 + 17 = 55$
Count $= 5$
Mean $= \dfrac{55}{5} = 11$
Answer: C) 11

A) 10 is the median (middle value of the sorted list).
D) 17 is the maximum, not the mean.

 

Worked Example 4 — Frequency Table

The scores of 11 students on a quiz are: 7, 8, 9, 9, 7, 10, 8, 9, 8, 10, 9. Which frequency table correctly summarizes this data?

The correct table is:

Score	Frequency
7	2
8	3
9	4
10	2

How to check: The frequencies must add to the total count of data points ($2 + 3 + 4 + 2 = 11$ ✓). The "Score" column lists the distinct values from the data, not products or counts.

 

Common Gotchas

• Confusing the mean with the median. The mean is the sum divided by the count. The median is the positional middle. They are often different, and wrong answer choices exploit the swap.
• On box plots, reading the wrong feature. Don't confuse Q1, Q3, or the min/max with the median. The median is always the line inside the box.
• Using the max value as the mean. The largest value in the data set is always a wrong-answer trap.
• Arithmetic errors in summing. The SAT designs wrong answers to match common addition mistakes (e.g., summing to 60 instead of 55 yields a mean of 12 instead of 11).

 

What to Do on Test Day

• Mean $= \dfrac{\text{sum}}{\text{count}}$. Always double-check your addition.
• Median: sort the values first, then find the middle position.
• Box plot median: look for the vertical line inside the box — not the edges, not the whiskers.
• For frequency tables: the left column is the value, the right column is the count. Make sure they are not swapped.
• If two answer choices are suspiciously close (e.g., 11 vs. 12), re-add the numbers carefully — one of them is a deliberate arithmetic-error trap.
• When working from a frequency table, remember: sum $= \sum(\text{value} \times \text{frequency})$, count $= \sum \text{frequency}$.