12.3 One-Variable Data: Center and Spread

College Board's Assessment Framework describes PSAT 8/9 one-variable work as calculating, comparing, and interpreting mean, median, and range from middle-school and first-year algebra courses, plus reading distributions in tables and other displays. This independent OpenExamPrep section also names mode, because a frequency table makes the most common value obvious. It does not teach standard deviation, margin of error, inference from a sample to a population, or how to judge an observational study versus an experiment. Those appear on later SAT Suite tests, not on PSAT 8/9.

A calculator is allowed. For a short list, add on paper anyway so you can see the arithmetic. SPR may want a decimal mean such as 10.5; type 10.5, not x=10.5. Two-variable scatterplots are the next chapter, 13.1 Two-Variable Data: Models and Scatterplots.

Mean, median, mode, and range on one list

Use this seven-number set all the way through the definitions: 3, 5, 5, 7, 10, 12, 14.

n = 7 (count the numbers; do not skip the repeated 5).

Mean (arithmetic average): add, then divide by n.

3 + 5 = 8 8 + 5 = 13 13 + 7 = 20 20 + 10 = 30 30 + 12 = 42 42 + 14 = 56

Mean = 56 / 7 = 8.

Median (middle after sorting). The list is already sorted. With odd n, the median is the single middle value: position (n + 1)/2 = 4th number, which is 7. The mean was 8; the median is 7. They need not match.

Mode (most frequent value): 5 appears twice; every other value appears once.

Range (a measure of spread): largest minus smallest: 14 − 3 = 11.

If every value appears once, there is no mode (or every value is equally common). The set 12, 15, 18, 20, 25 has sum 90, mean 18, median 18, range 25 − 12 = 13, and no mode.

Even n: median is the mean of two middles

List: 6, 8, 8, 10, 12, 16. n = 6.

Sum: 6 + 8 = 14, +8 = 22, +10 = 32, +12 = 44, +16 = 60. Mean = 60/6 = 10.

The two middle positions are 3rd and 4th: 8 and 10. Median = (8 + 10)/2 = 9.

Mode = 8. Range = 16 − 6 = 10.

Do not pick 8 as the median just because it is a mode. Median cares about position after sorting, not frequency.

What an outlier does

Start again with 3, 5, 5, 7, 10, 12, 14 (mean 8, median 7, range 11). Add 28:

New list: 3, 5, 5, 7, 10, 12, 14, 28. n = 8. Sum = 56 + 28 = 84. Mean = 84/8 = 10.5.

Median: 4th and 5th values are 7 and 10, so (7 + 10)/2 = 8.5.

Mode is still 5. Range = 28 − 3 = 25.

The mean jumped more than the median. The range jumped the most. The mode did not move. That is the PSAT 8/9 idea: a far-away value pulls the mean toward itself and stretches the range; the median only cares which value sits in the middle, so it moves less.

Second outlier example, test scores 80, 82, 84, 86, 88.

Sum: 80 + 82 = 162, +84 = 246, +86 = 332, +88 = 420. Mean = 420/5 = 84. Median = 84. Range = 88 − 80 = 8.

Replace 88 with 20 (a mistyped score, or a student who left early): 20, 80, 82, 84, 86.

Sum = 420 − 88 + 20 = 352. Mean = 352/5 = 70.4. Median = 82 (the third value). Range = 86 − 20 = 66.

The mean dropped from 84 to 70.4 (a drop of 13.6). The median dropped only from 84 to 82. If a stem asks which measure of center changed more, the answer is the mean. If it asks which measure of spread exploded, the answer is the range.

Frequency tables

When values repeat, a table saves listing them all. Multiply each value by its frequency, add those products, then divide by the total count.

Total students n = 3 + 5 + 4 + 2 = 14.

Total minutes = 20×3 + 25×5 + 30×4 + 40×2 = 60 + 125 + 120 + 80 = 385.

Mean = 385 / 14. 14 × 27 = 378, remainder 7, so 385/14 = 27.5 minutes.

Ordered list: 20, 20, 20, 25, 25, 25, 25, 25, 30, 30, 30, 30, 40, 40. For n = 14, the median is the mean of the 7th and 8th values. Both are 25, so median = 25.

Mode = 25 (frequency 5). Range = 40 − 20 = 20.

Relative frequency of the 25-minute row is 5/14 of the class. That fraction is a description of this table. It is not a claim about every eighth grader in the state, and this test does not ask you for a margin of error around 5/14.

Mean 27.5 is greater than median 25 because the 40-minute values sit on the high side. When a tail of large values pulls the mean above the median, the distribution is skewed right in a loose, visual sense. You do not need a formula for skewness on PSAT 8/9; you do need to notice that mean and median told different stories.

Same mean, different spread

Class A scores: 70, 75, 80, 85, 90. Sum = 70 + 75 = 145, +80 = 225, +85 = 310, +90 = 400. Mean = 400/5 = 80. Median = 80. Range = 90 − 70 = 20.

Class B scores: 60, 80, 80, 80, 100. Sum = 60 + 80 = 140, +80 = 220, +80 = 300, +100 = 400. Mean = 80. Median = 80. Mode = 80. Range = 100 − 60 = 40.

Both classes have mean 80 and median 80. Class B is more spread out. If a stem asks which class is more consistent, use the range (or look at how clustered the numbers are), not the mean. The mean cannot see that difference.

Removing a value also changes the measures. From 12, 15, 18, 20, 25 (mean 18, median 18), drop 25: remaining sum 90 − 25 = 65, n = 4, mean = 65/4 = 16.25, median = (15 + 18)/2 = 16.5, range = 20 − 12 = 8. The mean and the median both moved; the range shrank because the max left the list.

Choosing a measure, and what this test will not ask

• Use the mean when every value should count, including the extremes, and the stem asks for the average.
• Use the median when a far-away value would distort the typical number (house prices, one very long homework night).
• Use the mode when the question is which value appears most (most common shoe size in the table).
• Use the range when the question is how spread out the numbers are.

Do not invent a margin of error, a confidence interval, or a conclusion about a larger population from these seven homework times. Do not decide whether a survey was an experiment. Later SAT Suite PSD points cover inference from sample statistics, margin of error (SAT), and evaluating observational studies and experiments (SAT). PSAT 8/9 stops at describing the distribution you are given.

Desmos can add a list, but a frequency table still needs you to multiply 20×3 yourself unless you type all 14 numbers. On scratch paper, write the ordered list or the running sum. Math modules still run easiest to hardest inside a module; a five-number mean can be an early gift, and a frequency-table median can sit later. Bank the clean arithmetic, then spend time on the outlier-comparison sentence.

/practice/psat-89Practice questions with detailed explanations