1.5 Basic Statistics & Data Analysis
Key Takeaways
- Mean = sum of all values divided by how many values; the median is the middle value of an ordered list, and the mode is the value that appears most often.
- For an even count of ordered values, the median is the average of the two middle numbers, not a single listed value.
- A single extreme value (outlier) pulls the mean far off but barely moves the median, so the median better represents skewed PMAEE data sets.
- Probability of an event equals favorable outcomes over total equally-likely outcomes, and P(not A) = 1 - P(A).
- On no-calculator chart items, read the axis or table exactly, add mentally, and turn part-over-total into a fraction before converting to a percent.
Measures of Central Tendency
The PMAEE Mathematics subtest expects you to summarize a list of numbers with a single value. The three tools are the mean, the median, and the mode. Because no calculator is allowed, learn to compute each quickly by hand.
Mean (Average)
The mean is the sum of all values divided by the number of values.
Example: A cadet applicant scores 82, 85, 88, 90, and 95 on five practice quizzes. Add them: 82 + 85 + 88 + 90 + 95 = 440. There are 5 scores, so the mean is 440 / 5 = 88.
No-calculator shortcut: subtract a convenient base from every value, average the small leftovers, then add the base back. Using base 88, the leftovers are -6, -3, 0, +2, +7, which sum to 0; 0 / 5 = 0, so the mean is 88 + 0 = 88.
Median (Middle Value)
The median is the middle value once the numbers are placed in order. First sort, then locate the center.
- Odd count: the median is the single middle number. For 4, 7, 9, 12, 20 the median is 9.
- Even count: average the two middle numbers. For 12, 15, 18, 21, 24, 30 the two middle values are 18 and 21, so the median is (18 + 21) / 2 = 19.5.
Exam trap: many test-takers forget to sort first, or they pick a middle value without averaging when the count is even. Always order the list before you point at the center.
Mode (Most Frequent)
The mode is the value that occurs most often. In 3, 5, 5, 7, 8, 8, 8, 9 the number 8 appears three times, more than any other, so the mode is 8. A set can have no mode (all values unique) or more than one mode (a tie).
When the Mean Misleads: Outliers
An outlier is a value far from the rest. Consider 20, 22, 24, 26, and 200. The mean is (20 + 22 + 24 + 26 + 200) / 5 = 292 / 5 = 58.4, yet four of the five numbers are below 27. The median is the middle value, 24, which represents the data far better. The PMAEE often asks which measure best describes a skewed set — choose the median when a lone extreme value is present.
| Measure | Definition | Best used when |
|---|---|---|
| Mean | Sum divided by count | Data is evenly spread, no extreme values |
| Median | Middle of the ordered list | Data has outliers or is skewed |
| Mode | Most frequent value | You need the most common category or score |
Simple Probability
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes:
Probability = (favorable outcomes) / (total possible outcomes)
Example: A bag holds 4 red, 6 blue, and 2 green marbles, so there are 4 + 6 + 2 = 12 marbles in total. The probability of drawing a red marble is 4 / 12 = 1/3.
The Complement Rule
The chance an event does NOT happen is 1 minus the chance it does: P(not A) = 1 - P(A). Using the same bag, P(green) = 2 / 12 = 1/6, so P(not green) = 1 - 1/6 = 5/6. You can check this directly: 10 of the 12 marbles are not green, and 10 / 12 = 5/6.
Counting Two-Event Outcomes
When two dice are rolled there are 6 x 6 = 36 equally likely outcomes. A sum of 7 comes from (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1) — six ways. So P(sum = 7) = 6 / 36 = 1/6. Always simplify fractions; 6/36 reduces to 1/6.
Exam trap: watch for the words "and," "or," and "not." "Not" signals the complement rule, while "or" with mutually exclusive events means you add the separate probabilities.
Reading Graphs, Tables, and Charts
The subtest includes data analysis: pulling numbers from a table, bar chart, line graph, or pie chart, then computing with them.
Study this table of PMAEE applicants by region:
| Region | Applicants |
|---|---|
| A | 120 |
| B | 200 |
| C | 80 |
| D | 150 |
| E | 50 |
Total applicants: 120 + 200 + 80 + 150 + 50 = 600.
- Fraction/percent of a part: Region B has 200 of 600 applicants. As a fraction that is 200 / 600 = 1/3, or about 33.3%. On a pie chart, one-third of the circle equals 120 degrees (since 360 x 1/3 = 120).
- Difference: Region B minus Region C = 200 - 80 = 120 more applicants.
- Ratio: applicants in Region D to Region E is 150 : 50 = 3 : 1.
Reading tips for no-calculator speed: read the axis scale and units first (a bar reaching "3" on an axis marked in tens means 30, not 3); on a line graph the steepest segment shows the fastest change; on a pie chart the whole circle equals 100% or 360 degrees, so a quarter slice is 25% or 90 degrees. Estimate before you finalize — if a slice looks like half the pie, your answer should be near 50%.
A cadet applicant scores 78, 84, 90, and 88 on four math drills. What is the mean score?
A jar contains 5 red, 3 white, and 2 black marbles. If one marble is drawn at random, what is the probability that it is NOT red?
Find the median of the data set 7, 3, 9, 5, 11, 3.