Example 1 of 5
The middle of five
Quiz scores: 4, 10, 7, 6, 8. What is the median?
| Ordered scores |
|---|
| 4 |
| 6 |
| 7 |
| 8 |
| 10 |
- Put them in order: 4, 6, 7, 8, 10.
- Five numbers, so the middle is the third one.
- median = 7
Answer: 7
Data
Put scores in order, find a typical value, and see how spread out the numbers are.
In plain words
Sort the values from smallest to largest. The median is the middle value. If there is an even number of values, the median is the average of the two middle ones. At least half the values are at or below the median, and at least half are at or above it. The mean is what you get when you add all the values and divide by how many there are, like sharing evenly. The range is the largest value (maximum) minus the smallest (minimum).
A box plot shows five summary values: the smallest (minimum), lower quartile (Q1), median, upper quartile (Q3), and largest (maximum). Quartiles help divide ordered data into roughly four quarters. Here, when the list has an odd number of values, leave the overall median out of both halves. Q1 is the median of the lower half; Q3 is the median of the upper half. If a half has two middle values, average them. The interquartile range (IQR) is Q3 − Q1, the width of the box. Some books use a different rule for splitting the list, so follow the rule in the question.
The population is the whole group you want to learn about, such as every student in a school. A sample is a smaller group you study. For a simple random sample of 10 students, every possible group of 10 has the same chance of being picked. One way is to mix one identical name slip per student and draw 10 without looking. Each student then has the same chance of inclusion. The fraction in the sample can estimate the fraction in the school. Another sample may give a different estimate, and neither is guaranteed to equal the actual school count. A circle graph shows parts of a whole. To compare groups, look at a typical central value (mean or median) and how spread out the values are (range or IQR).
Examples
Try it. Then open the steps.
Example 2 of 5
Five-number summary
Data: 2, 4, 4, 6, 8, 10, 12. Find min, Q1, median, Q3, max, and the IQR. Leave the median out of both halves.
- Seven numbers, already ordered. Median is the 4th: 6. Leave 6 out of both halves.
- Lower half: 2, 4, 4. Q1 is the middle, 4. Upper half: 8, 10, 12. Q3 is 10.
- min 2, Q1 4, median 6, Q3 10, max 12
- IQR = 10 − 4 = 6
Answer: min 2, Q1 4, med 6, Q3 10, max 12; IQR 6
Example 3 of 5
The circle of sports
20 students voted. 8 soccer, 5 basketball, 4 track, 3 other. What percent chose soccer?
| Sport | Students |
|---|---|
| soccer | 8 |
| basketball | 5 |
| track | 4 |
| other | 3 |
- Part-to-whole. Soccer is 8 out of 20.
- 8 ÷ 20 = 0.40 = 40%
Answer: 40%
Example 4 of 5
Two classes
Class A scores: 3, 5, 5, 6, 8. Class B scores: 4, 7, 8, 9, 10. How much greater is Class B's median?
| Class A | Class B |
|---|---|
| 3 | 4 |
| 5 | 7 |
| 5 | 8 |
| 6 | 9 |
| 8 | 10 |
- Each class has 5 scores. Median is the third.
- A median = 5. B median = 8.
- 8 − 5 = 3
Answer: 3
Example 5 of 5
A fair sample
A random sample of 40 students found 15 who prefer tacos. The school has 600 students. About how many prefer tacos?
- The sample is the 40 students asked; the population is all 600 students. Use the sample’s fraction who prefer tacos, 15/40, to estimate the fraction for the school. A different random sample may give a different estimate, so our answer is not a guaranteed count.
- 15 / 40 = 0.375
- 0.375 × 600 = 225
Answer: 225 students
Things to watch for
Easy to miss: Finding the median without putting the numbers in order first.
Do this: Sort small to large, then pick the middle (or average the two middles).
Easy to miss: On an even list, grabbing one of the two middle numbers. Or, on an odd list, leaving the median inside a half when you find Q1 and Q3.
Do this: Even count: average the two middles. Odd count: the median sits out. The lower half is everything before it. The upper half is everything after it.
24 problems · Camp, Trail, Summit