One-variable data
Find the mean, median and range from lists, tables and charts. See why outliers drag the mean but barely touch the median, and compare spread without ever computing a standard deviation.
Find the balance point
This dot plot shows seven values. The mean is where the plot would balance on a finger. The median is the middle dot once the dots are in order. Drag the dots, or tap one and use the arrow keys, and watch both move.
Mean, median, mode and range
- Mean: add the values and divide by how many there are
- Median: the middle value in order. With an even count, the mean of the two middle values
- Mode: the value that appears most often
- Range: largest minus smallest
Frequency tables and charts
A frequency table, dot plot or bar chart tells you how many times each value appears. To find the mean, multiply each value by its frequency, add, and divide by the total frequency.
| Pets | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Students | 3 | 5 | 6 | 4 | 2 |
- Value × frequency: 0 + 5 + 12 + 12 + 8 = 37
- Total frequency: 3 + 5 + 6 + 4 + 2 = 20
- Mean: 37 ÷ 20 = 1.85 pets
- Median: the 10th and 11th values. The first 3 are 0, the next 5 are 1 (up to the 8th), the next 6 are 2. Both are 2, so the median is 2
Measuring spread
Two data sets can have the same mean and look nothing alike. Spread measures how far the values are from the centre.
- Range: largest minus smallest, so it depends on just two values
- Standard deviation: roughly the typical distance from the mean. Values bunched close to the mean give a small standard deviation
- Adding the same number to every value moves the centre but leaves the spread unchanged
Outliers
An outlier is a value far from the rest. It pulls the mean towards it, while the median, which only depends on the middle, hardly changes. That is why typical incomes and house prices are usually reported with the median.
Finding a missing value
When a question gives you a mean and asks for a missing value, work with totals. The total is the mean times the number of values.
- Total so far: 85 × 4 = 340
- Total needed: 87 × 5 = 435
- Score needed: 435 − 340 = 95
Box plots and quartiles
A box plot shows five numbers: minimum, first quartile (Q1), median, third quartile (Q3) and maximum. Each section holds about a quarter of the data, so the box holds the middle half. The interquartile range is IQR = Q3 − Q1.
Let Desmos do the arithmetic
For a long list, type it into Desmos inside square brackets. It returns the mean, median and more instantly, which saves time and slips.
Spot the trap
Each solution has one faulty step. All three slips appear as wrong answer choices on the SAT.