The girls in this picture vary in height. The shortest girl has a height of 52 cm, and the tallest girl has a height of 64 cm. The other two girls fall in between these two extremes. How could you describe the heights of all four girls with a single number? How could express how they vary in height with another number?

### Using Statistics to Describe a Sample

The girls in the picture above make up a small sample—there are only four of them. In scientific investigations, samples may include hundreds or even thousands of people or other objects of study. Especially when samples are very large, it’s important to be able to summarize their overall characteristics with a few numbers. That’s where descriptive statistics come in. Descriptive statistics are measures that show the central tendency, or center, of a sample or the variation in a sample.

### Describing the Center

The central tendency of a sample can be represented by the mean, median, or mode.

- The
**mean**is the average value. It is calculated by adding the individual measurements and dividing the sum by the total number of measurements. - The median is the middle value. To find the median, rank all the measurements from smallest to largest and then find the measurement that is in the middle.
- The mode is the most common value. It is the value that occurs most often.

**Q:** A sample of five children have the following heights: 60 cm, 58 cm, 54 cm, 62 cm, and 58 cm. What are the mean, median, and mode of this sample?

**A:** The mean is (60 cm + 58 cm + 54 cm + 62 cm + 58 cm) ÷ 5 = 58 cm. The median and mode are both 58 cm as well. The mean, median, and mode are not always the same, as they are for this sample. In fact, sometimes these three statistics are very different from one another for the same sample.

### Describing the Range

Many samples have a lot of variation in measurements. Variation can be described with a statistic called the range. The **range** is the total spread of values in a sample. It is calculated by subtracting the smallest value from the largest value.

**Q:** What is the range of heights in the sample of children in the previous question?

**A:** The range is 62 cm – 54 cm = 8 cm.

### Summary

- Descriptive statistics are measures that summarize the characteristics of a sample.
- The central tendency, or center, of a sample can be represented by the mean, median, or mode.
- The variation in a sample can be represented by the range, or the total spread of values.

### Review

- What are descriptive statistics, and why are they useful?
- Find the mean, median, and mode of this set of values: 12 g, 9 g, 13 g, 12 g, 20 g, 17 g, 15 g.
- What is the range of the set of values in question 2?