Describing Data Numerically: 5-Number Summary and Calculator Use

Describing Data Numerically Using a Graphing Calculator

5-Number Summary

The 5-number summary is a concise way to describe the distribution of a dataset using five key values. These values provide insight into the center, spread, and overall range of the data.

• Minimum: The smallest data value.

• First Quartile (Q1): The value below which 25% of the data fall.

• Median (Q2): The middle value, dividing the data into two equal halves.

• Third Quartile (Q3): The value below which 75% of the data fall.

• Maximum: The largest data value.

These five statistics are useful for summarizing the distribution and identifying potential outliers.

Example: Student Ages in Statistics Class

Given the ages of students in a college statistics class:

Student Ages
19, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 25, 26, 28

The 5-number summary for this dataset is:

• Minimum: 19

• Q1: 21

• Median: 22

• Q3: 24

• Maximum: 28

Using a TI-84 Calculator to Find the 5-Number Summary

For large datasets, calculating quartiles and medians by hand is tedious. A graphing calculator can quickly compute these values.

• Enter the data into a list (e.g., L1).

• Press STAT, then select CALC and choose 1-Var Stats.

• Specify the list containing your data (e.g., L1).

• Press ENTER to view the summary statistics, including the 5-number summary.

Note: The calculator may display extra information depending on whether the data is a sample or population.

Example: Monthly Salaries

Suppose you have the monthly salaries (in dollars) of 20 employees at a medium-sized company:

Monthly Salaries ($)
4195, 2698, 2698, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 2700, 3435, 3435, 3435, 3995

The calculator provides the following 5-number summary:

• Minimum: 2698

• Q1: 2700

• Median: 2700

• Q3: 3435

• Maximum: 4195

Formulas and Interpretation

• Quartiles: Divide the ordered data set into four equal parts. Q1 is the median of the lower half, Q3 is the median of the upper half.

• Interquartile Range (IQR): Measures the spread of the middle 50% of the data. $IQR = Q_3 - Q_1$

• Range: Measures the overall spread of the data. $\text{Range} = \text{Maximum} - \text{Minimum}$

Example Application: The 5-number summary is often used to construct boxplots, which visually display the distribution, center, and spread of the data, as well as potential outliers.